Uma classi cação de brados de Fell estáveis

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

  

❯♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ✜❜r❛❞♦s

❞❡ ❋❡❧❧ ❡stá✈❡✐s

❈❛♠✐❧❛ ❋❛❜r❡ ❙❡❤♥❡♠

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘✉② ❊①❡❧ ❋✐❧❤♦

  ❋❧♦r✐❛♥ó♣♦❧✐s ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✹

  ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛

  ❯♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧ ❡stá✈❡✐s

  ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛✲ ❞❛✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❋ís✐❝❛s ❡ ▼❛t❡✲ ♠át✐❝❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥✲ t❛ ❈❛t❛r✐♥❛✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❝♦♠ ár❡❛ ❞❡ ❝♦♥✲ ❝❡♥tr❛çã♦ ❡♠ ❆♥á❧✐s❡✳

  ❈❛♠✐❧❛ ❋❛❜r❡ ❙❡❤♥❡♠ ❋❧♦r✐❛♥ó♣♦❧✐s

  ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✹

  

através do Programa de Geração Automática da Biblioteca Universitária da UFSC.

Fabre Sehnem ; orientador, Ruy Exel Filho - Florianópolis,

Uma classificação de fibrados de Fell estáveis / Camila

Sehnem, Camila Fabre

Ficha de identificação da obra elaborada pelo autor,

Programa de Pós-Graduação em Matemática Pura e Aplicada.

Catarina, Centro de Ciências Físicas e Matemáticas.

Dissertação (mestrado) - Universidade Federal de Santa

137 p. SC, 2014. de Pós-Graduação em Matemática Pura e Aplicada. III. Título.

Exel. II. Universidade Federal de Santa Catarina. Programa

Ações parciais de grupos. 4. Produtos smash. I. Filho, Ruy

1. Matemática Pura e Aplicada. 2. Fibrados de Fell. 3.

Inclui referências

  

❯♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧

❡stá✈❡✐s

  ❈❛♠✐❧❛ ❋❛❜r❡

  

  ❊st❛ ❉✐ss❡rt❛çã♦ ❢♦✐ ❥✉❧❣❛❞❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ✑▼❡str❡✑✱ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ❆♥á❧✐s❡✱ ❡ ❛♣r♦✈❛❞❛ ❡♠ s✉❛

  ❢♦r♠❛ ✜♥❛❧ ♣❡❧♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✳

  Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ●♦♥ç❛❧✈❡s ❈♦♦r❞❡♥❛❞♦r

  ❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛ Pr♦❢✳ ❉r✳ ❘✉② ❊①❡❧ ❋✐❧❤♦

  ✭❖r✐❡♥t❛❞♦r ✲ ❯❋❙❈✮ Pr♦❢✳ ❉r✳ ▼✐❝❤❛❡❧ ❉♦❦✉❝❤❛❡✈

  ✭❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ✲ ❯❙P✮ Pr♦❢✳ ❆❧❝✐❞❡s ❇✉ss

  ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲❯❋❙❈✮ Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ●♦♥ç❛❧✈❡s

  ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲ ❯❋❙❈✮ Pr♦❢✳ ❉r✳ ●✐✉❧✐❛♥♦ ❇♦❛✈❛

  ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲ ❯❋❙❈✮ ❋❧♦r✐❛♥ó♣♦❧✐s✱ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✹✳

  ✶

❇♦❧s✐st❛ ❞❛ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r ✲

❈❆P❊❙

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s ♣❡❧♦ ❛♠♦r✱ ❝❛r✐♥❤♦ ❡ ❛♣♦✐♦ r❡❝❡❜✐❞♦s ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✳ ❆❣r❛❞❡ç♦ ✐♥✜♥✐t❛♠❡♥t❡ ♣♦r tê✲❧♦s ❛♦ ♠❡✉ ❧❛❞♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ q✉❡ ✈♦❝ês tê♠ ❡♠ ♠✐♠✱ ♣♦r ♥ã♦ ♠❡❞✐r❡♠ ❡s❢♦rç♦s ♣❛r❛ ✈❡r s❡✉s ✜❧❤♦s ❢❡❧✐③❡s✳ ❱ê✲❧♦s ❢❡❧✐③❡s ❡ ♦r❣✉❧❤♦s♦s é ♠✐♥❤❛ ♠❛✐♦r ♠♦t✐✈❛çã♦ ♣❛r❛ ❧✉t❛r ♣♦r ♠❡✉s ♦❜❥❡t✐✈♦s✳

  ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❘✉② ❊①❡❧ ❋✐❧❤♦✱ ♣♦r t❡r ❛❝❡✐t♦ ♦r✐❡♥t❛r✲♠❡ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦ ❡ ♣❡❧❛ s✉❣❡stã♦ ❞♦ t❡♠❛ ♣❛r❛ ❡st❛ ❞✐s✲ s❡rt❛çã♦✳ ❆❧é♠ ❞❡ t✉❞♦ ♦ q✉❡ ❛♣r❡♥❞✐✱ ♦ ♣r❛③❡r q✉❡ t✐✈❡ ❡♠ ❡st✉❞❛r ❝❛❞❛ t❡♦r✐❛ ♣❛r❛ ❛❧❝❛♥ç❛r ♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❢♦rt❛❧❡❝❡✉ ♠✉✐t♦ ♦ ♠❡✉ ❞❡s❡❥♦ ❞❡ ❝♦♥t✐♥✉❛r ❛ ❝❛rr❡✐r❛ ❛❝❛❞ê♠✐❝❛ ❡ ✐♥❣r❡ss❛r ❡♠ ✉♠ ♣r♦❣r❛♠❛ ❞❡ ❞♦✉t♦r❛❞♦✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ♣❡❧❛ ❞✐s♣♦s✐çã♦ ❡ ♣r♦♥t✐❞ã♦ ♣❛r❛ r❡✲ s♦❧✈❡r ♠✐♥❤❛s ❞ú✈✐❞❛s ❡ ❞✐s❝✉t✐r ♦ tr❛❜❛❧❤♦ ❡ ♣♦r t❡r ❝♦♠♣❛rt✐❧❤❛❞♦ ❝♦♠✐❣♦ ✉♠ ♣♦✉❝♦ ❞❡ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s✳ ❆♣r❡♥❞✐ ♠✉✐t♦ ♥❡ss❡s ❞♦✐s ❛♥♦s ❞❡ ♠❡str❛❞♦ ❝♦♠ ♦ ♠❛t❡♠át✐❝♦ ❡ ♣r♦✜ss✐♦♥❛❧ ❛❞♠✐rá✈❡❧ q✉❡ ✈♦❝ê é✳

  ❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❆❧❝✐❞❡s ❇✉ss✱ ❉❛♥✐❡❧ ●♦♥ç❛❧✈❡s✱ ●✐✉❧✐❛♥♦ ❇♦❛✈❛ ❡ ▼✐❝❤❛❡❧ ❉♦❦✉❝❤❛❡✈ ♣♦r t♦❞❛s ❛s ❝♦rr❡çõ❡s✱ s✉❣❡stõ❡s ❡ ♣♦r t❡✲ r❡♠ ❞❡❞✐❝❛❞♦ ✉♠ ♣❡rí♦❞♦ ❞❡ s❡✉s t❡♠♣♦s ♣❛r❛ ❛ ❧❡✐t✉r❛ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❆❧❝✐❞❡s ❇✉ss ♣♦r t❡r ❞❛❞♦ ✐❞❡✐❛s ❞❡ ❡①❡♠♣❧♦s ❡ r❡s✉❧t❛❞♦s ♣❛r❛ ❛❝r❡s❝❡♥t❛r ♥♦ tr❛❜❛❧❤♦ ✜♥❛❧✱ ❡ ♣❡❧♦s ❝♦♠❡♥tár✐♦s ❡ s✉❣❡stõ❡s q✉❡ ♠❡ ✜③❡r❛♠ ❛♣r❡♥❞❡r ♠❛✐s ❛✐♥❞❛✳ ❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ●✐✉❧✐❛♥♦ ❇♦❛✈❛ ♣♦r ♥ã♦ ❞❡✐①❛r ♣❛ss❛r ❞❡s♣❡r❝❡❜✐❞♦ ♥❡♠ ✉♠❛ ✈ír❣✉❧❛ ❢♦r❛ ❞❛ ♠❛r❣❡♠✳ ❖❜r✐❣❛❞❛ ♣❡❧❛ ❛t❡♥çã♦ ✐♠♣r❡ss✐♦♥❛♥t❡ q✉❡ t❛♠❜é♠ t❡✈❡ ❝♦♠ ❛ ♣❛rt❡ ❡stét✐❝❛ ❞♦ tr❛❜❛❧❤♦✳

  ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ ♠❛t❡♠át✐❝❛✱ ❙❛r❛ P✐♥t❡r✱ ❉❡✐✈✐❞✐ ❘✐❝❛r❞♦ P❛♥s❡r❛✱ ●✉st❛✈♦ ❋❡❧✐s❜❡rt♦ ❱❛❧❡♥t❡✱ ❙♦②❛r❛ ❇✐❛③♦tt♦ ❡ ▼❛ír❛ ●❛✉❡r✱ ♣❡❧♦s ❝❛❢és✱ r✐s❛❞❛s✱ ❛❧♠♦ç♦s✱ ❝♦♥s❡❧❤♦s ❡ ❛♠✐③❛❞❡✳

  ❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s✱ ✐♥❝❧✉✐♥❞♦ ♦s ❥á ❝✐t❛❞♦s✱ ♣♦r ❝❛❞❛ ♠♦♠❡♥t♦ ❞❡ ❞✐str❛çã♦✱ ♣❡❧♦ ❛♣♦✐♦ ❡♠ t♦❞❛s ❛s ❤♦r❛s ❡ ♣♦r t♦r❝❡r❡♠ ♣♦r ♠✐♠ s❡♠♣r❡✳ ➱ ♠✉✐t♦ ❜♦♠ s❛❜❡r q✉❡ t❡♥❤♦ ❛♠✐❣♦s ❞❡ ✈❡r❞❛❞❡✱

  ❝♦♠ ♦s q✉❛✐s ♣♦ss♦ ❝♦♥t❛r ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✱ ❡ q✉❡ t♦r♥❛♠ ♠✐♥❤❛ ✈✐❞❛ ♠✉✐t♦ ♠❛✐s ❛❧❡❣r❡ ❡ ❡s♣❡❝✐❛❧✳

  ❆❣r❛❞❡ç♦ às ♠✐♥❤❛s q✉❡r✐❞❛s ❛♠✐❣❛s ❡ ❝♦❧❡❣❛s ❞❡ ❝❛s❛✱ ❆♥❛ ▲ú❝✐❛ ❉❛♥✐❡❧❡✇✐❝③ ❡ ❈❛r❧❛ ❉❛♥✐❡❧❡✇✐❝③✱ ♣♦r ♠❡ ❞❡✐①❛r❡♠ ♣r❛t✐❝❛♠❡♥t❡ t♦♠❛r ♣♦ss❡ ❞❛ ♠❡s❛ ❞❛ s❛❧❛ ♣❛r❛ ♦s ❡st✉❞♦s ❞❡st❛ ❞✐ss❡rt❛çã♦✳

  ❆❣r❛❞❡ç♦ à ❊❧✐s❛✱ s❡❝r❡tár✐❛ ❞❛ ♣ós✱ ♣♦r s✉❛ ❝♦♠♣❡tê♥❝✐❛ ❡ ♣r♦♥t✐❞ã♦ ♣❛r❛ r❡s♦❧✈❡r t♦❞❛s ❛s q✉❡stõ❡s ❜✉r♦❝rát✐❝❛s ♥❡❝❡ssár✐❛s✳

  P♦r ú❧t✐♠♦✱ ♠❛s ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✱ ❛❣r❛❞❡ç♦ à ❈❆P❊❙ ✭❈♦♦r✲ ❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r✮ ♣❡❧❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦s ❢♦r♥❡❝✐❞❛✱ s❡♠ ❛ q✉❛❧ ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ❡st❛ ❞✐ss❡r✲ t❛çã♦✳

  ❘❡s✉♠♦

  ❉❛❞❛ ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ B ♣♦r ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ G✱ ❞❡✲ ✜♥✐♠♦s ❛ C ✲á❧❣❡❜r❛ ♣r♦❞✉t♦ s♠❛s❤ ❝♦♠♦ ✉♠❛ ❝❡rt❛ s✉❜á❧❣❡❜r❛ ❞❡ 2 B ⊗ K(l (G))

  ✳ ❯s❛♠♦s ❛ C ✲á❧❣❡❜r❛ ♣r♦❞✉t♦ s♠❛s❤ ♣❛r❛ ♠♦str❛r q✉❡✱ ❞❛❞♦ q✉❛❧✲ q✉❡r ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡stá✈❡❧ s♦❜r❡ ✉♠ ❣r✉♣♦ ❡♥✉♠❡rá✈❡❧ t❛❧ q✉❡ ❛ á❧❣❡❜r❛

  ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ é s❡♣❛rá✈❡❧✱ ❡①✐st❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❜❛s❡ ♥❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ ❝✉❥♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦ é ✐s♦♠♦r❢♦ ❛♦ ✜❜r❛❞♦ ✐♥✐❝✐❛❧✳

  ❆❜str❛❝t

  ●✐✈❡♥ ❛ ❣r❛❞❡❞ C ✲❛❧❣❡❜r❛ B ❜② ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ G✱ ✇❡ ❞❡✜♥❡ ∗ ∗ (G) t❤❡ s♠❛s❤ ♣r♦❞✉❝t C ✲❛❧❣❡❜r❛ B#C ❛s ❛ ❝❡rt❛✐♥ s✉❜❛❧❣❡❜r❛ ♦❢ 2 B ⊗ K(l (G))

  ✳ ❲❡ ✉s❡ t❤❡ s♠❛s❤ ♣r♦❞✉❝t C ✲❛❧❣❡❜r❛ t♦ s❤♦✇ t❤❛t ❣✐✈❡♥ ❛♥② st❛❜❧❡

  ❋❡❧❧ ❜✉♥❞❧❡ ♦✈❡r ❛ ❝♦✉♥t❛❜❧❡ ❣r♦✉♣ s✉❝❤ t❤❛t t❤❡ ✉♥✐t ✜❜❡r ❛❧❣❡❜r❛ ✐s s❡♣❛r❛❜❧❡✱ t❤❡r❡ ✐s ❛ ♣❛rt✐❛❧ ❛❝t✐♦♥ ♦❢ t❤❡ ❜❛s❡ ❣r♦✉♣ ♦♥ t❤❡ ✉♥✐t ✜❜❡r ❛❧❣❡❜r❛ ✇❤♦s❡ ❛ss♦❝✐❛t❡❞ ❋❡❧❧ ❜✉♥❞❧❡ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❣✐✈❡♥ ♦♥❡✳

  ❮♥❞✐❝❡

   ✻

   ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

   ✳ ✳ ✳ ✸✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

   ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

   ✾✺

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✶✵✻

   ✶✶✻

   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻

  ■♥tr♦❞✉çã♦

  ▼✉✐t♦ ❡♠❜♦r❛ ❛ t❡♦r✐❛ ❞❡ ✜❜r❛❞♦s C ✲❛❧❣é❜r✐❝♦s✱ ❛t✉❛❧♠❡♥t❡ ♠❛✐s ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ✜❜r❛❞♦s ❞❡ ❋❡❧❧✱ s❡❥❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦ ❝♦♥t❡①t♦ ♠❛✐s ❣❡r❛❧ ❞❡ ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s ✭✈❡❥❛ ❡st❛ t❡♦r✐❛ ❡stá ❡str❡✐t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ ❞❡ C ✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s q✉❛♥❞♦ ❧✐❞❛✲ ♠♦s ❝♦♠ ❣r✉♣♦s ❞✐s❝r❡t♦s✳ ❯♠❛ C ✲á❧❣❡❜r❛ B é ❞✐t❛ s❡r ❣r❛❞✉❛❞❛ ♣♦r g B g g ✉♠ ❣r✉♣♦ G s❡ B = ⊕ ∈G ✱ ❡♠ q✉❡✱ ♣❛r❛ ❝❛❞❛ g✱ B é ✉♠ s✉❜❡s♣❛ç♦

  = B −1 g B h ⊆ B gh ❢❡❝❤❛❞♦ ❞❡ B✱ B g g ✱ ❡ B ✱ ♣❛r❛ q✉❛✐sq✉❡r g, h ∈ G✳ ❊♠ t❡r♠♦s ❣❡r❛✐s✱ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B s♦❜r❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ G é ✉♠❛ t } t

  ❝♦❧❡çã♦ ❞❡ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ {B ∈G ❝♦♠ ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ · : B t × B s → B ts

  ❡ ✐♥✈♦❧✉çã♦ ∗ : B t → B t −1 s❛t✐s❢❛③❡♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ s❡r✐❛♠ s❛t✐s❢❡✐t❛s s❡ ❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s✲ t } t

  ♣❛ç♦s {B ∈G ❢♦ss❡✱ ❞❡ ❢❛t♦✱ ✉♠❛ ❣r❛❞✉❛çã♦ ♣❛r❛ ❛❧❣✉♠❛ C ✲á❧❣❡❜r❛✳ ❯♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❞á ♦r✐❣❡♠ ❛ C ✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s ♣❡❧❛ ❝♦❧❡✲ t } t

  çã♦ ❞❡ s✉❜❡s♣❛ç♦s {B ∈G ✉♠ t❛♥t♦ ❡s♣❡❝✐❛✐s✱ ❛ s❛❜❡r ❛s C ✲á❧❣❡❜r❛s s❡❝❝✐♦♥❛✐s ❝❤❡✐❛ ❡ r❡❞✉③✐❞❛✳ ❆ ♣r✐♠❡✐r❛ ❞❡❧❛s ♣♦ss✉✐ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❡ é ❞❡✜♥✐❞❛ ❞❡ ❢♦r♠❛ ♠❛✐s ❛❜str❛t❛✱ ❝♦♠♦ ❛ C ✲á❧❣❡❜r❛ ❡♥✲ ✈♦❧✈❡♥t❡ ❞❡ ✉♠❛ ❝❡rt❛ ∗✲á❧❣❡❜r❛✱ q✉❡ é ♦❜t✐❞❛ ♥❛t✉r❛❧♠❡♥t❡ ❛ ♣❛rt✐r ❞♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❏á ❛ ú❧t✐♠❛ é ❞❡✜♥✐❞❛ ❛ ♣❛rt✐r ❞❡ ✉♠❛ r❡♣r❡s❡♥t❛✲ çã♦ ❝♦♥❝r❡t❛ ❞❡st❛ ∗✲á❧❣❡❜r❛ ❡✱ ❞❡ ❝❡rt❛ ❢♦r♠❛✱ é ❛ ✏♠❡♥♦r✑ C ✲á❧❣❡❜r❛ t } t ∈G ❣r❛❞✉❛❞❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s {B ✭✈❡❥❛ ■st♦ ♥♦s ❧❡✈❛ ❛♦ ❢❛t♦ q✉❡ ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ♣♦❞❡ ♥ã♦ s❡r ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛ C

  ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ♦✉ r❡❞✉③✐❞❛ ❞♦ s❡✉ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦ ✭✈❡❥❛ ❊①❡♠♣❧♦ ❊♥tr❡t❛♥t♦✱ ♣♦❞❡♠♦s ✈ê✲❧❛ ❝♦♠♦ ♦ ❝♦♠♣❧❡t❛✲ g B g ♠❡♥t♦ ❞❛ ∗✲á❧❣❡❜r❛ ⊕ ∈G ❡♠ ✉♠❛ ❞❛❞❛ C ✲♥♦r♠❛ ❡✱ ♥❡st❡ ❝♦♥t❡①t♦✱ ❛♣r❡s❡♥t❛ ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ❡st❛s C ✲á❧❣❡❜r❛s s❡❥❛♠ ✐s♦♠♦r❢❛s✳

  ∗

  C ∗ ∗ ✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s t❛♠❜é♠ s✉r❣❡♠ ❛ ♣❛rt✐r ❞❡ ❛çõ❡s ❞❡ ❣r✉♣♦s ❡♠ C ✲á❧❣❡❜r❛s✳ P♦r ❡①❡♠♣❧♦✱ ✉♠❛ C ✲á❧❣❡❜r❛ ❛❞♠✐t✐♥❞♦ ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ❞❡ ✉♠ ❣r✉♣♦ ❝♦♠♣❛❝t♦ ❛❜❡❧✐❛♥♦ Γ é ❣r❛❞✉❛❞❛ ♣❡❧♦ ❣r✉♣♦ bΓ✳ ❏á ✉♠❛ ❛çã♦ ❣❧♦❜❛❧ α ❞❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ G ❡♠ ✉♠❛ C ✲á❧❣❡❜r❛ A α G ❞á ♦r✐❣❡♠ ❛♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ A⋊ ✱ q✉❡ é ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ♣♦r ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s q✉❡ sã♦ ❝ó♣✐❛s ❞❡ A✱ ♣❡❧♦ ♠❡♥♦s ❝♦♠♦ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ α é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✱ t❡♠♦s ✉♠❛ C

  ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ♣♦r ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❡s♣❛ç♦s q✉❡ sã♦ ❝ó♣✐❛s ❞❡ ✐❞❡❛✐s ❞❡ A✱ q✉❡ ❝❤❛♠❛♠♦s ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧✳

  ❖ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ ♣♦r ✉♠ ú♥✐❝♦ ❛✉t♦♠♦r✜s♠♦ ♣❛r❝✐❛❧ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ❡♠ ❡✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ❢♦✐ ❣❡♥❡r❛❧✐③❛❞♦ ❡♠ ♣❛r❛ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ♣♦r ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ q✉❛❧q✉❡r✳ ❊♠ s✉♠❛✱ ♥❛ ♣r✐♠❡✐r❛ ❝♦♥str✉çã♦✱ ♦ ❛✉t♦♠♦r✜s♠♦ ✉s❛❞♦ ♥❛ ❞❡✜♥✐çã♦ ❞♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ✉s✉❛❧ ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ ♣❡❧♦ ❣r✉♣♦ ❞♦s ✐♥t❡✐r♦s ❢♦✐ s✉❜st✐t✉í❞♦ ♣♦r ✉♠ ∗✲✐s♦♠♦r✜s♠♦ ❡♥tr❡ ❞♦✐s ✐❞❡✲ ❛✐s✱ ❡♥q✉❛♥t♦ ♥❛ ú❧t✐♠❛✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✱ ❢♦✐ ❞❡✜♥✐❞♦ ✉♠❛ ❡str✉t✉r❛ ❞❡ ∗✲á❧❣❡❜r❛ ❞❡ ❇❛♥❛❝❤ ❡♠ ✉♠ ❝❡rt♦ s✉❜❡s♣❛ç♦ ❞❛s ❢✉♥✲ çõ❡s ✐♥t❡❣rá✈❡✐s ❞♦ ❣r✉♣♦ ♥❛ C ✲á❧❣❡❜r❛✳ ❖ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❢♦✐ ❞❡✜♥✐❞♦ ❝♦♠♦ ❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ t❛❧ ∗✲á❧❣❡❜r❛ ❞❡ ❇❛♥❛❝❤✱ ❣❡♥❡r❛❧✐③❛♥❞♦ ❛ss✐♠ ❛ ♥♦çã♦ ❞❡ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ✉s✉❛❧✳

  ❯♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ G ❡♠ ✉♠❛ C ✲á❧❣❡❜r❛ A é ✉♠ ♣❛r α = ({D g } g ∈G , {α g } g ∈G ) g

  ❡♠ q✉❡✱ ♣❛r❛ ❝❛❞❛ g ∈ G✱ D é ✉♠ ✐❞❡❛❧ ❞❡ A g : D −1 → D g

  ✱ α g é ✉♠ ∗✲✐s♦♠♦r✜s♠♦ ❡✱ ♣❡❧♦ ♠❡♥♦s q✉❛♥❞♦ ♣♦ssí✈❡❧✱ t❡♠♦s ✉♠❛ ❝❡rt❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♠♣♦s✐çã♦ ❞♦s ∗ g

  ✲✐s♦♠♦r✜s♠♦s α ✬s ❡ ❛ ♦♣❡r❛çã♦ ❞♦ ❣r✉♣♦✳ ❊♠ ✉♠ ❝♦♥t❡①t♦ ❛✐♥❞❛ ♠❛✐s ❣❡r❛❧✱ t❡♠♦s ❛çõ❡s ♣❛r❝✐❛✐s t♦r❝✐❞❛s✱ ❝✉❥❛ ❞❡✜♥✐çã♦✱ ❛❧é♠ ❞♦s ✐❞❡❛✐s ❡ ∗✲ ✐s♦♠♦r✜s♠♦s ✐♥❞❡①❛❞♦s ❡♠ G✱ ❡♥✈♦❧✈❡ ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ✐♥❞❡①❛❞♦s ❡♠ G × G✳

  ❊♠ ✱ ❆✳ ❇✉ss✱ ❘✳ ▼❡②❡r ❡ ❈✳ ❩❤✉ ♠♦str❛r❛♠ q✉❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧ s❛t✉r❛❞♦s✱ ♥♦ ❝♦♥t❡①t♦ ♠❛✐s ❣❡r❛❧ ❞❡ ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s✱ ❝♦r✲ r❡s♣♦♥❞❡♠ ❛ ❛çõ❡s ✭❣❧♦❜❛✐s✮ ❞❡ ❣r✉♣♦s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r❛♠♦s ✜❜r❛❞♦s ❞❡ ❋❡❧❧ s♦❜r❡ ❣r✉♣♦s ❞✐s❝r❡t♦s ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ s❛t✉r❛❞♦s✮ ❡✱ s♦❜ ❝❡rt❛s ❤✐♣ót❡s❡s✱ ♦❜t❡♠♦s ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧ s♦❜r❡ ✉♠ ❣r✉♣♦ G ❡ ❛çõ❡s ♣❛r❝✐❛✐s ❞❡ G✳ ❆ss✐♠✱ t♦r♥❛♠♦s ♠❛✐s ♣r❡❝✐s❛ ❛ ✐❞❡✐❛ ❞❡ q✉❡ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ é ✉♠❛ ❡s♣é❝✐❡ ❞❡ ❛çã♦ ❞❡ ❣r✉♣♦✱ ❡♠❜♦r❛ ❥á s❛✐❜❛♠♦s ❞❡ q✉❡ ❞❛❞♦ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B s♦❜r❡ ✉♠ ❣r✉♣♦ ❡♥✉♠❡rá✈❡❧ ❝✉❥❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ é ❡stá✈❡❧ ❡ s❡♣❛rá✈❡❧✱ ❡♥tã♦ B ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ t♦r❝✐❞❛ ♦✉✱ ❝♦♠♦ t❛♠❜é♠ é ❝♦♥❤❡❝✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛✱ B ♣♦❞❡ s❡r ❡①✐❜✐❞♦ ❝♦♠♦ ✉♠ ✜❜r❛❞♦ ♣r♦❞✉t♦ s❡♠✐❞✐r❡t♦✳ ❆q✉✐✱ ♠❡❧❤♦r❛♠♦s ❡st❡ r❡s✉❧t❛❞♦✱ ❡①✐❜✐♥❞♦ ✉♠ ✜❜r❛❞♦ s❛✲ t✐s❢❛③❡♥❞♦ ❛s ♠❡s♠❛s ❤✐♣ót❡s❡s ❛ss✉♠✐❞❛s ❡♠ ❝♦♠♦ ♦ ✜❜r❛❞♦ ❞❡

  ❋❡❧❧ ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ♥ã♦ t♦r❝✐❞❛ ❞♦ ❣r✉♣♦ ❜❛s❡ ♥❛ s✉❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡✳ ❉❡st❛ ❢♦r♠❛✱ ❝♦♠ ♦ q✉❡ ❢♦✐ ❢❡✐t♦ ❡♠ t❡♠♦s ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ s❡❥❛ ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧✳

  ❖r❣❛♥✐③❛♠♦s ♦ tr❛❜❛❧❤♦ ❝♦♠♦ s❡❣✉❡✿ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❡♠❜❛s❛❞♦s ❡♠ ❞❡✜♥✐♠♦s C ✲♠ó❞✉✲

  ❧♦s ❞❡ ❍✐❧❜❡rt ❡ s✉❛ C ✲á❧❣❡❜r❛ ❞❡ ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s✳ ❋❡✐t♦ ✐st♦✱ ✐♥tr♦❞✉③✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❜✐♠ó❞✉❧♦s ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❡✱ ❡♥tã♦✱ ❞❡✲ ✜♥✐♠♦s ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ C ✲á❧❣❡❜r❛s✳ P♦r ✜♠✱ ♠♦str❛♠♦s q✉❡ ❡st❛ r❡❧❛çã♦✱ ❝♦♠♦ ♦ ♣ró♣r✐♦ ♥♦♠❡ s✉❣❡r❡✱ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ C ✲á❧❣❡❜r❛s ❡ ❡♥❝❡rr❛♠♦s ❝♦♥str✉✐♥❞♦ ❛ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦ ❞❡ ✉♠ ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✱ q✉❡ ❛❧é♠ ❞❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❡st✉❞♦ ❞❡ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛✱ ♥❡st❡ tr❛❜❛❧❤♦ t❛♠❜é♠ s❡rá ✉s❛❞❛ ♣❛r❛ ♦❜t❡r ✐♠♣♦rt❛♥t❡s r❡s✉❧t❛❞♦s s✉❜s❡q✉❡♥t❡s✳

  ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ s♦❜r❡ ❣r✉♣♦s ❞✐s❝r❡t♦s ❡ ❝♦♠❡ç❛♠♦s ❝♦♥str✉✐♥❞♦ ✉♠❛ ∗✲á❧❣❡❜r❛ r❡❧❛❝✐♦♥❛❞❛ ❛ ✉♠ ✜❜r❛❞♦✳ ▼♦str❛♠♦s q✉❡ t❛❧ ∗✲á❧❣❡❜r❛ é ❛❞♠✐ssí✈❡❧ ❡✱ ❛ss✐♠✱ ❞❡✜✲ ∗ ∗ ♥✐♠♦s ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❝♦♠♦ s❡♥❞♦ s✉❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✲ ✈❡♥t❡✳ ❊♠ s❡❣✉✐❞❛✱ ❝♦♥str✉í♠♦s ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ✐♥❥❡t✐✈❛ ❞♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✱ q✉❡ ♥♦s ❧❡✈❛ ❛ ❞❡✜♥✐r ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ r❡❞✉③✐❞❛✱ ❛❧é♠ ❞❡ ❝♦♥❝❧✉✐r ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❞❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛✳

  ◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐♠♦s ❛çõ❡s ♣❛r❝✐❛✐s ❞❡ ❣r✉♣♦s ❞✐s❝r❡t♦s ❡ ♠♦str❛♠♦s q✉❡✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✱ é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❆❧é♠ ❞✐ss♦✱ ❛ ✜♠ ❞❡ ✐❧✉str❛r ❞❡✜♥✐çõ❡s ♣♦st❡r✐♦r❡s✱ ✐♥tr♦❞✉③✐♠♦s ❜r❡✈❡♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❛çõ❡s ❝♦♥tí♥✉❛s ❞❡ ❣r✉♣♦s ❧♦✲ ❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s ❡♠ C ✲á❧❣❡❜r❛s ❡ ❞❡✜♥✐♠♦s ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ❛ss♦❝✐❛❞♦✳ ▼♦str❛♠♦s t❛♠❜é♠ q✉❡✱ ❞❛❞❛ ✉♠❛ ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦ ❞✐s✲ ❝r❡t♦ ❛❜❡❧✐❛♥♦✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ❞♦ s❡✉ ❣r✉♣♦ ❞✉❛❧ ♥♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♦❜t✐❞♦✳ ❊st❡ ❝❛s♦ s❡rá s✉✜❝✐❡♥t❡ ♣❛r❛ ♦ q✉❡ ♣r❡✲ ❝✐s❛♠♦s✱ ♠✉✐t♦ ❡♠❜♦r❛ ✐st♦ t❛♠❜é♠ s❡❥❛ ✈❡r❞❛❞❡ q✉❛♥❞♦ ♦ ❣r✉♣♦ ❡♠ q✉❡stã♦ é ✉♠ ❣r✉♣♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❛❜❡❧✐❛♥♦ q✉❛❧q✉❡r✳

  ◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ ✐♥tr♦❞✉③✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ C ✲á❧❣❡❜r❛s ❡stá✈❡✐s ❡✱ ❝♦♠ ✐ss♦✱ ❞❡✜♥✐♠♦s ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡stá✈❡❧ ❝♦♠♦ s❡♥❞♦ ✉♠ ✜❜r❛❞♦ ❝✉❥❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ é ✉♠❛ C ✲á❧❣❡❜r❛ ❡stá✈❡❧✳ ❉❡s❡♥✈♦❧✈❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♥❡st❛ t❡♦r✐❛✱ t❡♥❞♦ ♣♦r ♦❜❥❡t✐✈♦ ♦❜t❡r ❛s ❢❡rr❛♠❡♥t❛s ♥❡❝❡ssár✐❛s ♣❛r❛ ♦ ❝❛♣ít✉❧♦ ✜♥❛❧✳ P♦r ✜♠✱ ❡♠❜❛s❛❞♦s ❡♠ ❛♣r❡s❡♥t❛♠♦s ♦ t❡♦r❡♠❛ ❞❡ ❇r♦✇♥✲●r❡❡♥✲❘✐❡✛❡❧✳

  ◆♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ✜♥❛❧♠❡♥t❡ ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞♦ tr❛❜❛❧❤♦✳ ❈♦♠❡ç❛♠♦s ❞❡✜♥✐♥❞♦ C ✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s ❡✱ ♣❛r❛ t❛✐s✱ ❞❡✜♥✐♠♦s ❛ C ✲á❧❣❡❜r❛ ♣r♦❞✉t♦ s♠❛s❤✱ q✉❡ t❛♠❜é♠ é ❝♦♥❤❡❝✐❞❛ ♥❛ ❧✐✲ t❡r❛t✉r❛ ❝♦♠♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦✱ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❝♦❛çõ❡s ❞❡ ❣r✉♣♦s ✭✈❡❥❛

  

  ▼♦str❛♠♦s q✉❡✱ ❞❛❞❛ ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛✱ s✉❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ é ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ✐❞❡❛❧ ❞❛ C ✲á❧❣❡❜r❛ ♣r♦❞✉t♦ s♠❛s❤✳ ❚❛❧ ✐❞❡❛❧ ❛❞♠✐t❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❜❛s❡✱ ❝✉❥♦ ♣r♦❞✉t♦

  ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ♦❜t✐❞♦ é ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡ à C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ❡♠ q✉❡stã♦✱ q✉❛♥❞♦ ❡st❛ é ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❞❡ s❡✉ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦✳ ❈♦♠ ✐st♦ ❡♠ ♠ã♦s ❡ ♦ t❡♦r❡♠❛ ❞❡ ❇r♦✇♥✲●r❡❡♥✲❘✐❡✛❡❧✱ ❛ss✉♠✐♠♦s ❝❡rt❛s ❤✐♣ót❡s❡s s♦❜r❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡ ♦❜t❡♠♦s ♦ ♣r✐♥✲ ❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞♦ tr❛❜❛❧❤♦✳

  ◆♦ ❛♣ê♥❞✐❝❡✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✉s❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦ ❡♥✈♦❧✈❡♥❞♦ ❛ á❧❣❡❜r❛ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛✱ ❛♣r❡✲ s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡q✉✐✈❛❧❡♥t❡s ♣❛r❛ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ∗ ∗ ♣♦s✐t✐✈♦ ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❝♦♥str✉í♠♦s ❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ ✉♠❛ ∗✲á❧❣❡❜r❛ ❛❞♠✐ssí✈❡❧✳

  ❋✐①❡♠♦s ♥♦t❛çõ❡s ✉s❛❞❛s ❛♦ ❧♦♥❣♦ t❡①t♦✳ ❉❛❞❛ ✉♠❛ s❡♥t❡♥ç❛ ❧ó❣✐❝❛ P

  ✱ ♦ sí♠❜♦❧♦ [P ] t❡♠ ✈❛❧♦r 1 s❡ ❛ s❡♥t❡♥ç❛ P ❢♦r ✈❡r❞❛❞❡✐r❛✳ ❈❛s♦ ❝♦♥trár✐♦✱ ♦ sí♠❜♦❧♦ [P ] ♣♦ss✉✐ ✈❛❧♦r 0✳ P♦r ❡①❡♠♣❧♦✱ ♦ sí♠❜♦❧♦ [s = t] t❡♠ ✈❛❧♦r 1 s❡ s = t✱ ❡ ♣♦ss✉✐ ✈❛❧♦r 0 s❡ s 6= t✳ ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ ♦ sí♠❜♦❧♦ [n ≥ k] t❡♠ ✈❛❧♦r 1 s❡ n ≥ k ❡✱ ♥♦ ❝❛s♦ ❡♠ q✉❡ n < k✱ t❡♠♦s [n ≥ k] = 0

  ✳ ❈♦♠ r❡❧❛çã♦ ❛ ♣ré✲r❡q✉✐s✐t♦s✱ ❛ t❡♦r✐❛ ❞❡ ✐♥t❡❣r❛çã♦ ❞❡ ❣r✉♣♦s ❝♦♠

  ✈❛❧♦r❡s ❡♠ ✉♠❛ C ✲á❧❣❡❜r❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❖ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ C ✲á❧❣❡❜r❛s é ✉s❛❞♦ ❝♦♠ ❜❛st❛♥t❡ ❢r❡q✉ê♥❝✐❛✱ ❡ é ❛❜♦r❞❛❞♦ ❡♠ ✱ ❥á ❛ t❡♦r✐❛ ❞❡ ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s é ❛♣r❡s❡♥t❛❞❛ ❡♠ ❆♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✱ ❝✐t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s út❡✐s ❡ s✉❛s r❡❢❡rê♥❝✐❛s✱ à ♠❡❞✐❞❛ q✉❡ ✐ss♦ ❢♦r ♥❡❝❡ssár✐♦✳ ❊♥tr❡t❛♥t♦✱ ❛❝r❡❞✐t❛♠♦s q✉❡✱ ❡♠ s✉❛ ♠❛✐♦r✐❛✱ ❡st❡s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠

  ❈❛♣ít✉❧♦ ✶ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ ◆❡st❡ ❝❛♣ít✉❧♦✱ ❡♠❜❛s❛❞♦s ❡♠ ❝♦♠❡ç❛♠♦s ✐♥tr♦❞✉③✐♥❞♦

  C ✲♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt✱ ❛♣r❡s❡♥t❛♥❞♦ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡

  ❝♦♥str✉✐♥❞♦ ❛ á❧❣❡❜r❛ ❞❡ ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s✳ ❋❡✐t♦ ✐st♦✱ ✐♥tr♦❞✉③✐✲ ♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ C ✲á❧❣❡❜r❛s ❡ ♠♦str❛♠♦s q✉❡ ✐st♦✱ ❞❡ ❢❛t♦✱ ❞❡✜♥❡ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❆ r❡❢❡rê♥❝✐❛ t❛♠❜é♠ ❢♦✐ ❛♠♣❧❛♠❡♥t❡ ✉s❛❞❛✳

  ✶✳✶ ▼ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt

  ❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ ♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s C ❡ A ✉♠❛ C ✲á❧❣❡❜r❛✳ ❉✐③❡♠♦s q✉❡ X é ✉♠ A✲♠ó❞✉❧♦ ✭à A ❞✐r❡✐t❛✮✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r X ✱ s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ X × A → X✱ (x, a) 7→ xa s❛t✐s❢❛③❡♥❞♦✱ ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ X✱ a, b ∈ A ❡ λ ∈ C✱ ✭✐✮ x(ab) = (xa)b❀ ✭✐✐✮ λ(xa) = (λx)a = x(λa)❀ ✭✐✐✐✮ x(a + b) = xa + xb❀ ✭✐✈✮ (x + y)a = xa + ya✳ A A ❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❙❡❥❛ X ✉♠ A✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉✐③❡♠♦s q✉❡ X A : é ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ h·, ·i X × X → A t❛❧ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r x, y, z ∈ X✱ a ∈ A ❡ ❡s❝❛❧❛r❡s λ, µ ∈ C

  ✱ s❛t✐s❢❛③ ♦s s❡❣✉✐♥t❡s ♣♦st✉❧❛❞♦s✿

  A = λhx, yi A + µhx, zi A

  ✭✐✮ hx, λy + µzi ❀ A = hx, yi A a ✭✐✐✮ hx, yai ❀

  = hy, xi A ✭✐✐✐✮ hx, yi A ❀ A ≥ 0 A ✭✐✈✮ hx, xi ✱ ✐✳❡✳✱ hx, xi é ♣♦s✐t✐✈♦ ❝♦♠♦ ✉♠ ❡❧❡♠❡♥t♦ ❞❛ C ✲ á❧❣❡❜r❛ A❀ A = 0 ✭✈✮ hx, xi ✐♠♣❧✐❝❛ q✉❡ x = 0✳ A : X × X → A

  ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ h·, ·i é ✉♠ A

  ✲♣r♦❞✉t♦ ✐♥t❡r♥♦✳ A ❖❜s❡r✈❛çã♦ ✶✳✶✳✸✳ ❖s ❛①✐♦♠❛s ✭✐✮ ❡ ✭✐✐✐✮ ✐♠♣❧✐❝❛♠ q✉❡ h·, ·i é ❝♦♥❥✉❣❛❞♦✲ ❧✐♥❡❛r ♥❛ ♣r✐♠❡✐r❛ ✈❛r✐á✈❡❧✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♠ ❡❢❡✐t♦✱ t❡♠♦s ∗ ∗ hλx + µy, zi A = hλz, λx + µyi A = (λhz, xi A + µhz, yi A ) ∗ ∗

  ¯ = λhz, xi + ¯ µhz, yi A A

  ¯ = λhx, zi A + ¯ µhy, zi A . A = ❖❜s❡r✈❛çã♦ ✶✳✶✳✹✳ ❆s ❝♦♥❞✐çõ❡s ✭✐✐✮ ❡ ✭✐✐✐✮ ✐♠♣❧✐❝❛♠ q✉❡ hxa, yi a hx, yi A

  ✱ ❞♦♥❞❡ s❡❣✉❡ q✉❡ hX, Xi A := span{hx, yi A : x, y ∈ X} é ✉♠ ✐❞❡❛❧ ❡♠ A✳ A

  X ❖❜s❡r✈❛çã♦ ✶✳✶✳✺✳ ❙❡ é ✉♠ A✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✱ ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ s✐♠✐❧❛r♠❡♥t❡✳ ◆❡st❡ ❝❛s♦✱ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ A✲❧✐♥❡❛r ♥❛ ♣r✐♠❡✐r❛ ✈❛r✐á✈❡❧✱ ♦✉ s❡❥❛✱ A hλx + µy, zi = λ A hx, yi + µhy, zi A hax, yi = a A hx, yi,

  ❡ ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ X✱ a ∈ A✱ ❡ λ, µ ∈ C✳ ❉❡✜♥✐çã♦ ✶✳✶✳✻✳ ❙❡❥❛ X ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳ ❉✐③❡♠♦s A = A q✉❡ X é ❝❤❡✐♦ s❡ hX, Xi ✳

  ❊①❡♠♣❧♦ ✶✳✶✳✼✳ ❚♦❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ✭♥ã♦✲♥✉❧♦✮ ❝♦♠ ♣r♦✲ ❞✉t♦ ✐♥t❡r♥♦ ❧✐♥❡❛r ♥❛ s❡❣✉♥❞❛ ✈❛r✐á✈❡❧ é ✉♠ C✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝❤❡✐♦✳ ❊①❡♠♣❧♦ ✶✳✶✳✽✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛✳ ❊♥tã♦ A é ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝❤❡✐♦ ❝♦♠ ❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❞❛❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ A = a b ♣❡❧❛ ❞✐r❡✐t❛ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ha, bi ✱ ♣❛r❛ a, b ∈ A✳ ❙❡ I é ✉♠ ✐❞❡❛❧ ♣ró♣r✐♦ ❞❡ A✱ ❡♥tã♦ I é ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡✜♥✐❞♦s ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✳ ◆♦ ❡♥t❛♥t♦✱ I ♥ã♦ é ❝❤❡✐♦✳ ❉❡♠♦♥str❛çã♦✿ ❖s ✐t❡♥s ✭✐✮✲✭✐✈✮ ❞❛ ❉❡✜♥✐çã♦ s❡❣✉❡♠ ❞✐r❡t❛✲ ♠❡♥t❡ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❡ ♣♦st✉❧❛❞♦s r❡❧❛t✐✈♦s às ♦♣❡r❛çõ❡s ❞❡ ✐♥✈♦❧✉çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ A✳ ❖ ✐t❡♠ ✭✐✈✮ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ C ✲❛①✐♦♠❛✳ A = A

  ❏á ❛ ✐❣✉❛❧❞❛❞❡ hA, Ai ✱ s❡❣✉❡ ❞♦ ❢❛t♦ q✉❡ λ ) λ ∈Λ a = lim u λ a = lim hu λ , ai A , λ λ ❡♠ q✉❡ a ∈ A ❡ (u é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ A✳ A : X → R A k 2 +

  ❋❛③❡♥❞♦ ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ❝❛s♦ ❡s❝❛❧❛r✱ ♣♦❞❡rí❛♠♦s ♥♦s ♣❡r❣✉♥✲ 1 t❛r s❡ ❛ ❛♣❧✐❝❛çã♦ k · k ✱ x 7→ khx, xi é ✉♠❛ ♥♦r♠❛ ❡♠ A

  ✳ P❛r❛ ♦❜t❡r ✉♠❛ r❡s♣♦st❛ ❛✜r♠❛t✐✈❛✱ r❡st❛ ♣r♦✈❛r♠♦s q✉❡ ❛ ❞❡s✐❣✉❛❧✲ ❞❛❞❡ tr✐❛♥❣✉❧❛r é s❛t✐s❢❡✐t❛ ❡ ❡st❡ é✱ ❞❡ ❢❛t♦✱ ♥♦ss♦ ♣ró①✐♠♦ ♦❜❥❡t✐✈♦✳ ▲❡♠❛ ✶✳✶✳✾ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✮✳ ❙❡❥❛ X ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡ s❡❥❛♠ x, y ∈ X✳ ❊♥tã♦ hx, yi hx, yi A ≤ khx, xi A khy, yi A A

  ❝♦♠♦ ❡❧❡♠❡♥t♦s ❞❛ C ✲á❧❣❡❜r❛ A✳ A k = 1 ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ khx, xi ✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ a ∈ A ∗ ∗ 0 ≤ hxa − y, xa − yi A = a hx, xi A a − hy, xi A a − a hx, yi A + hy, yi A ∗ ∗

  ≤ a a − hy, xi A a − a hx, yi A + hy, yi A , ∗ ∗ ba ≤ kbka a ❡♠ q✉❡ ♥❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡ a ✱ ♣❛r❛ + t♦❞♦ b ∈ A ✳ ❈♦❧♦❝❛♥❞♦ a = hx, yi A ♦❜t❡♠♦s hx, yi hx, yi A ≤ hy, yi A , A

  ❝♦♠♦ ❞❡s❡❥❛❞♦✳ P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ x = 0 ♥ã♦ ❤á ♥❛❞❛ ❛ ❢❛③❡r✳ P❛r❛ ♦ ❝❛s♦ ❣❡r❛❧✱

  1 1 ❜❛st❛ ❛♣❧✐❝❛r♠♦s ♦ q✉❡ ❥á ❢♦✐ ❢❡✐t♦ ♣❛r❛ z = λx✱ ❡♠ q✉❡ λ = ✳ 2 khx, xi A k

  ❈♦r♦❧ár✐♦ ✶✳✶✳✶✵✳ ❙❡ X é ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ ❡♥tã♦ 2 1 kxk A := khx, xi A k A ≤ kakkxk A ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡♠ X t❛❧ q✉❡ kxak ✳ ▼❛✐s ❛✐♥❞❛✱

  XhX, Xi A := span{xhy, zi A : x, y, z ∈ X} é ✉♠ s✉♣❡s♣❛ç♦ ❞❡♥s♦ ❡♠ X✳ A ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❡❥❛♠♦s q✉❡ k · k é ✉♠❛ ♥♦r♠❛ ❡♠

  X ✳

  P❛r❛ λ ∈ C ❡ x ∈ X t❡♠♦s 2 1 2 2 1 2 1 kλxk A = khλx, λxi A k = k|λ| hx, xi A k = |λ|khx, xi A k = |λ|kxk A . A = 0 A = 0 ❙❡ kxk ✱ ❡♥tã♦ hx, xi ❡ ❞❛ ❝♦♥❞✐çã♦ ✭✈✮ ❞❛ ❉❡✜♥✐çã♦

  ✈❡♠ q✉❡ x = 0✳ A ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ k · k s❛t✐s❢❛③ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✳ ❖ A k ≤ kxk A kyk A

  ▲❡♠❛ ❡ ♦ C ✲❛①✐♦♠❛ ♥♦s ❞✐③❡♠ q✉❡ khx, yi ✱ ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ X✳ ❆ss✐♠✱ 2 kx + yk ≤ khx, xi A k + khx, yi A k + khy, xi A k + khy, yi A k A 2 2 ≤ kxk + 2kxk A kyk A + kyk A A 2 = (kxk A + kyk A ) . A

  P♦rt❛♥t♦✱ k · k é ✉♠❛ ♥♦r♠❛ ❡♠ X✳ ❆❧é♠ ❞✐ss♦✱ 2 ∗ kxak = khxa, xai A k = ka hx, xi A ak, A ∗ ∗ hx, xi A a ≤ khx, xi A ka a ❡♠ q✉❡ a ∈ A✳ ❯♠❛ ✈❡③ q✉❡ a ✱ ♦❜t❡♠♦s ❛ A ≤ kakkxk A ❞❡s✐❣✉❛❧❞❛❞❡ kxak ✳ A λ ) λ

  P♦r ✜♠✱ ♠♦str❡♠♦s q✉❡ XhX, Xi é ❞❡♥s♦ ❡♠ X✳ ❙❡♥❞♦ (u ∈Λ A ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ ♦ ✐❞❡❛❧ ❢❡❝❤❛❞♦ hX, Xi ✱ t❡♠♦s 2 kx − xu λ k = khx, xi A − hx, xi A u λ − u λ hx, xi A + u λ hx, xi A u λ k. A λ k A < ε

  ❉❛í✱ ❞❛❞♦ ε > 0✱ ❡①✐st❡ λ t❛❧ q✉❡ kx − xu ✳

  2 A

  ❆❣♦r❛✱ s❡❥❛ y ❡♠ hX, Xi t❛❧ q✉❡ ε ku λ − yk < .

  2 A λ − ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ♣❛r❛ k · k ❡ ♦ ❢❛t♦ q✉❡ kx(u y)k A ≤ kxk A ku λ − yk

  ✱ ❝♦♥❝❧✉í♠♦s q✉❡ A kx − xyk A < ε. ❉♦♥❞❡ XhX, Xi é ❞❡♥s♦ A✳

  ■ss♦ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛ ❞♦ ❝♦r♦❧ár✐♦✳ ❉❡✜♥✐çã♦ ✶✳✶✳✶✶✳ ❯♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt é ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦✲ A ❞✉t♦ ✐♥t❡r♥♦ X q✉❡ é ❝♦♠♣❧❡t♦ ♥❛ ♥♦r♠❛ k · k ✳ ❊①❡♠♣❧♦ ✶✳✶✳✶✷✳ ❚♦❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❧✐♥❡❛r ♥❛ s❡❣✉♥❞❛ ✈❛r✐á✈❡❧ é ✉♠ C✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✳ ❊①❡♠♣❧♦ ✶✳✶✳✶✸✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛✳ ❊♥tã♦ A é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✱ ❝♦♠ ❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡✜♥✐❞♦s ♥♦ ❊①❡♠♣❧♦ ❙❡ I é ✉♠ ✐❞❡❛❧ ✭❢❡❝❤❛❞♦✮ ❞❡ A✱ ❡♥tã♦ I é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝♦♠ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝♦♠♦ ♥♦ ❊①❡♠♣❧♦ A ❉❡♠♦♥str❛çã♦✿ ■ss♦ s❡❣✉❡ ❞♦ ❢❛t♦ q✉❡ ❛ ♥♦r♠❛ k · k ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ C

  ✲♥♦r♠❛ k · k✳ ❊①❡♠♣❧♦ ✶✳✶✳✶✹✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛ ❡ p ✉♠❛ ♣r♦❥❡çã♦ ♥❛ á❧❣❡❜r❛ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s M(A)✳ ❊♥tã♦ Ap = {ap : a ∈ A} é ✉♠ pAp✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦ ❝♦♠ ❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❞❛❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣❡❧❛ pAp = pa bp ❞✐r❡✐t❛ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡✜♥✐❞♦ ♣♦r hap, bpi ✱ ♣❛r❛ a, b ∈ A

  ✳ ❉❡♠♦♥str❛çã♦✿ ❆s ♣r♦♣r✐❡❞❛❞❡s ❛❧❣é❜r✐❝❛s sã♦ ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛❞❛s ❛ ♣❛rt✐r ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦ ❞❡ A

  ✳ pAp ◆♦✈❛♠❡♥t❡✱ ❛ ♥♦r♠❛ k · k ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ♥♦r♠❛ ❞❡ Ap ❤❡r❞❛❞❛

  ❞❡ A✱ ♣♦✐s 2 ∗ 2 kapk = khpa api pAp k = kapk . pAp ❙❡❣✉❡ q✉❡ Ap é ❝♦♠♣❧❡t♦✱ ❥á q✉❡ Ap = (Ap)p ❡✱ ♣♦rt❛♥t♦✱ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ❡♠ Ap ❝♦♥✈❡r❣❡♥t❡ ❡♠ A✱ ♣♦ss✉✐ ♦ ❧✐♠✐t❡ ❡♠ Ap✳ ❆❧é♠ ❞✐ss♦✱ λ ) λ s❡ (u ∈Λ é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ A ❡ a ∈ A✱ ♦❜s❡r✈❛♠♦s q✉❡ pap = lim pu λ ap = lim hu λ p, api pAp , λ λ pAp

  ❞♦♥❞❡ hAp, Api é ❞❡♥s♦ ❡♠ pAp✳ ▲♦❣♦✱ Ap é ✉♠ pAp✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦✳

  ❊①❡♠♣❧♦ ✶✳✶✳✶✺ ✭❙♦♠❛ ❞✐r❡t❛✮✳ ❙✉♣♦♥❤❛ q✉❡ X ❡ Y s❡❥❛♠ A✲♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt✳ ❊♥tã♦ Z = X ⊕ Y := {(x, y) : x ∈ X, y ∈ Y } é ✉♠ A

  ✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝♦♠ ❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❞❛❞❛ ♣♦r Z × A → Z✱ ((x, y), a) 7→ (xa, ya)

  ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡✜♥✐❞♦ ♣♦r ′ ′ ′ ′ h(x, y), (x , y )i A := hx, x i A + hy, y i A . ❉❡♠♦♥str❛çã♦✿ ❚♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❛❧❣é❜r✐❝❛s s❡❣✉❡♠ ❞❛ ❞❡✜♥✐çã♦ ❞❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ ❡ ❞♦ ❢❛t♦ q✉❡ X ❡ Y sã♦ A✲ ♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt✳ A

  ❱❛♠♦s ♠♦str❛r q✉❡ Z é ❝♦♠♣❧❡t♦ ❝♦♠ ❛ ♥♦r♠❛ k · k ✳ ❈♦♠ ❡❢❡✐t♦✱ hx, xi A ≤ hx, xi A + hy, yi A ,

  ❡ ✐ss♦ ♥♦s ❞✐③ q✉❡ 2 2 2 2 kxk ≤ khx, xi A + hy, yi A k = k(x, y)k ≤ kxk + kyk . A A A A ❙✐♠✐❧❛r♠❡♥t❡✱ 2 2 2 2 kyk ≤ k(x, y)k ≤ kxk + kyk . A A A A

  ❖✉ s❡❥❛✱ q 2 2 n n max{kxk A , kyk A } ≤ k(x, y)k A ≤ kxk + kyk . A A ✭†✮ n ) ∈N

  = ❙❡❥❛ (z ✉♠ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ Z✳ ❊s❝r❡✈❡♠♦s z

  (x n , y n ) ✱ ♣❛r❛ ❝❛❞❛ n✳ ❈♦♠♦ X ❡ Y sã♦ ❝♦♠♣❧❡t♦s✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ n → x

  ❧❛❞♦ ❡sq✉❡r❞♦ ❡♠ ♥♦s ❞✐③ q✉❡ ❡①✐st❡ x ∈ X ❡ y ∈ Y t❛✐s q✉❡ x n → y ❡ y ✳ ❆❣♦r❛✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✐♠♣❧✐❝❛ q✉❡ z n = (x n , y n ) → (x, y)

  ❡♠ Z✳ ▲♦❣♦✱ Z é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✳

  ✶✳✷ ❖♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s

  ◆❡st❛ s❡çã♦✱ ✈❛♠♦s ❝♦♥str✉✐r ✉♠❛ C ✲á❧❣❡❜r❛ ❛ ♣❛rt✐r ❞❡ ✉♠ A✲ ♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✱ ❛ s❛❜❡r✱ ❛ á❧❣❡❜r❛ ❞❡ ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s✳ ❚❛❧

  ∗

  C ✲á❧❣❡❜r❛ ♥♦s ♣❡r♠✐t✐rá ♦❜t❡r ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ✐♥❥❡t✐✈❛ ❞❡ ✉♠ ✜✲

  ❜r❛❞♦ ❞❡ ❋❡❧❧ ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✳ ❈♦♠❡ç❛♠♦s ❞❡✜♥✐♥✐♥❞♦ ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s ❡♠ ✉♠ C ✲♠ó❞✉❧♦

  ❞❡ ❍✐❧❜❡rt✳ ❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛♠ X ❡ Y A✲♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt✳ ❯♠❛ ❢✉♥çã♦ T : X → Y : Y → X

  é ❛❞❥✉♥tá✈❡❧ s❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ T t❛❧ q✉❡ hT (x), yi A = hx, T (y)i A , ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ A✳

  ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ T é ♦ ❛❞❥✉♥t♦ ❞❡ T ✳ P♦st❡r✐♦r♠❡♥t❡✱ ✈❡r❡♠♦s q✉❡✱ q✉❛♥❞♦ ❡①✐st❡✱ ♦ ❛❞❥✉♥t♦ é ú♥✐❝♦✳ ▲❡♠❛ ✶✳✷✳✷✳ ❚♦❞❛ ❛♣❧✐❝❛çã♦ ❛❞❥✉♥tá✈❡❧ T : X → Y ❡♥tr❡ A✲♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt é A✲❧✐♥❡❛r ✭✐st♦ é✱ T é ❧✐♥❡❛r ❡ T (xa) = T (x)a✱ ♣❛r❛ t♦❞♦ a ∈ A

  ✮ ❡ ❧✐♠✐t❛❞❛✳ ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❛♠♦s q✉❡ s❡ Z é ✉♠ A✲♠ó❞✉❧♦ A = 0 ❞❡ ❍✐❧❜❡rt ❡ x ∈ Z é t❛❧ q✉❡ hx, zi ✱ ♣❛r❛ t♦❞♦ z ∈ Z✱ ❡♥tã♦ ❛♦ ❡s❝♦❧❤❡r z = x ❝♦♥❝❧✉í♠♦s q✉❡ x = 0✳

  ❉❡st❛ ❢♦r♠❛✱ s❡♥❞♦ x ∈ X ❡ y ✉♠ ❡❧❡♠❡♥t♦ ❡s❝♦❧❤✐❞♦ ❛r❜✐tr❛r✐❛✲ ♠❡♥t❡ ❡♠ Y ✱ t❡♠♦s ∗ ∗ ∗ hT (xa), yi A = hxa, T (y)i A = a hx, T (y)i A = a hT (x), yi A = hT (x)a, yi A .

  ■ss♦ ♥♦s ❞✐③ q✉❡ hT (xa) − T (x)a, yi A = 0, ♣❛r❛ ❝❛❞❛ y ∈ Y ✱ ❞♦♥❞❡ T (xa) = T (x)a✳ 1 + x 2 ) = T (x 1 ) +

  ❙✐♠✐❧❛r♠❡♥t❡✱ ♣r♦✈❛✲s❡ q✉❡ T (λx) = λT (x) ❡ T (x T (x 2 ) 1 , x 2 ∈ X

  ✱ ❡♠ q✉❡ x ❡ λ ∈ C✳ ❘❡st❛ ♣r♦✈❛r♠♦s q✉❡ T é ❧✐♠✐t❛❞♦✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s ✉s❛r ♦ t❡♦r❡♠❛

  ❞♦ ❣rá✜❝♦ ❢❡❝❤❛❞♦✳ n ) n ∈N n ) → ❙❡❥❛ (x ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ X ❝♦♥✈❡r❣✐♥❞♦ ❛ x ❡ t❛❧ q✉❡ T (x y

  ❡♠ Y ✳ ❙❡❥❛ z ∈ Y ✳ P♦r ✉♠ ❧❛❞♦✱ hT (x n ), zi A → hy, zi A ❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ∗ ∗ hT (x n ), zi A = hx n , T (z)i A → hx, T (z)i A = hT (x), zi A ,

  A

  ❡♠ q✉❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❛♣❧✐❝❛çã♦ x → hx, zi é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✳

  P♦rt❛♥t♦✱ ❝♦♠♦ z é ❛r❜✐trár✐♦✱ ❞❡✈❡♠♦s t❡r y = T (x)✳ ❉♦♥❞❡ T é ❧✐♠✐t❛❞❛✳

  ◆❡♠ t♦❞♦ ♦♣❡r❛❞♦r A✲❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦ ❡♥tr❡ ♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt é ❛❞❥✉♥tá✈❡❧✳ ■st♦ s❡❣✉❡ ♥♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✳ ❊①❡♠♣❧♦ ✶✳✷✳✸✳ ❙❡❥❛ A = C([0, 1]) ❡ s❡❥❛ J = {f ∈ A : f(0) = 0}✳ ❉♦ ❊①❡♠♣❧♦ s❡❣✉❡ q✉❡ A ❡ J sã♦ A✲♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt✳ ❙❡❥❛ X := A ⊕ J

  ❡ s❡❥❛ T : X → X t❛❧ q✉❡ T (f, g) = (g, 0)✱ ♣❛r❛ f ∈ A ❡ g ∈ J ✳ ❊♥tã♦✱ T é A✲❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦✱ ♠❛s ♥ã♦ é ❛❞❥✉♥tá✈❡❧✳

  ❉❡♠♦♥str❛çã♦✿ ➱ ❢á❝✐❧ ✈❡r q✉❡ T é A✲❧✐♥❡❛r✳ P❛r❛ ✈❡r q✉❡ T é ❝♦♥tí✲ ♥✉♦✱ ♥♦t❡♠♦s q✉❡ 2 1 kT (f, g)k A = k(g, 0)k A = khg, gi A k 2 1 ≤ khf, f i A + hg, gi A k = k(f, g)k A . A = 1

  ❚♦♠❛♥❞♦ g ∈ J ❝♦♠ kgk ✱ ❝♦♥❝❧✉í♠♦s q✉❡ kT k = 1✳ ▲♦❣♦✱ T é A

  ✲❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦✳ (1, 0)

  ❙✉♣♦♥❤❛ q✉❡ T s❡❥❛ ❛❞❥✉♥tá✈❡❧ ❡ s❡❥❛ (f, g) := T ✳ P❛r❛ t♦❞♦ (h, k) ∈ X t❡♠♦s

  ¯ k = hT (h, k), (1, 0)i A = h(h, k), (f, g)i A = ¯ hf + ¯ kg.

  ✭†✮ ❉❛í✱ s❡❣✉❡ q✉❡ f(0) = 0✳ ❆❣♦r❛✱ ♣❛r❛ k ∈ J ❛r❜✐trár✐♦✱ ❛ ✐❣✉❛❧❞❛❞❡ ✐♠♣❧✐❝❛ q✉❡

  ¯ k(1 − f − g) = 0. ▲♦❣♦✱ ❞❡✈❡♠♦s t❡r f + g = 1✳

  ❆ss✐♠✱ 1 = f(0) + g(0) = 0 + g(0) = g(0)✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s g ∈ J✳ ❉❡✜♥✐çã♦ ✶✳✷✳✹✳ ❙❡❥❛♠ X ❡ Y A✲♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt✳ ❉❡♥♦t❛♠♦s ♣♦r L(X, Y ) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s ❞❡ X ❡♠ Y A )

  ✳ ◗✉❛♥❞♦ X = Y ✱ ❡s❝r❡✈❡♠♦s L(X)✱ ♦✉ ❛✐♥❞❛ L(X ✱ ❡♠ ✈❡③ ❞❡ L(X, X)

  ✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ ❛❣♦r❛ é ♠♦str❛r q✉❡✱ s❡ X é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧✲

  ❜❡rt✱ L(X) é ✉♠❛ C ✲á❧❣❡❜r❛✳ ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♠♦str❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s✳

  Pr♦♣♦s✐çã♦ ✶✳✷✳✺✳ ❙❡❥❛♠ X, Y, Z A✲♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt ❡ s❡❥❛♠ T : X → Y ✱ S : X → Y ❡ R : Y → Z ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s✳ ❊♥tã♦✿ ✭✐✮ T é ú♥✐❝♦❀ ∗ ∗∗

  = T ✭✐✐✮ T é ❛❞❥✉♥tá✈❡❧ ❡ T ❀ ∗ ∗ ∗

  = ¯ λT + S ✭✐✐✐✮ P❛r❛ ❝❛❞❛ λ ∈ C✱ λT + S é ❛❞❥✉♥tá✈❡❧ ❡ (λT + S) ❀ ∗ ∗ ∗

  = T R ✭✐✈✮ RT é ❛❞❥✉♥tá✈❡❧ ❡ (RT ) ✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙❡❥❛ U : Y → X t❛❧ q✉❡ hT (x), yi A = hx, U (y)i A , ♣❛r❛ q✉❛✐sq✉❡r x ∈ X ❡ y ∈ Y ✳ ❉❡st❛ ❢♦r♠❛✱ ✜①❛❞♦ y ❡♠ Y ❡ ♣❛r❛ x ❡s❝♦❧❤✐❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ ❡♠ X✱ s❡❣✉❡ hx, T (y)i A = hx, U (y)i A .

  (y) = U (y) P♦r ✉♠ ❛r❣✉♠❡♥t♦ ❥á ✉t✐❧✐③❛❞♦✱ ✐ss♦ ✐♠♣❧✐❝❛ T ✳ ✭✐✐✮ P❛r❛ x ∈ X ❡ y ∈ Y ✱ ✈❛❧❡ q✉❡ ∗ ∗ ∗ ∗ ∗∗ hT (y), xi A = hx, T (y)i = hT (x), yi = hy, T (x)i A . A A

  = T ❉♦♥❞❡ T ✳ ✭✐✐✐✮ ◆♦✈❛♠❡♥t❡✱

  ¯ h(λT + S)(x), yi A = λhT (x), yi A + hS(x), yi A ∗ ∗ ¯

  = λhx, T (x)i A + hx, S (y)i A ∗ ∗ = hx, ¯ λ(T + S )(y)i A . ❈♦♠♦ x é ❛r❜✐trár✐♦✱ ✜❝❛ ✈❡r✐✜❝❛❞♦ ♦ ✐t❡♠ ✭✐✐✐✮✳

  ✭✐✈✮ ❆ ♣r♦✈❛ é ❢❡✐t❛ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ ✐t❡♠ ✭✐✐✐✮✳ ❚❡♦r❡♠❛ ✶✳✷✳✻✳ ❙❡ X é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✱ ❡♥tã♦ L(X) é ✉♠❛ C

  ✲á❧❣❡❜r❛ ❝♦♠ r❡s♣❡✐t♦ à ♥♦r♠❛ ❤❡r❞❛❞❛ ❞❛ á❧❣❡❜r❛ ❞❡ ❇❛♥❛❝❤ B(X)✳ ❉❡♠♦♥str❛çã♦✿ ❉❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ L(X) é ✉♠❛ s✉❜á❧✲

  ❣❡❜r❛ ❞❡ B(X)✳ ❈♦♠♦ B(X) é ✉♠❛ á❧❣❡❜r❛ ❞❡ ❇❛♥❛❝❤✱ t❡♠♦s q✉❡ ∗ ∗ kT T k ≤ kT kkT k, ♣❛r❛ t♦❞♦ T ❡♠ L(X)✳

  ❆❧é♠ ❞✐ss♦✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ ∗ ∗ 2 kT T k ≥ sup khT T (x), xi A k = sup khT (x), T (x)i A k = kT k kxk≤1 kxk≤1 ∗ ∗∗ k = T ❡ ♦❜t❡♠♦s q✉❡ kT k ≤ kT ✳ ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ ♦❜s❡r✈❛♥❞♦ q✉❡ T ✱ ∗ ∗ k ≤ kT k k = kT k ❝♦♥❝❧✉í♠♦s q✉❡ kT ✳ ▲♦❣♦✱ kT ✳ 2 ∗ ∗ 2

  ≤ kT T k ≤ kT kkT k = kT k ❆ss✐♠✱ kT k ✱ ❞♦♥❞❡ s❡ ✈❡r✐✜❝❛ ♦ C

  ✲❛①✐♦♠❛ ♣❛r❛ L(X)✳ ❯♠❛ ✈❡③ q✉❡ ❛ ♦♣❡r❛çã♦ ❞❡ ✐♥✈♦❧✉çã♦ é ✉♠❛ ✐s♦♠❡tr✐❛✱ L(X) é✱ ❞❡ ❢❛t♦✱ ✉♠❛ C ✲á❧❣❡❜r❛✳ ❈♦r♦❧ár✐♦ ✶✳✷✳✼✳ ❙❡❥❛ X ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❡ T ∈ L(X)✳ ❊♥tã♦ 2 hT (x), T (x)i A ≤ kT k hx, xi A . 2 ∗ ∗

  − T T ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ kT k é ✉♠ ❡❧❡♠❡♥t♦ ♣♦s✐t✐✈♦ ❞❛ C ✲ 2 ∗ ∗

  − T T = S S á❧❣❡❜r❛ L(X)✱ ❡①✐st❡ S ∈ L(X) t❛❧ q✉❡ kT k ✳ ❉❡st❛ ❢♦r♠❛✱ 2 2 ∗ kT k hx, xi A − hT (x), T (x)i A = kT k hx, xi A − hT T (x), xi A 2 ∗ ∗

  = h(kT k − T T )(x), xi A = hS S(x), xi A A ≤ kT k hx, xi A . = hS(x), S(x)i A ≥ 0. 2 ❉♦♥❞❡ s❡❣✉❡ hT (x), T (x)i

  ✶✳✸ ❇✐♠ó❞✉❧♦s ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡

  ◆❡st❛ s❡çã♦✱ ❞❡✜♥✐♠♦s ♦ q✉❡ s❡r✐❛ ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐✲ t✐✈✐❞❛❞❡✱ ♣❛r❛ A ❡ B C ✲á❧❣❡❜r❛s✳ ❆♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♥❡st❡ s❡♥t✐❞♦✳ ❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s✳ ❯♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ é ✉♠ A✲B ❜✐♠ó❞✉❧♦ X t❛❧ q✉❡✿ ✭✐✮ X é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦ à ❡sq✉❡r❞❛ ❡ ✉♠ B✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦ à ❞✐r❡✐t❛❀

  ✭✐✐✮ P❛r❛ q✉❛✐sq✉❡r x, y, z ∈ X✱ A hx, yiz = xhy, zi B . ❊①❡♠♣❧♦ ✶✳✸✳✷✳ ❯♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H é ✉♠ K(H)✲C ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❝♦♠ ❛s ❛çõ❡s ❞❡ ♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ❡ à ❞✐r❡✐t❛ ❞❛❞❛s ♣♦r (T, h) 7→ T (h) ❡ (h, λ) 7→ λh✱ ♣❛r❛ T ∈ K(H)✱ λ ∈ C ❡ h ∈ H✱ ❡ ♣r♦❞✉t♦s ✐♥t❡r♥♦s ❞❡✜♥✐❞♦s ❝♦♠♦ s❡❣✉❡✿ K hh, ki := h ⊗ k, (H) ❡♠ q✉❡ (h ⊗ k)(z) = hk, zih, ♣❛r❛ t♦❞♦ z ∈ H✱ ❡ C hh, ki := hh, ki. ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ F (H) ♦ ❝♦♥❥✉♥t♦ ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ♣♦st♦ ✜♥✐t♦ s♦❜r❡ H✳ ❙❛❜❡♠♦s ❞♦ ❚❡♦r❡♠❛ ✷✳✹✳✺ ❞❡ q✉❡ F (H) é ❞❡♥s♦ ❡♠ K(H)

  ✳ ❏á ♦ ❚❡♦r❡♠❛ ✷✳✹✳✻✱ ♥♦✈❛♠❡♥t❡ ❡♠ ♥♦s ❞✐③ q✉❡ F (H) é ❣❡r❛❞♦ ♣♦r ♦♣❡r❛❞♦r❡s ❞❡ ♣♦st♦ 1✱ ❡ ❡st❡s sã♦ ♣r❡❝✐s❛♠❡♥t❡ ♦s ♦♣❡r❛❞♦✲ K hH, Hi r❡s ❞❛ ❢♦r♠❛ h ⊗ k✱ h, k ∈ H✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡ (H) é ❞❡♥s♦ ❡♠ K(H)

  ✳ ▼❛✐s ❛✐♥❞❛✱ ♦❜s❡r✈❛♠♦s 2 1 2 2 1 khk = k(h ⊗ h)k = (khk ) = khk, K (H)

  ❞♦♥❞❡ H é ✉♠ K(H)✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦✳ ❱❛♠♦s ✈❡r✐✜❝❛r ❛ ❝♦♥❞✐çã♦ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ❙❡❥❛♠ x, y, z ∈ H✳ ❊♥tã♦✱ C K (H) hx, yiz = hy, zix = xhy, zi .

  ▲♦❣♦✱ H é ✉♠ K(H)✲C ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❊①❡♠♣❧♦ ✶✳✸✳✸✳ ❙❡❥❛ A ✉♠ C ✲á❧❣❡❜r❛✳ ❊♥tã♦ A é ✉♠ A✲A ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ ❜✐♠ó❞✉❧♦ ❞❛❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ A ha, bi = ab A = a b ∗ ∗ ❡♠ A✱ ❡ ❝♦♠ ♣r♦❞✉t♦s ✐♥t❡r♥♦s ❡ ha, bi ✱ ♣❛r❛

  a, b ∈ A ✳

  ❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ✈❡r✐✜❝❛r ♦ ✐t❡♠ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ■st♦ s❡❣✉❡ ❞♦ s❡❣✉✐♥t❡ ❝á❧❝✉❧♦✿ A ha, bic = ab c = ahb, ci A ,

  ❡♠ q✉❡ a, b, c ∈ A✳ ❊①❡♠♣❧♦ ✶✳✸✳✹✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛ ❡ s❡❥❛♠ p, q ♣r♦❥❡çõ❡s ❡♠ M (A)

  ✳ ❙✉♣♦♥❤❛ q✉❡ ApA = AqA = A✳ ❊♥tã♦ pAq é ✉♠ pAp✲qAq ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ ❜✐♠ó❞✉❧♦ ❞❛❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ A✱ ❡ ♣r♦❞✉t♦s ✐♥t❡r♥♦s ❞❡✜♥✐❞♦s ♣♦r pAp hpaq, pbqi = paqb p hpaq, pbqi qAq = qa pbq, ♣❛r❛ a, b ∈ A✳ ❉❡♠♦♥str❛çã♦✿ ❖ ❢❛t♦ ❞❡ pAq s❡r ✉♠ pAp✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦ ❡ ✉♠ qAq✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞❛ ❤✐✲ ♣ót❡s❡ ApA = AqA = A✳ ❆ ❝♦♥❞✐çã♦ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ é ✈❡r✐✜❝❛❞❛ ♥♦ s❡❣✉✐♥t❡ ❝á❧❝✉❧♦✿ pAp hpaq, pbqipcq = paqb pcq = paqhpbq, pcqi qAq , ❡♠ q✉❡ a, b, c ∈ A✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛♦ ❡s❝♦❧❤❡r q = 1✱ s❡❣✉❡ q✉❡ pA é ✉♠ pAp✲A ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳

  ❯♠❛ ♣r♦❥❡çã♦ p ❡♠ M(A) s❛t✐s❢❛③❡♥❞♦ ❛ ❤✐♣ót❡s❡ ❞♦ ❡①❡♠♣❧♦ ❛♥✲ t❡r✐♦r ✭ApA = A✮ é ❞✐t❛ s❡r ❝❤❡✐❛✳ ❯♠❛ C ✲á❧❣❡❜r❛ ❞❛ ❢♦r♠❛ pAp✱ ❡♠ q✉❡ p ∈ M(A) é ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛✱ é ❝❤❛♠❛❞❛ ❝❛♥t♦ ❝❤❡✐♦✳ ❊st❡ t✐♣♦ ❞❡ ♣r♦❥❡çã♦ t❡rá ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ❡ s❡rá ❡st✉❞❛❞♦ ❝♦♠ ♠❛✐s ❞❡t❛❧❤❡s ♥❛ ❙❡çã♦ Pr♦♣♦s✐çã♦ ✶✳✸✳✺✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s ❡ s✉♣♦♥❤❛ q✉❡ X s❡❥❛ ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❊♥tã♦ ✭✐✮ P❛r❛ q✉❛✐sq✉❡r a ∈ A✱ b ∈ B✱ ❡ x, y ∈ X✱ A hxb, yi = A hx, yb i B = hx, a yi B ; ∗ ∗

  ❡ hax, yi ✭✐✐✮ P❛r❛ q✉❛✐sq✉❡r a ∈ A✱ b ∈ B✱ ❡ x ∈ X✱ 2 2 hax, axi B ≤ kak hx, xi B A hxb, xbi ≤ kbk A hx, xi. ❡

  ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙✉♣♦♥❤❛ q✉❡ X s❡❥❛ ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐✲

  ♠✐t✐✈✐❞❛❞❡✳ P❡❧♦ ✐t❡♠ ✭✐✐✐✮✱ ❞❛❞♦s x, y, z ∈ X✱ t❡♠♦s q✉❡ ∗ ∗ xhay, zi B = A hx, ayiz = A hx, yia z = xhy, a zi B .

  ❆ss✐♠✱ s❡ w é ♦✉tr♦ ❡❧❡♠❡♥t♦ ❞❡ X✱ ✈❛❧❡ q✉❡ hw, xi B hay, zi B = hw, xi B hy, a zi B . B ❯♠❛ ✈❡③ q✉❡ hX, Xi é ❞❡♥s♦ ❡♠ B✱ ❝♦♥❝❧✉í♠♦s q✉❡ bhay, zi B = bhy, a zi B , ♣❛r❛ t♦❞♦ b ∈ B✳ P♦r ♠❡✐♦ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛✱ ♦❜t❡♠♦s q✉❡ hay, zi B = hy, a zi B , ♣❛r❛ q✉❛✐sq✉❡r y, z ∈ X✳ A hyb, zi = A hy, zbi

  ❙✐♠✐❧❛r♠❡♥t❡✱ ♣r♦✈❛✲s❡ q✉❡ ✱ ♣❛r❛ q✉❛✐sq✉❡r y, z ∈

  X ✳

  ✭✐✐✮ P❡❧♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♥♦ ✐t❡♠ ✭✐✮✱ A ❛❣❡ ♣♦r ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s ❡♠

  X B B ) ✳ ❉❡st❛ ❢♦r♠❛✱ ✈❛♠♦s ♠♦str❛r q✉❡ ❛ ❛♣❧✐❝❛çã♦ ϕ : A → L(X ✱ a 7→ ϕ(a)

  é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦✱ ❡♠ q✉❡ ϕ(a)(x) = ax,

  ♣❛r❛ t♦❞♦ x ∈ X✳ ∗ ∗ = ϕ(a )

  ❉❡ ❢❛t♦✱ ❞♦ ✐t❡♠ ✭✐✮✱ ❝♦♥❝❧✉í♠♦s q✉❡ ϕ(a) ❡ ϕ é ✉♠ ∗✲ ❤♦♠♦♠♦r✜s♠♦✳ P❛r❛ ✈❡r q✉❡ ϕ é ✐♥❥❡t✐✈♦✱ s❡❥❛ a ∈ A t❛❧ q✉❡ ϕ(a) = 0✳ ❊♥tã♦ s❡❣✉❡ q✉❡ ax = 0✱ ♣❛r❛ t♦❞♦ x ∈ X✳ ▼❛s ✐ss♦ s✐❣♥✐✜❝❛ q✉❡✱ ♣❛r❛ ❝❛❞❛ y ∈ X✱ 0 = A hax, yi = a A hx, yi. A hX, Xi

  ❯♠❛ ✈❡③ q✉❡ ♦ ✐❞❡❛❧ é ❞❡♥s♦ ❡♠ A✱ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡ a = 0✳ ❆ss✐♠✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ 2 2 hax, axi B = hϕ(a)(x), ϕ(a)(x)i B ≤ kϕ(a)k hx, xi B = kak hx, xi B .

  ❯♠ ❛r❣✉♠❡♥t♦ ❛♥á❧♦❣♦ ♠♦str❛ q✉❡ A hxb, xbi ≤ kbk A hx, xi. 2 ❖❜s❡r✈❛çã♦ ✶✳✸✳✻✳ ❉♦ ▲❡♠❛ s❡❣✉❡ q✉❡ (ax)b = a(xb) ❡ (λa)(xb) = a(x(λb))✳ ❆ss✐♠✱ é r❡❞✉♥❞❛♥t❡ ❡①✐❣✐r q✉❡ X s❡❥❛ ✉♠ A✲B ❜✐♠ó❞✉❧♦✳ ❈♦r♦❧ár✐♦ ✶✳✸✳✼✳ ❙❡❥❛ X ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❊♥✲ tã♦✱ kxk A = kxk B ,

  ♣❛r❛ t♦❞♦ x ∈ X✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ x ∈ X✳ ❚❡♠♦s q✉❡ 4 2 kxk = k A hx, xik = k A hx, xi A hx, xik A = k A h A hx, xix, xik = k A hxhx, xi B , xik.

  ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❡ ✐t❡♠ ✭✐✐✮ ❞❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ 4 2 2 A ≤ kxk B kxk ≤ kxhx, xi B k A kxk A ≤ kxk kxk . A B A

  ▲♦❣♦✱ kxk ✳ ❯♠ ❝á❧❝✉❧♦ ❛♥á❧♦❣♦ ♣r♦✈❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❝♦♥trár✐❛✳ A = kxk B

  P♦rt❛♥t♦✱ kxk ✳

  ✶✳✹ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛

  ◆❡st❛ s❡çã♦✱ t❡♥❞♦ ❡♠ ♠ã♦s ♦ q✉❡ ❢♦✐ ❢❡✐t♦ ❛té ❛q✉✐✱ ❞❡✜♥✐♠♦s ▼♦✲ r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ C ✲á❧❣❡❜r❛s✳ ❚❛❧ r❡❧❛çã♦ é r❡✢❡①✐✈❛ ❡ s✐♠étr✐❝❛✱ ❡ ❝♦♠ ✉♠ ❡s❢♦rç♦ ✉♠ ♣♦✉❝♦ ♠❛✐♦r✱ ♠♦str❛♠♦s q✉❡ ❛ tr❛♥s✐t✐✈✐❞❛❞❡ t❛♠❜é♠ é s❛t✐s❢❡✐t❛✳ ❊♥❝❡rr❛♠♦s ❝♦♥str✉✐♥❞♦ ❛ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦ ❞❡ ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✱ q✉❡ t❡♠ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❡st✉❞♦ ❞❡ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❡❧❛ q✉❡ s❡rá ✉s❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ❉❡✜♥✐çã♦ ✶✳✹✳✶✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s✳ ❉✐③❡♠♦s q✉❡ A é ▼♦r✐t❛ M B ❡q✉✐✈❛❧❡♥t❡ ❛ B✱ ❡ ❞❡♥♦t❛♠♦s A ∼ ✱ s❡ ❡①✐st❡ ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳

  ❆❧é♠ ❞♦s ❡①❡♠♣❧♦s ❛♣r❡s❡♥t❛❞♦s ♥❛ ❙❡çã♦ tr❛r❡♠♦s ❛ s❡❣✉✐r ✉♠ ♦✉tr♦ ❡①❡♠♣❧♦✱ q✉❡ ♥♦s ❞✐③ q✉❡ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ é ✉♠ ❝♦♥❝❡✐t♦ ♠❛✐s ❢r❛❝♦ q✉❡ ✐s♦♠♦r✜s♠♦✳ ◆♦ ❡♥t❛♥t♦✱ ✈❡r❡♠♦s ♥♦ ❈❛♣ít✉❧♦ q✉❡✱ s♦❜ ❝❡rt❛s ❤✐♣ót❡s❡s ❞❡ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡✱ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ ✐♠♣❧✐❝❛ ✐s♦♠♦r✜s♠♦ ❡stá✈❡❧✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡ A é ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡ ❛

  ∗

  B ❡ K ❞❡♥♦t❛ ❛ C ✲á❧❣❡❜r❛ ❞♦s ♦♣❡r❛❞♦r❡s ❝♦♠♣❛❝t♦s s♦❜r❡ ✉♠ ❡s♣❛ç♦

  ❞❡ ❍✐❧❜❡rt s❡♣❛rá✈❡❧ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✱ ❡♥tã♦ A ⊗ K ∼ = B ⊗ K ✳ ❊①❡♠♣❧♦ ✶✳✹✳✷✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s✳ ❙❡❥❛ ψ : A → B ✉♠ ✐s♦♠♦r✜s♠♦✳ ❊♥tã♦ A é ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡ ❛ B✳ ❉❡♠♦♥str❛çã♦✿ ❈♦❧♦❝❛♠♦s X = B ❡ ❞❡✜♥✐♠♦s ❛ ❡str✉t✉r❛ ❞❡ ❜✐♠ó✲ ❞✉❧♦ ♣♦r

  (a, x) 7→ ψ(a)x ❡ (x, b) 7→ xb

  ❡ ♣r♦❞✉t♦s ✐♥t❡r♥♦s ❞❛❞♦s ♣♦r A hx, yi = ψ (xy ) B = x y, −1 ∗ ∗ ❡ hx, yi

  ♣❛r❛ a ∈ A✱ b ∈ B ❡ x, y ∈ X✳ ❆✜r♠❛♠♦s q✉❡ X é ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ B ➱ ❝❧❛r♦ q✉❡ hX, Xi é ❞❡♥s♦ ❡♠ B✳ ❯♠❛ ✈❡③ q✉❡ ψ é ✉♠ ✐s♦♠♦r✲ A hX, Xi

  ✜s♠♦✱ t❛♠❜é♠ ✈❛❧❡ q✉❡ é ❞❡♥s♦ ❡♠ A✳ ❆ ❝♦♥❞✐çã♦ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ s❡❣✉❡ ❞♦ s❡❣✉✐♥t❡ ❝á❧❝✉❧♦✿ A hx, yiz = ψ(ψ (xy ))z = xy z = xhy, zi B . −1 ∗ ∗ ▲♦❣♦✱ X é ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ◆♦ss♦ ♣ró①✐♠♦ ♦❜❥❡t✐✈♦ é ♠♦str❛r q✉❡ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ é✱ ❝♦♠♦ ♦

  ♣ró♣r✐♦ ♥♦♠❡ s✉❣❡r❡✱ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐✲ çã♦ ♠♦str❛ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ s✐♠❡tr✐❛ ❞❛ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❆♥t❡s ❞❡ ❡♥✉♥❝✐á✲❧❛✱ ❧❡♠❜r❛♠♦s q✉❡ s❡ V é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧

  V = {˜ v : v ∈ V } ❝♦♥❥✉❣❛❞♦ ❞❡ V é ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ e ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ❞❛❞❛s ♣♦r

  ¯ ˜ x + ˜ y := ] x + y, λ˜ x := f λx. ❆ ♣❛rt✐r ❞❡ ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✱ é ♣♦ssí✈❡❧ ❞❡✜♥✐r

  ✉♠❛ ❡str✉t✉r❛ ❞❡ B✲A ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ♥♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧

  X ❝♦♥❥✉❣❛❞♦ e ✳ ❈♦♠♦ s❡❣✉❡✿ Pr♦♣♦s✐çã♦ ✶✳✹✳✸✳ ❙❡❥❛ X ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❉❡✲

  X ✜♥✐♠♦s ✉♠❛ ❡str✉t✉r❛ ❞❡ B✲A✲❜✐♠ó❞✉❧♦ ❡♠ e ♣♦r ∗ ∗

  (b, ˜ x) 7→ g xb a x ❡ (˜x, a) 7→ g

  ❡ ♣r♦❞✉t♦s ✐♥t❡r♥♦s ♣♦r B h˜ x, ˜ yi := hx, yi B A := A hx, yi, ❡ h˜x, ˜yi

  X X ♣❛r❛ a ∈ A✱ b ∈ B ❡ ˜x, ˜y ∈ e ✳ ❊♥tã♦✱ e é ✉♠ B✲A ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❉❡♠♦♥str❛çã♦✿ ❉✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❞♦s ♣r♦❞✉t♦s ✐♥t❡r♥♦s ❡ ❞♦ B h e X, e Xi ❢❛t♦ q✉❡ X é ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✱ s❡❣✉❡ q✉❡

  X, e Xi A é ❞❡♥s♦ ❡♠ B ❡ h e é ❞❡♥s♦ ❡♠ A✳

  X ❖s s❡❣✉✐♥t❡s ❝á❧❝✉❧♦s ♣r♦✈❛♠ q✉❡ e é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt à

  ❞✐r❡✐t❛✿ h˜ x, ˜ yai A = A hx, a yi = A hx, yia = h˜ x, ˜ yi A a. ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ ♣r♦✈❛✲s❡ q✉❡ X é ✉♠ B✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt à

  ❡sq✉❡r❞❛✳ ❆❧é♠ ❞✐ss♦✱ B h˜ x, ˜ yi˜ z = zhy, xi B ^

  ^ = A hz, yix = ˜ xh˜ y, ˜ zi A .

  X ▲♦❣♦✱ ❝♦♠ ❡st❛s ♦♣❡r❛çõ❡s e é ✉♠ B✲A ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐✲

  ❞❛❞❡✳ P❛r❛ ♣r♦✈❛r ❛ ♣r♦♣r✐❡❞❛❞❡ tr❛♥s✐t✐✈❛ ❞❛ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛✱ ♣r❡✲

  ❝✐s❛♠♦s ❡st✉❞❛r ❝♦♠♣❧❡t❛♠❡♥t♦s ❞❡ C ✲♠ó❞✉❧♦s ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳ ❆ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ ✈❛♠♦s ❞❡s❡♥✈♦❧✈❡r ✉♠ ♣♦✉❝♦ ❞❛ t❡♦r✐❛ ♥❡ss❡ s❡♥t✐❞♦✳ ❉❡✜♥✐çã♦ ✶✳✹✳✹✳ ❙❡❥❛ A ✉♠❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡♥s❛ ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ A

  ❡ s❡❥❛ X ✉♠ A ✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉✐③❡♠♦s q✉❡ X é ✉♠ A ✲♠ó❞✉❧♦ : X ×X → A

  ❝♦♠ ♣ré✲♣r♦❞✉t♦ ✐♥t❡r♥♦ s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ h·, ·i ≥ 0 q✉❡ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ✭✐✮✲✭✐✈✮ ❞❛ ❉❡✜♥✐çã♦ ✭❡♠ q✉❡ hx, xi

  é ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ❡❧❡♠❡♥t♦ ❞❡ A✮✳ ❖❜s❡r✈❛çã♦ ✶✳✹✳✺✳ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✭▲❡♠❛ ❝♦♠ ✉♠❛ ❞❡♠♦♥str❛çã♦ ✐❞ê♥t✐❝❛✳

  Pr♦♣♦s✐çã♦ ✶✳✹✳✻✳ ❙❡❥❛ A ✉♠❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡♥s❛ ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ A

  ❡ s❡❥❛ X ✉♠ A ✲♠ó❞✉❧♦ ❝♦♠ ♣ré✲♣r♦❞✉t♦ ✐♥t❡r♥♦ h·, ·i ✳ ❊♥tã♦ ❡①✐st❡ → X

  ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt X ❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r q : X t❛❧ q✉❡ q(X ) é ❞❡♥s♦ ❡♠ X✱ q(xa) = q(x)a ♣❛r❛ t♦❞♦ x ∈ X ✱ a ∈ A ✱ ❡ hq(x), q(y)i A = hx, yi

  ✳ : hx, xi = 0}

  ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ N = {x ∈ X ✳ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ hx, yi = 0 = hy, xi , s❡♠♣r❡ q✉❡ y ∈ X ❡ x ∈ N. ✭†✮

  ▲♦❣♦✱ N é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ X ❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❡s♣❛ç♦ /N

  ✈❡t♦r✐❛❧ q✉♦❝✐❡♥t❡ X ✳ → X /N = ❙❡❥❛ q : X ❛ ❛♣❧✐❝❛çã♦ q✉♦❝✐❡♥t❡✳ ❈♦♠♦ hxa, xai a hx, xi a

  ✱ s❡❣✉❡ q✉❡ N é ✉♠ A ✲s✉❜♠ó❞✉❧♦✳ ❉❡st❛ ❢♦r♠❛✱ s❡❣✉❡ q✉❡ ′ ′ ′ )

  ) s❡ q(x) = q(x ✱ t❡♠✲s❡ ax = ax ✳ ▼❛✐s ❛✐♥❞❛✱ ♣♦r s❡ q(y) = q(y ✈❛❧❡ q✉❡ ′ ′ ′ hx, y − y i = 0 , y i = 0 ′ ′ ❡ hx − x

  = hx , y i ❞♦♥❞❡ hx, yi ✳

  P♦rt❛♥t♦✱ ❛s ❢ór♠✉❧❛s hq(x), q(y)i A := hx, yi ❡ q(x)a := q(xa)

  /N ❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s ❡ ❢❛③❡♠ ❞❡ X ✉♠ A ✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳

  ❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ ❈♦r♦❧ár✐♦ 2 1 kq(x)k := khx, xi k /N

  é ✉♠❛ ♥♦r♠❛ ❡♠ X ✱ ❞♦♥❞❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r s❡✉ ❝♦♠♣❧❡t❛♠❡♥t♦ X /N

  ✳ ■❞❡♥t✐✜❝❛♥❞♦ X ❝♦♠ ♦ s✉❜❡s♣❛ç♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡♠ X✱ t❡♠♦s q✉❡ 2 ∗ 2 2 2 kq(x)ak = khxa, xai k = ka hx, xi ak ≤ khx, xi kak = kq(x)k kak .

  ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣❡❧❛ ❞✐r❡✐t❛ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ a /N

  ❞❡ A é ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ s♦❜r❡ X ✱ ❡ ❛ss✐♠ s❡ ❡st❡♥❞❡ ❛ ✉♠ ♦♣❡r❛❞♦r s♦❜r❡ X s❛t✐s❢❛③❡♥❞♦ kaxk ≤ kakkxk✱ ♣❛r❛ t♦❞♦ x ∈ X✳ ◆♦✲ ✈❛♠❡♥t❡ ✉s❛♥❞♦ ❛ ❝♦♥t✐♥✉✐❞❛❞❡✱ ♣♦❞❡♠♦s ❡st❡♥❞❡r ❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ à C

  ✲á❧❣❡❜r❛ A✳ n )} n ∈N n )} n ∈N ❙✐♠✐❧❛r♠❡♥t❡✱ s❡♥❞♦ x, y ∈ X✱ ❡ {q(x ✱ {q(x s❡q✉ê♥✲

  /N n ) → x n ) → y ❝✐❛s ❡♠ X t❛✐s q✉❡ q(x ❡ q(y ✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝♦❧♦❝❛♥❞♦ hx, yi A := lim hq(x n ), q(y n )i A . n

  ❖ ❧✐♠✐t❡ ❛❝✐♠❛ ❡①✐st❡ ♣♦✐s khx n , y n i − hx m , y m i k = khx n − x m , y n i + hx m , y n − y m i k ≤ kq(x n ) − q(x m )kkq(y n )k +kq(x m )kkq(y n ) − q(y m )k. ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ ♣r♦✈❛✲s❡ q✉❡ ♦ ❧✐♠✐t❡ ✐♥❞❡♣❡♥❞❡ ❞❛ s❡q✉ê♥❝✐❛ q✉❡

  A

  t♦♠❛r♠♦s ❝♦♥✈❡r❣✐♥❞♦ ❛ x ❡ ❛ y✳ ❖✉ s❡❥❛✱ h·, ·i ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❯♠❛ ✈❡③ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ♣♦s✐t✐✈♦s ❞❡ A é ❢❡❝❤❛❞♦✱ ✜❝❛ A

  ❢á❝✐❧ ✈❡r q✉❡ h·, ·i s❛t✐s❢❛③ (i) − (iv) ❞❛ ❉❡✜♥✐çã♦ ❆❧é♠ ❞✐ss♦✱ hx, xi = 0 n kq(x n )k = 0 ✐♠♣❧✐❝❛ lim ✳ ▼❛s✱ ❡st❡ ❧✐♠✐t❡ é ❡①❛t❛♠❡♥t❡ kxk A

  ✳ ❖✉ s❡❥❛✱ x = 0✱ ♦ q✉❡ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛ ❞❡ q✉❡ h·, ·i é ✉♠ A

  ✲♣r♦❞✉t♦ ✐♥t❡r♥♦✳ P♦rt❛♥t♦✱ X é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✳ ❱❛♠♦s ♥♦s r❡❢❡r✐r ❛♦ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt X ❝♦♥str✉í❞♦ ❛❝✐♠❛ ❝♦♠♦

  ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞♦ A ✲♠ó❞✉❧♦ X ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳

  ❉❡✜♥✐çã♦ ✶✳✹✳✼✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s ❡ s❡❥❛♠ A ❡ B ✲s✉❜á❧❣❡✲ ❜r❛s ❞❡♥s❛s ❞❡ A ❡ B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❯♠ A ✲B ♣ré✲❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ X q✉❡ é ✉♠ A ✲B ❜✐♠ó❞✉❧♦ s❛t✐s❢❛③❡♥❞♦✿ ✭✐✮ X é ✉♠ A ✲♠ó❞✉❧♦ ❝♦♠ ♣ré✲♣r♦❞✉t♦ ✐♥t❡r♥♦ à ❡sq✉❡r❞❛ ❡ ✉♠ B ✲ ♠ó❞✉❧♦ ❝♦♠ ♣ré✲♣r♦❞✉t♦ ✐♥t❡r♥♦ à ❞✐r❡✐t❛❀ A hX , X i , X i B ✭✐✐✮ ❡ hX sã♦ ✐❞❡❛✐s ❞❡♥s♦s ❡♠ A ❡ B✱ r❡s♣❡❝t✐✈❛✲ ♠❡♥t❡✳ ✭✐✐✐✮ P❛r❛ q✉❛✐sq✉❡r a ∈ A ✱ b ∈ B ✱ ❡ x ∈ X ✱ 2 2 hax, axi B ≤ kak hx, xi B A hxb, xbi ≤ kbk A hx, xi;

  ❡ ✭✐✈✮ P❛r❛ q✉❛✐sq✉❡r x, y, z ∈ X ✱ A hx, yiz = xhy, zi B . ∗ ∗

  C Pr♦♣♦s✐çã♦ ✶✳✹✳✽✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s ❡ s❡❥❛♠ A ❡ B ✲ á❧❣❡❜r❛s ❞❡♥s❛s ❞❡ A ❡ B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛ X ✉♠ A ✲B ♣ré✲ ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❊♥tã♦ ❡①✐st❡♠ ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠✲

  → X ♣r✐♠✐t✐✈✐❞❛❞❡ X ❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❜✐♠ó❞✉❧♦s q : X ✭✐st♦ é✱ q é ❧✐♥❡❛r ❡ q(axb) = aq(x)b✱ ♣❛r❛ q✉❛✐sq✉❡r x ∈ X ✱ a ∈ A ❡ b ∈ B )

  ✮ t❛❧ q✉❡ q(X é ❞❡♥s♦ ❡♠ X ❡ A hq(x), q(y)i = A hx, yi B = hx, yi B ❡ hq(x), q(y)i s❡♠♣r❡ q✉❡ x, y ∈ X ✳

  ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ❛♣❧✐❝❛♥❞♦ ♦ ✐t❡♠ ✭✐✐✐✮ ❞❛ ❉❡✜♥✐çã♦ A kxk = kxk B t❡♠♦s q✉❡ ✱

  ♣❛r❛ t♦❞♦ x ∈ X ✱ ❝♦♠ ✉♠❛ ❞❡♠♦♥str❛çã♦ ✐❞ê♥t✐❝❛ à q✉❡ ❢♦✐ ❢❡✐t❛ ♥♦ ❈♦r♦❧ár✐♦ ❝♦✐♥❝✐❞❡♠✳ A = N B A = {x ∈ X : A hx, xi = 0} ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ N ✱ ❡♠ q✉❡ N B = {x ∈ X : hx, xi B = 0} ❡ N ✳ ❆ss✐♠✱ ❝♦♥t✐♥✉❛♠♦s ✉s❛♥❞♦ ❛ ♥♦t❛çã♦ q → X /N.

  ♣❛r❛ ❛♣❧✐❝❛çã♦ q✉♦❝✐❡♥t❡ q : X A k · k = k · k B ◆♦✈❛♠❡♥t❡ ✉s❛♥❞♦ ♦ ❢❛t♦ q✉❡ ✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦

  /N, k · k B ) ❝♦♠♣❧❡t❛♠❡♥t♦ X ❞❡ (X é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦ ❡ ✉♠ B✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦✳ ▼❛✐s ❛✐♥❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r a ∈ A ✱ b ∈ B

  ✱ ❡ x ∈ X ✱ q(axb) = q(ax)b = aq(x)b. A hx, yiz = xhy, zi B ❈♦♠♦ ♣❛r❛ q✉❛✐sq✉❡r x, y, z ∈ X ✈❛❧❡ q✉❡ ✱ t❡♠✲s❡ q✉❡ A hq(x), q(y)iq(z) = A hx, yiq(z) = q( A hx, yiz)

  = q(xhy, zi B ) = q(x)hy, zi B ❡ ❛ss✐♠ t❛♠❜é♠ é ✈❡r❞❛❞❡ ♣❛r❛ q✉❛✐sq✉❡r x, y, z ∈ X✳

  P♦rt❛♥t♦✱ ✜❝❛ ♣r♦✈❛❞♦ q✉❡ X é ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐✲ ❞❛❞❡✳ ❈♦r♦❧ár✐♦ ✶✳✹✳✾✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s ❡ s❡❥❛ X ✉♠ A✲B ❜✐♠ó❞✉❧♦ s❛t✐s❢❛③❡♥❞♦ ♦s ✐t❡♥s ✭✐✮✱ ✭✐✐✮ ❡ ✭✐✈✮ ❞❛ ❉❡✜♥✐çã♦ ❊♥tã♦ X é ✉♠ A

  ✲B ♣ré✲❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ s❡ ❡ s♦♠❡♥t❡ s❡ ✭✐✐✐✮✬ ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ X✱ a ∈ A✱ ❡ b ∈ B✱ ∗ ∗ hax, yi B = hx, a yi B A hxb, yi = A hx, yb i.

  ❡ ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ X s❡❥❛ ✉♠ A✲B ❜✐♠ó❞✉❧♦ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ✭✐✮✱ ✭✐✐✮ ❡ ✭✐✈✮ ❞❛ ❉❡✜♥✐çã♦ ❡ ❛ ❝♦♥❞✐çã♦ ✭✐✐✐✮✬✳ ❙❡❥❛

  ˜ A = {a + λ1 : a ∈ A, λ ∈ C}

  ❛ ✉♥✐t✐③❛çã♦ ❞❡ A✳ ❊♥tã♦✱ ❛ ❛çã♦ ❞❡ A ❡♠ X s❡ ❡st❡♥❞❡ ❛ ✉♠❛ ❛çã♦ ❞❡ ˜

  A ♣♦r (a + λ1, x) 7→ ax + λx✳ ◆♦t❡♠♦s q✉❡ h(a + λ1)x, yi B = hax + λx, yi B

  = hax, yi B + hλx, yi B = hx, a yi B + hλx, ¯ λyi B = hx, (a + λ1) yi B .

  ∗

  d A ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ c = d é ✉♠ ❡❧❡♠❡♥t♦ ♣♦s✐t✐✈♦ ❡♠ ˜ ✱ ♦❜t❡♠♦s q✉❡ 2 ∗ hx, cxi B = hx, d dxi B = hdx, dx ≥ 0.

  1 − a a A ❈♦♠♦ kak é ✉♠ ❡❧❡♠❡♥t♦ ♣♦s✐t✐✈♦ ❡♠ ˜ ✱ ❝♦♥❝❧✉í♠♦s q✉❡ 2 2 ∗ kak hx, xi B − hax, axi B = hx, (kak 1 − a a)xi B ≥ 0.

  ❈♦♥t❛s ❛♥á❧♦❣❛s ♠♦str❛♠ q✉❡ ♦ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ ♦ B✲♣r♦❞✉t♦ ✐♥✲ 2 A hx, xi − A hxb, xbi ≥ 0 t❡r♥♦✱ ♦✉ s❡❥❛✱ kbk ✱ ♣❛r❛ q✉❛✐sq✉❡r x ∈ X✱ ❡ b ∈ B

  ✳ ▲♦❣♦✱ X s❛t✐s❢❛③ ✭✐✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ❡ ♣♦rt❛♥t♦✱ é ✉♠ A✲B ♣ré✲❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳

  ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ X s❡❥❛ ✉♠ A✲B ♣ré✲❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐✲ ❞❛❞❡ ❡ ✈❛♠♦s ♠♦str❛r q✉❡ X s❛t✐s❢❛③ ✭✐✐✐✮✬✳

  ❙❡❥❛♠ x, y, z, w ∈ X✳ ❊♥tã♦✱ h A hx, yiz, wi B = hxhy, zi B , wi B = hz, yi B hx, wi B = hz, yhx, wi B i B = hz, A hx, yi wi B . A hx, yi ▲♦❣♦✱ s❡ a ∈ A é ❞❛ ❢♦r♠❛ ✱ ♣❛r❛ ❛❧❣✉♥s x, y ∈ X✱ ✈❛❧❡ q✉❡ haz, wi B = hz, a wi B

  ✳ P♦r ❈❛✉❝❤②✲❙❝❤✇❛r③ ❡ ♣❡❧❛ ❝♦♥❞✐çã♦ ✭✐✐✐✮✱ t❡♠♦s q✉❡ khaz, wi B k ≤ kazk B kwk B ≤ kakkzk B kwk B . A hX, Xi

  ❆ss✐♠✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ♦❜t❡r ❡ q✉❡ B = hz, a wi B é ❞❡♥s♦ ❡♠ A✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ haz, wi s❡ ✈❡r✐✜❝❛ ♣❛r❛ t♦❞♦ a ∈ A✳ A hzb, wi = A hz, wb i

  ❙✐♠✐❧❛r♠❡♥t❡✱ ♣r♦✈❛✲s❡ q✉❡ ✱ ♣❛r❛ t♦❞♦ b ∈ B✳ P♦rt❛♥t♦✱ X s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭✐✐✐✮✬✳ ❙❡❥❛♠ X ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ✭à ❞✐r❡t❛✮ ❡ Y ✉♠ B✲♠ó❞✉❧♦ ❞❡

  ❍✐❧❜❡rt ✭à ❞✐r❡✐t❛✮✳ ❙❡❥❛ φ : A → L(Y ) ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦✳ ◗✉❡r❡✲ φ Y ♠♦s ❝♦♥str✉✐r ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ X ⊗ ♣♦ss✉✐♥❞♦ ✉♠❛ ❡str✉t✉r❛ ❞❡ ✉♠ B✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✳ alg Y

  ❙❡❥❛ X ⊗ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❛❧❣é❜r✐❝♦ ❞♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s X ❡ Y ✳ ❈♦♥s✐❞❡r❛♠♦s Y ❝♦♠♦ ✉♠ A✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❛tr❛✈és ❞❛ ❛çã♦ (a, y) 7→ φ(a)y

  ❡ ❛ss✐♠ ♣♦❞❡♠♦s ❝♦♥str✉✐r ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❛❧❣é❜r✐❝♦ A Y A Y ❞❡ X ❡ Y s♦❜r❡ A✱ ❞❡♥♦t❛❞♦ ♣♦r X ⊙ ✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ X ⊙ alg Y é ♦ q✉♦❝✐❡♥t❡ ❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ X ⊗ s♦❜r❡ C ♣❡❧♦ s✉❜❡s♣❛ç♦ N := span{xa ⊗ y − x ⊗ φ(a)y : x ∈ X, y ∈ Y, a ∈ A}.

  A Y

  ❈♦♠♦ ♣♦❞❡♠♦s ✈❡r ♥♦ ▲❡♠❛ ❛✱ ❞❛ ❙❡çã♦ ✾✳✺ ❞❡ X ⊙ ♣♦s✲ s✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ B✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❝♦♠ ❛ ❛çã♦ ❡♠ ✉♠ t❡♥s♦r ❡❧❡♠❡♥t❛r ❞❛❞❛ ♣♦r (x ⊗ y, b) 7→ x ⊗ yb. A Y

  ◆❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦✱ ✈❛♠♦s ✈❡r q✉❡ X⊙ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ ✉♠ B✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳ Pr♦♣♦s✐çã♦ ✶✳✹✳✶✵✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s✱ X ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❡ Y ✉♠ B✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✳ ❙❡❥❛ φ : A → L(Y ) ✉♠ ∗✲ A Y ❤♦♠♦♠♦r✜s♠♦✳ ❊♥tã♦ X ⊙ é ✉♠ B✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳ ▼❛✐s ❛✐♥❞❛✱ ❡♠ t❡♥s♦r❡s ❡❧❡♠❡♥t❛r❡s ♦ B✲♣r♦❞✉t♦ ✐♥t❡r♥♦ é ❞❛❞♦ ♣♦r hx 1 ⊗ y 1 , x 2 ⊗ y 2 i B = hy 1 , φ(hx 1 , x 2 i A )y 2 i B .

  ❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ❞❡✜♥✐r ✉♠ B✲♣ré✲♣r♦❞✉t♦ ✐♥t❡r♥♦ ♥♦ ♣r♦❞✉t♦ alg Y = t❡♥s♦r✐❛❧ ❛❧❣é❜r✐❝♦ X ⊗ ❡✱ ❢❡✐t♦ ✐st♦✱ ✈❛♠♦s ♠♦str❛r q✉❡ N {z ∈ X ⊗ alg Y : hz, zi B = 0}

  é ❡①❛t❛♠❡♥t❡ ♦ s✉♣❡s♣❛ç♦ ✈❡t♦r✐❛❧ N q✉❡ ❞❡✜♥✐♠♦s ❛❝✐♠❛✳ ❖✉ s❡❥❛✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ N = span{xa ⊗ y − x ⊗ φ(a)y : x ∈ X, y ∈ Y, a ∈ A}. ′ ′

  , y ) 7→ ❙❡❥❛♠ x ∈ X ❡ y ∈ Y ✜①❛❞♦s✳ ❆ ❛♣❧✐❝❛çã♦ X × Y → B✱ (x ′ ′ hy, φ(hx, x i A )y i B x,y : X ⊗ alg Y → B é ❜✐❧✐♥❡❛r ❡✱ ♣♦rt❛♥t♦✱ s❡ ❡st❡♥❞❡ ❛ ✉♠❛ ❛♣❧✐❝❛çã♦

  ❧✐♥❡❛r T t❛❧ q✉❡ ′ ′ ′ ′ ′ ′ ∗ T x,y (x ⊗ y ) = hy, φ(hx, x i A )y i B , ∈ X ∈ Y

  ♣❛r❛ q✉❛✐sq✉❡r x ❡ y ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛ T x,y ❛ tr❛♥s✲ alg Y x,y (z)) ∗ ∗ ❢♦r♠❛çã♦ ❞❡ X ⊗ ❡♠ B ❞❛❞❛ ♣♦r z 7→ (T ✳ ❊♥tã♦ T x,y é ❝♦♥❥✉❞❛❞♦✲❧✐♥❡❛r✳ ❆❣♦r❛✱ ❛ ❛♣❧✐❝❛çã♦ (x, y) 7→ T x,y é ✉♠❛ ❜✐❧✐♥❡❛r ❞❡ X ×Y alg Y

  ♥♦ ❡s♣❛ç♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❝♦♥❥✉❣❛❞♦✲❧✐♥❡❛r❡s ❞❡ X ⊗ ❡♠ B alg Y, B)

  ✳ ❙❡ CL(X ⊗ ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❝♦♥❥✉❣❛❞♦✲ alg Y ❧✐♥❡❛r❡s ❞❡ X ⊗ ❡♠ B✱ s❡❣✉❡ q✉❡ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r T : X ⊗ alg Y → CL(X ⊗ alg Y, B) ∗ ∗ t❛❧ q✉❡

  T (x ⊗ y) = T , x,y ♣❛r❛ q✉❛✐sq✉❡r x ∈ X ❡ y ∈ Y ✳

  P♦r ✜♠✱ ❝♦❧♦❝❛♠♦s ∗ ∗ alg Y ⊗ y , x ⊗ y ∈ X ⊗ alg Y hz, wi B := (T (z)(w)) , ♣❛r❛ z, w ∈ X ⊗ ✳ ◆♦t❡♠♦s q✉❡ ♣❛r❛ x 1 1 2 2 t❡♥s♦r❡s ❡❧❡♠❡♥t❛r❡s✱ t❡♠♦s ∗ ∗ ∗ ∗ hx ⊗ y , x ⊗ y i B = (T (x ⊗ y )(x ⊗ y )) = (T (x ⊗ y )) 1 1 2 2 1 1 2 ∗ ∗ 2 x ,y 1 1 2 2 = (T x ,y (x ⊗ y ) ) = T x ,y (x ⊗ y ) 1 1 2 2 1 1 2 2 = hy , φ(hx , x i A )y i B . 1 1 2 2

  ❆❧é♠ ❞✐ss♦✱ s❡ b ∈ B✱ hx ⊗ y , (x ⊗ y )bi B = hx ⊗ y , x ⊗ (y b)i B 1 1 2 2 1 1 2 2 = hy , φ(hx , x i A )y bi B 1 1 2 2

  = hy , φ(hx , x i A )y i B b 1 1 2 2 = hx ⊗ y , x ⊗ y i B 1 1 B 2 2 b.

  ❱❛♠♦s ✈❡r✐✜❝❛r ❛❣♦r❛ ❛ ♣♦s✐t✐✈✐❞❛❞❡ ❞❡ h·, ·i ✳ P n x i ⊗ y i ∈ X ⊗ alg Y

  ❙❡❥❛ z = ✳ ❊♥tã♦✱ i =1

  X hz, zi B = hy i , φ(hx i , y i i A )y j i B i,j (n) = hy, φ (M )yi, n

  , y , . . . , y n ) ∈ Y n (A) ❡♠ q✉❡ y = (y 1 2 ✱ M ∈ M é ❛ ♠❛tr✐③ ❝✉❥❛s ❡♥✲ ij = hx i , x j i A tr❛❞❛s sã♦ ❞❛❞❛s ♣♦r a ✱ ♣❛r❛ ❝❛❞❛ i, j ∈ {1, 2, . . . , n}✱ ❡ h·, ·i

  ❞❡♥♦t❛ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✉s✉❛❧ ❞❡✜♥✐❞♦ ♥♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❡ ♠ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt ✭✈❡❥❛ ❊①❡♠♣❧♦

  ❖ ▲❡♠❛ ✷✳✻✺ ❞❡ ♥♦s ❞✐③ q✉❡ ❛ ♠❛tr✐③ M é ♣♦s✐t✐✈❛✳ ❈♦♠♦ φ é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ❡♥tr❡ C ✲á❧❣❡❜r❛s✱ φ é ❝♦♠♣❧❡t❛♠❡♥t❡ ♣♦s✐t✐✈♦✱ (n) (n)

  (M ) ≥ 0 B = hy, φ (M )yi ≥ 0 ❞♦♥❞❡ φ ✳ ❈♦♠♦ r❡s✉❧t❛❞♦✱ hz, zi ✳

  ◆♦ss♦ ú❧t✐♠♦ ♣❛ss♦ é ♠♦str❛r q✉❡ N ❡ N ❝♦✐♥❝✐❞❡♠✱ ❡♠ q✉❡ N = span{xa ⊗ y − x ⊗ φ(a)y : x ∈ X, y ∈ Y, a ∈ A},

  ❡ N = {z ∈ X ⊗ alg Y : hz, zi B = 0}.

  ❙❡❥❛ z ∈ N ❞❛ ❢♦r♠❛ xa ⊗ y − x ⊗ φ(a)y✳ ❊♥tã♦✱ ✉s❛♥❞♦ q✉❡ φ é ✉♠ ∗

  ✲❤♦♠♦♠♦r✜s♠♦✱ s❡❣✉❡ hz, zi B = hy, φ(hxa, xai A )yi B + hφ(a)y, φ(hx, xi A )φ(a)yi B −hy, φ(hxa, xi A )φ(a)yi B − hφ(a)y, φ(hx, xai A )yi B

  = hy, φ(hxa, xai A )yi B + hφ(a)y, φ(hx, xai A )yi B −hy, φ(hxa, xai A )yi B − hφ(a)y, φ(hx, xai A )yi B = 0.

  ′

  ❆ss✐♠✱ ♦❜t❡♠♦s ❛ ✐♥❝❧✉sã♦ N ⊆ N ✳ P n x i ⊗ y i ∈ X ⊗ alg Y

  ◆♦✈❛♠❡♥t❡✱ s❡❥❛ z = ❡✱ ❝♦♠♦ ✜③❡♠♦s ❛♥t❡✲ i (n) (n) =1 (M )yi B (M ) r✐♦r♠❡♥t❡✱ ❡s❝r❡✈❡♠♦s hy, φ ♣❛r❛ hz, zi ✳ ❙❡❥❛ T = φ ✳ 1 2 n

  ∈ L(Y ) ❊♥tã♦✱ T ≥ 0 ❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r T s✉❛ ú♥✐❝❛ r❛✐③ q✉❛✲ 2 1

  (y) = 0 ❞r❛❞❛ ♣♦s✐t✐✈❛✳ ❚❡♠♦s q✉❡ T ✱ ♣♦✐s 2 1 2 1 hT (y), T (y)i = hy, T (y)i. 4 1

  (y) = 0 ❆♥❛❧♦❣❛♠❡♥t❡✱ t❡♠♦s q✉❡ T ✳ n n (A)

  ❈♦♥s✐❞❡r❛♠♦s ❛❣♦r❛ X ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ M ✲♠ó❞✉❧♦ ❞❡ ❍✐❧✲ p , x , . . . , x n ) hm, mi M

  ❜❡rt✳ ❊♥tã♦✱ s❡♥❞♦ m = (x 1 2 1 2 ❡ s❡ |m| := (A) ✱ n t❡♠♦s q✉❡ |m| = M ✳ ❱❛♠♦s ✉s❛r ❛❣♦r❛ ♦ ▲❡♠❛ ✹✳✹ ❞❡ ❙❡❥❛ X ✉♠ A✲♠ó❞✉❧♦ ❞❡

  ❍✐❧❜❡rt✱ x ∈ X ❡ 0 < α < 1✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ w ❡♠ X t❛❧ q✉❡ α x = w|x| ✳ 1 , w 2 , . . . , w n ) ∈ X n

  P♦r ❡st❡ r❡s✉❧t❛❞♦✱ ♦❜t❡♠♦s w = (w t❛❧ q✉❡ 1 4 4 1 = m ij wM 4 1 ✳ ❊s❝r❡✈❡♠♦s c ♣❛r❛ ❛ ❡♥tr❛❞❛ i, j ❞❛ ♠❛tr✐③ M ✳ ❯♠❛ (n) 4 1 1 4

  = φ (M ) ij )) i,j ✈❡③ q✉❡ T ✱ ❛ ♠❛tr✐③ ❞❡ T é (φ(c ✳ P♦rt❛♥t♦✱

  X X x j = w i c ij φ(c ij )y j = 0. i j ❡ P w i ⊗ φ(c ij )y j = 0

  ❉❛í✱ i,j ❡ ✈❡♠ q✉❡

  X X

  X j i,j i,j x j ⊗ y j = w i c ij ⊗ y j = (w i c ij ⊗ y j − w i ⊗ φ(c ij )y j ), ′ ′ q✉❡ é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ N ✳ ❉♦♥❞❡ N = N ✳ A Y ❉❡st❛ ❢♦r♠❛✱ X ⊙ é✱ ❞❡ ❢❛t♦✱ ✉♠ B✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳

  ❉❡✜♥✐çã♦ ✶✳✹✳✶✶✳ ❖ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞♦ B✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥✲ A Y φ Y t❡r♥♦ X ⊙ ✱ ❞❡♥♦t❛❞♦ ♣♦r X ⊗ ✱ é ❝❤❛♠❛❞♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ✐♥t❡r♥♦ ❞❡ X ❡ Y ✭r❡❧❛t✐✈♦ ❛ φ✮✳

  P❛r❛ ✜♥s ❞❛ ♣ró①✐♠❛ ❞❡♠♦♥str❛çã♦✱ t❡r❡♠♦s Y ✉♠ B✲C ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❡ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ✐♥t❡r♥♦ r❡❧❛✲ C ) t✐✈♦ à ✐♥❝❧✉sã♦ ❞❡ B ♥❛ C ✲á❧❣❡❜r❛ ❞♦s ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s L(Y ✳ B B (b)y = by

  ❖✉ s❡❥❛✱ ♥❡st❡ ❝❛s♦✱ t❡r❡♠♦s φ = ι ✱ ❡♠ q✉❡ ι ✱ ♣❛r❛ t♦❞♦ y ∈ Y ❡ b ∈ B✳ ◆❡st❡ ❝❛s♦✱ ✈❛♠♦s ❞❡♥♦t❛r ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ✐♥t❡r♥♦

  X ⊗ ι Y B Y B s✐♠♣❧❡s♠❡♥t❡ ♣♦r X ⊗ ✳

  ∗

  Pr♦♣♦s✐çã♦ ✶✳✹✳✶✷✳ ❙❡❥❛♠ A✱ B ❡ C C ✲á❧❣❡❜r❛s ❡ s✉♣♦♥❤❛ q✉❡ X s❡❥❛ ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❡ Y s❡❥❛ ✉♠ B −C✲❜✐♠ó❞✉❧♦ B Y ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❊♥tã♦ Z = X ⊙ é ✉♠ A✲C ❜✐♠ó❞✉❧♦ ❡ ❡①✐st❡♠ ú♥✐❝♦s ♣r♦❞✉t♦s ✐♥t❡r♥♦s ❛✈❛❧✐❛❞♦s ❡♠ A ❡ C✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s❛t✐s❢❛✲ ③❡♥❞♦ hx ⊗ y , x ⊗ y i C = hy , hx , x i B y i C 1 1 2 2 1 1 2 2

  ❡ A hx ⊗ y , x ⊗ y i = A hx hy , y i, x i, 1 1 2 2 1 B 1 2 2 , x ∈ X , y ∈ Y B Y

  ❡♠ q✉❡ x 1 2 ❡ y 1 2 ✳ ❖ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ✐♥t❡r♥♦ X ⊗ é ✉♠ A✲C ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❉❡♠♦♥str❛çã♦✿ P♦r ♠❡✐♦ ❞♦s ♠❡s♠♦s ❛r❣✉♠❡♥t♦s q✉❡ ✉t✐❧✐③❛♠♦s ♥❛ Pr♦♣♦s✐çã♦ ❝♦♥❝❧✉í♠♦s q✉❡ A ❛❣❡ s♦❜r❡ Z à ❡sq✉❡r❞❛ ❡ t❛❧ ❛çã♦ é ❞❛❞❛ ❡♠ ✉♠ t❡♥s♦r ❡❧❡♠❡♥t❛r ♣♦r (a, x ⊗ y) 7→ (ax) ⊗ y. B Y ▼❛✐s ❛✐♥❞❛✱ ❡①✐st❡ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ X ⊙ ❝♦♠ ✈❛❧♦r❡s ❡♠ A s❛t✐s❢❛③❡♥❞♦ 1 , x A hx ⊗ y , x ⊗ y i = A hx hy , y i, x i, 2 ∈ X 1 1 1 , y 2 2 ∈ Y 2 1 B 1 2 2 s❡♠♣r❡ q✉❡ x ❡ y ✳

  ❊①❛t❛♠❡♥t❡ ❝♦♠♦ ❝♦♥str✉í♠♦s ♥❛ Pr♦♣♦s✐çã♦ C ❛❣❡ à ❞✐r❡✐t❛ B Y ❡♠ X ⊙ ❡ t❛❧ ❛çã♦ é ❞❛❞❛ ❡♠ t❡♥s♦r❡s ❡❧❡♠❡♥t❛r❡s ♣♦r

  (x ⊗ y, c) 7→ x ⊗ (yc), B Y ❡♠ q✉❡ c ∈ C✳ ❆❧é♠ ❞✐ss♦✱ ❡①✐st❡ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ X ⊙ ❝♦♠ ✈❛❧♦r❡s ❡♠ C s❛t✐s❢❛③❡♥❞♦ hx ⊗ y , x ⊗ y i C = hy , hx , x i B y i C , 1 1 2 2 1 1 2 2

  , x ∈ X , y ∈ Y ♣❛r❛ q✉❛✐sq✉❡r x 1 2 ❡ y B Y 1 2 ✳

  ❱❛♠♦s ♠♦str❛r q✉❡ X ⊙ é ✉♠ A✲C ♣ré✲❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✲ ✈✐❞❛❞❡✳ A hX, Xi C P❡❧♦ ❈♦r♦❧ár✐♦ XB é ❞❡♥s♦ ❡♠ X ❡ BY é ❞❡♥s♦ ❡♠ Y ✳ ❈♦♠♦ A X ⊙ B Y B Y C ❡ hY, Y i sã♦ ❞❡♥s♦s ❡♠ A ❡ C✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s❡❣✉❡ q✉❡

  ❡ X ⊙ sã♦ ❝❤❡✐♦s✳ ❙❡❥❛ a ∈ A✳ ❊♥tã♦ ha(x ⊗ y ), x ⊗ y i C = h(ax ) ⊗ y , x ⊗ y i C 1 1 2 2 1 1 2 2

  = hy , hax , x i B y i C 1 1 2 2 = hy , hx , a x i B y i C 1 1 2 2

  = hx ⊗ y , (a x ) ⊗ y i C 1 1 2 2 = hx ⊗ y , a (x ⊗ y )i C . 1 1 2 2 ❙✐♠✐❧❛r♠❡♥t❡ ♣r♦✈❛✲s❡ q✉❡ A h(x ⊗ y )c, x ⊗ y i = A hx ⊗ y , (x ⊗ y )c i, 1 1 2 2 1 1 2 2

  ♣❛r❛ t♦❞♦ c ∈ C✳ ❆❧é♠ ❞✐ss♦✱ s❡ z ∈ X ❡ w ∈ Y ✱ A hx ⊗ y , x ⊗ y i(z ⊗ w) = ( A hx hy , y i, x iz) ⊗ w 1 1 2 2 1 B 1 2 2

  = (x hx hy , y i, zi B ) ⊗ w 1 2 B 2 1 = x ⊗ (hx hy , y i, zi B w) 1 2 B 2 1

  = x ⊗ ( B hy , y ihx , zi B w) 1 1 2 2 = x ⊗ ( B hy , hz, x i B y iw) 1 1 2 2

  = x ⊗ (y hhz, x i B y , wi C ) 1 1 2 2 = x 1 ⊗ (y 1 hy 2 , hx 2 , zi B wi C ) = (x 1 ⊗ y 1 )hx 2 ⊗ y B Y 2 , z ⊗ wi C .

  ▲♦❣♦✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ❝♦♥❝❧✉í♠♦s q✉❡ X ⊙ é ✉♠ A✲C ♣ré✲ B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❆ss✐♠✱ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ✐♥t❡r♥♦ X ⊗ Y

  é ✉♠ A✲C ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❆❣♦r❛✱ t❡♠♦s t♦❞❛s ❛s ❢❡rr❛♠❡♥t❛s ♣❛r❛ ♠♦str❛r q✉❡ ▼♦r✐t❛ ❡q✉✐✲

  ✈❛❧ê♥❝✐❛ é ❞❡ ❢❛t♦ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❈♦♠♦ s❡❣✉❡✿ Pr♦♣♦s✐çã♦ ✶✳✹✳✶✸✳ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥✲ ❝✐❛ ❡♥tr❡ C ✲á❧❣❡❜r❛s✳ ❉❡♠♦♥str❛çã♦✿ ❆❝❛❜❛♠♦s ❞❡ ♣r♦✈❛r ❛ tr❛♥s✐t✐✈✐❞❛❞❡ ♥❛ Pr♦♣♦s✐çã♦ ❡ q✉❡ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ é ✉♠❛ r❡❧❛çã♦ s✐♠étr✐❝❛✱ s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ n (C) m (C) ❊①❡♠♣❧♦ ✶✳✹✳✶✹✳ ❙❡❥❛♠ m, n ∈ N✳ ❊♥tã♦ M ❡ M sã♦ ▼♦✲ r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✳ n (C) ) m (C) ) n m

  ❉❡♠♦♥str❛çã♦✿ ■❞❡♥t✐✜❝❛♥❞♦ M ❝♦♠ B(C ❡ M ❝♦♠ B(C ✱ n (C) ∼ M m (C) ∼ M C C s❛❜❡♠♦s ❞♦ ❊①❡♠♣❧♦ q✉❡ M ❜❡♠ ❝♦♠♦ M ✳

  ❯♠❛ ✈❡③ q✉❡ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ s❡❣✉❡ n (C) m (C) q✉❡ M ❡ M sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✳ n (C) ❯♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ♠♦str❛r ❛ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ M m (C) n +m (C) e ii P n

  ❡ M ✱ s❡r✐❛ ❝♦♥s✐❞❡r❛r♠♦s A = M ✱ p = i ❡ q = =1 P n +m i ✱ ❡♠ q✉❡ ❞❡♥♦t❛ ❛ ♠❛tr✐③ ❝✉❥❛ ❡♥tr❛❞❛ a é ✐❣✉❛❧ ❛ 1✱ ❡ t♦❞❛s =n+1 e ii ii n (C) ∼ = pAp ❛s ♦✉tr❛s sã♦ ♥✉❧❛s✳ ❊♥tã♦ p ❡ q sã♦ ♣r♦❥❡çã♦ ❝❤❡✐❛s ❡ M ❡ M m (C) ∼ = qAq n (C) ∼ M M m (C)

  ✱ ❡ ♣❡❧♦ ❊①❡♠♣❧♦ ♦❜t❡♠♦s M ✳ ❖✉ ❛✐♥❞❛✱ ∗ ∗

  ▲❡♠❜r❡♠♦s q✉❡ ✉♠❛ C ✲s✉❜á❧❣❡❜r❛ B ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ A é ❞✐t❛ s❡r ✉♠ ❝❛♥t♦ ❝❤❡✐♦ s❡ ❡①✐st❡ ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛ ❡♠ M(A) ✭ApA = A✮ t❛❧ q✉❡ B = pAp✳ ❉♦✐s ❝❛♥t♦s ❝❤❡✐♦s pAp ❡ qAq sã♦ ❞✐t♦s ❝♦♠♣❧❡♠❡♥✲ t❛r❡s s❡ p + q = 1✳ ❱✐♠♦s ♥♦ ❊①❡♠♣❧♦ q✉❡ ❝❛♥t♦s ❝❤❡✐♦s ❞❡ ✉♠❛

  C ✲á❧❣❡❜r❛ sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✳ ◆♦ ♣ró①✐♠♦ t❡♦r❡♠❛✱ ✈❛♠♦s ✈❡r

  ❞✉❛s C ✲á❧❣❡❜r❛s ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s A ❡ B ❝♦♠♦ ❝❛♥t♦s ❝♦♠♣❧❡♠❡♥✲ t❛r❡s ❝❤❡✐♦s ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛✱ ❝❤❛♠❛❞❛ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦✳ ❚❡♦r❡♠❛ ✶✳✹✳✶✺✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s✳ ❊♥tã♦✱ A ❡ B sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ C ✲á❧❣❡❜r❛ C ❝♦♠ ❝❛♥t♦s ❝♦♠♣❧❡♠❡♥t❛r❡s ❝❤❡✐♦s ✐s♦♠♦r❢♦s ❛ A ❡ B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ X ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❡ s❡❥❛ e X s❡✉ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♥❥✉❣❛❞♦ ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ B✲A ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✱ ❞❡✜♥✐❞❛ ♥❛ Pr♦♣♦s✐çã♦ ❙❡❥❛ M = X ⊕ B✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❊①❡♠♣❧♦ M é ✉♠ B✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✳

  P❛r❛ a ∈ A✱ b ∈ B✱ x, y ∈ X✱ ❝♦❧♦❝❛♠♦s a x L =

  ✭‡✮ y ˜ b ♣❛r❛ ❞❡♥♦t❛r ❛ ❛♣❧✐❝❛çã♦ ❞❡ M → M ❞❛❞❛ ♣♦r a x z az + xc

  = , y ˜ b c hy, zi B + bc ❡♠ q✉❡ z ∈ X ❡ c ∈ B✳

  ❙❡ C ❞❡♥♦t❛ ❛ ❝♦❧❡çã♦ ❞❡ t♦❞♦s ❛s ❛♣❧✐❝❛çõ❡s ❞❡ss❛ ❢♦r♠❛✱ q✉❡r❡♠♦s ♠♦str❛r q✉❡ C é ✉♠❛ C ✲s✉❜á❧❣❡❜r❛ ❞❡ L(M)✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ ♦ ♦♣❡r❛❞♦r L ❡♠ é ❛❞❥✉♥tá✈❡❧ ❡ a y L = . x ˜ b

  ❙❡❥❛♠ z, z ∈ X

  = aa z + ax c + xhy , zi B + xb c hy, a z + x ci B + bhy , zi B + bb c = (△).

  ∈ B ❡ x, x

  ∈ A ✱ b, b

  , ♣❛r❛ q✉❛✐sq✉❡r a, a

  ˜ y b = aa + A hx, y i ax + xb ˜ ya + b˜ y hy, x i B + bb

  ❖✉ s❡❥❛✱ C é ✉♠❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡ L(M) ❡ a x ˜ y b a x

  = aa + A hx, y i ax + xb ˜ ya + b˜ y hy, x i B + bb z c .

  (aa + A hx, y i)z + (ax + xb )c (ha ′∗ y, zi B + hy b , zi B ) + (hy, x i B + bb )c

  ❘❡❛rr❛♥❥❛♥❞♦ ♦s t❡r♠♦s✱ (△) =

  = a x ˜ y b a z + x c hy , zi B + b c

  ❡ c, c ∈ B

  ˜ y b a x ˜ y b z c

  ❱❛♠♦s ✈❡r✐✜❝❛r ❛❣♦r❛ q✉❡ C é ✉♠❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡ L(M)✳ ❈♦♠ ❡❢❡✐t♦✱ a x

  ▲♦❣♦✱ C é✱ ❞❡ ❢❛t♦✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ L(M) ❢❡❝❤❛❞♦ ❡♠ r❡❧❛çã♦ à ♦♣❡r❛çã♦ ❞❡ ✐♥✈♦❧✉çã♦✳ ❆❧é♠ ❞✐ss♦✱ é ❢á❝✐❧ ✈❡r q✉❡ C é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ L(M)✱ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ✉s✉❛✐s ❞❡ ♠❛tr✐③❡s✳

  , a y ˜ x b z c B .

  , a z + yc hx, z i B + b c B = z c

  , z c B = haz + xc, z i B + (hy, zi B + bc) c = hz, a z + yc i B + c hx, z i B + c b c = z c

  , z c B = az + xc hy, zi B + bc

  ✳ ❊♥tã♦✱ a x ˜ y b z c

  , y, y ∈ X ✳

  ❘❡st❛ ♠♦str❛r♠♦s q✉❡ C é ❢❡❝❤❛❞♦ ❡♠ L(M)✱ ❡ ❝♦♠♦ r❡s✉❧t❛❞♦ t❡r❡♠♦s q✉❡ C é ✉♠❛ C ✲á❧❣❡❜r❛✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ s❡ L ∈ L(M) é ❞❛❞♦ ♣♦r ❡♥tã♦ max{kak, kxk B , kyk B , kbk} ≤ kLk ≤ kak + kxk B + kyk B + kbk.

  ✭✶✳✶✮ ❊s❝r❡✈❡♥❞♦ a x 0 0

  • L =

  , y ˜ b ❡ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r♠♦s q✉❡ ❝❛❞❛ ♣❛r❝❡❧❛ ♣♦ss✉✐ ♥♦r♠❛ ♠❡♥♦r ♦✉ ✐❣✉❛❧ q✉❡ ❛ ♥♦r♠❛ ❞❡ s✉❛ ❡♥tr❛❞❛ ♥ã♦ ♥✉❧❛✳ ❉❡ ❢♦r♠❛ ♠❛✐s ♣r❡❝✐s❛✱ t❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡✳ a 0 0

  = kak, = kbk, b ❡ 0 x B , B . = kxk = kyk

  ˜ y B ) ❉❡ ❢❛t♦✱ ❝♦♠♦ ❛ ✐♥❝❧✉sã♦ ❞❡ A ❡♠ L(X é ✐♥❥❡t✐✈❛✱ s❡❣✉❡ q✉❡ kak = sup kaxk B kxk B ≤1 ✱ ❞♦♥❞❡ a

  = kak. B ≤ ❯s❛♥❞♦ ♦ ❢❛t♦ q✉❡ B ♣♦ss✉✐ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ❡ q✉❡ kxck kxk B kck

  ✱ s❡❣✉❡ q✉❡ 0 x B . = kxk B k ≤ kyk B kzk B

  P♦r ❈❛✉❝❤②✲❙❝❤✇❛r③✱ t❡♠♦s q✉❡ khy, zi ✳ ❈♦❧♦❝❛♥❞♦ y z = B = 1

  ✱ t❡♠♦s q✉❡ kzk ❡ kyk B 2 2 khy, zi hy, zi B k = khy, zi B k = khy, yi B k = kyk . B B ❉❛í✱ ♦❜t❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡ B .

  = kyk y ˜

  ❈❧❛r❛♠❡♥t❡ t❡♠♦s 0 0 = kbk. b

  P❛r❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❡ ♦❜s❡r✈❛♠♦s q✉❡ 4 2 y kyk = khy, yi B k = , L B hy, yi B 3 B ≤ khy, yi B kkLkkyk B = kLkkyk . B ❙✐♠✐❧❛r♠❡♥t❡✱ 4 2 x kxk = khx, xi B k = , L B 3 hx, xi B B ≤ kxk B kLkkhx, xi B k = kLkkxk . B B B

  ❆ss✐♠✱ kLk ❞♦♠✐♥❛ kxk ❡ kyk ✳ ◆♦✈❛♠❡♥t❡ ✉s❛♥❞♦ q✉❡ ❛ ✐♥❝❧✉sã♦ B )

  ❞❡ A ❡♠ L(X é ✐♥❥❡t✐✈❛ ❡✱ ♣♦rt❛♥t♦✱ ✐s♦♠étr✐❝❛✱ ♣♦❞❡♠♦s ♦❜t❡r x ′ ′ X k B = 1 k B t❛❧ q✉❡ kx ❡ kax é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ kak✳ ❉❡st❛ ❢♦r♠❛✱ 2 ′ 2 ′ ′ kak ∼ kax k = khax , ax i B k B ′ ′ ax x

  = , L ′ ′ B ≤ kax k B kLkkx k B 2

  ≤ kakkLkkx k B = kakkLk,

  ❡ ♣♦rt❛♥t♦ kak ≤ kLk✳ λ ) λ P♦r ✜♠✱ s❡♥❞♦ (u ∈Λ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ B✱ 2 ∗ kbk = lim kb bu λ k = lim , L λ λ ≤ kbkkLk. b u λ B

  ▲♦❣♦✱ ✜❝❛ ♣r♦✈❛❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ C é ✉♠❛ C ✲s✉❜á❧❣❡❜r❛ ❞❡ L(M)✳

  ❘❡st❛ ♣r♦✈❛r♠♦s q✉❡ A ❡ B sã♦ ❝❛♥t♦s ❝♦♠♣❧❡♠❡♥t❛r❡s ❝❤❡✐♦s ❞❡ C

  ✳ P❡❧♦ q✉❡ ✜③❡♠♦s ❛té ❛q✉✐✱ ✜❝❛ ❝❧❛r♦ q✉❡ ❛s ❛♣❧✐❝❛çõ❡s a 0 0 a 7→ ,

  ❡ b 7→ b sã♦ ∗✲❤♦♠♦♠♦r✜s♠♦s ✐♥❥❡t✐✈♦s ❞❡ A ❡ B ❡♠ C✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛ p ♦ ♦♣❡r❛❞♦r ❛❞❥✉♥tá✈❡❧ s♦❜r❡ M ❞❛❞♦ ♣♦r z z p = . 2 c

  = p ➱ ❢á❝✐❧ ✈❡r q✉❡ p = p ✳ ❊s❝r❡✈❡♠♦s 1 0

  0 0 ♣❛r❛ p✱ ❡♠ q✉❡ ✏1✑ é ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ s♦❜r❡ X✳

  ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡❥❛ q ♦ ♦♣❡r❛❞♦r L(M) ❞❡✜♥✐❞♦ ♣♦r z q = c c

  ❡ ❡s❝r❡✈❡♠♦s 0 0 0 1

  ♣❛r❛ q✱ ❡♠ q✉❡ ✏1✑ é ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ s♦❜r❡ B✳ ◆♦✈❛♠❡♥t❡ t❡♠♦s 2 q = q = q ✳

  ❆❧é♠ ❞✐ss♦✱ ♣r♦✈❛✲s❡ q✉❡ a x a x a x a p = , p =

  ˜ y b y ˜ b y ˜ ❜❡♠ ❝♦♠♦ a x a x x q = , q = .

  ˜ y b y ˜ b y ˜ b b ▲♦❣♦✱ ❛s ♣r♦❥❡çõ❡s p ❡ q sã♦ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ C ❡ t❡♠♦s pCp ∼ = A

  ❡ qCq ∼ = B ✳ ▼❛✐s ❛✐♥❞❛✱ p + q = 1✳ ❆❣♦r❛✱ ♣❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦✱ só ♣r❡❝✐s❛♠♦s ♠♦str❛r q✉❡ p

  ❡ q sã♦ ♣r♦❥❡çõ❡s ❝❤❡✐❛s✳ ▼❛s✱ ′ ′ aa ax ′ ′ CpC = : a, a ∈ A, y, x ∈ X ′ ′

  ˜ ya hy, x i B ❡ ❛ss✐♠✱ ❝♦♠♦ X é ✉♠ B✲♠ó❞✉❧♦ ❝❤❡✐♦✱ AX é ❞❡♥s♦ ❡♠ X ❝♦♠ ❛ ♥♦r♠❛ k · k A A B

  ❡ ❛s ♥♦r♠❛s k · k ❡ k · k ❝♦✐♥❝✐❞❡♠ ✭❈♦r♦❧ár✐♦ ✉s❛♠♦s ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣❛r❛ ♦❜t❡r q✉❡ p é ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛✳

  ❈♦♠ ❛r❣✉♠❡♥t♦s s❡♠❡❧❤❛♥t❡s✱ ♣r♦✈❛✲s❡ q✉❡ A hx, y i xb ′ ′ ′ ′ CqC = : b, b ∈ B, x, y ∈ X ′ ′ b˜ y bb

  é ❞❡♥s♦ ❡♠ C✱ ❞♦♥❞❡ q é ♣r♦❥❡çã♦ ❝❤❡✐❛✳ P♦rt❛♥t♦✱ A ❡ B sã♦ ❝❛♥t♦s ❝♦♠♣❧❡♠❡♥t❛r❡s ❝❤❡✐♦s ❞❡ C✳ ❆ r❡❝í♣r♦❝❛ s❡❣✉❡ ❞♦ ❊①❡♠♣❧♦

  ❉❡✜♥✐çã♦ ✶✳✹✳✶✻✳ ❙❡❥❛ X ✉♠ A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❆ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦ ❞❡ X é ❛ C ✲á❧❣❡❜r❛ ❞❡ ♠❛tr✐③❡s ❝♦♥str✉í❞❛ ♥♦ ❚❡✲ ♦r❡♠❛

  ❆ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦ é tã♦ ✐♠♣♦rt❛♥t❡ q✉❛♥t♦ ♦ r❡s✉❧t❛❞♦ ❛♣r❡s❡♥✲ t❛❞♦ ♥♦ ❚❡♦r❡♠❛ P♦r ❡①❡♠♣❧♦✱ t❛❧ C ✲á❧❣❡❜r❛ t❡♠ ✉♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ t❡♦r❡♠❛ ❞❡ ❇r♦✇♥✲●r❡❡♥✲❘✐❡✛❡❧✱ q✉❡ ❛q✉✐ s❡ ❡♥❝♦♥tr❛ ♥❛ ❙❡çã♦

  ❈❛♣ít✉❧♦ ✷ ❋✐❜r❛❞♦s ❞❡ ❋❡❧❧

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐♠♦s ✜❜r❛❞♦s ❞❡ ❋❡❧❧ s♦❜r❡ ❣r✉♣♦s ❞✐s❝r❡t♦s ❡ ❝♦♥str✉í♠♦s s✉❛s C ✲á❧❣❡❜r❛s s❡❝❝✐♦♥❛✐s ❝❤❡✐❛ ❡ r❡❞✉③✐❞❛✳ ❆ ♣r✐♠❡✐r❛✱ é ♦❜t✐❞❛ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s ❛❜str❛t❛✱ ❛ ♣❛rt✐r ❞❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ ✉♠❛ ❝❡rt❛ ∗✲á❧❣❡❜r❛ ❡ ❛ s❡❣✉♥❞❛✱ é ❞❡✜♥✐❞❛ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❝♦♥✲ ❝r❡t❛✱ ❝♦♠♦ ❛ C ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧❛ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ✐♥❥❡t✐✈❛ ❞❛ ∗✲á❧❣❡❜r❛ r❡❧❛❝✐♦♥❛❞❛ ❛ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ✉s❛❞❛s ❢♦r❛♠

  ❆♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ ❝❛♣ít✉❧♦✱ G é ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❝♦♠ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ e✳

  ∗

  

✷✳✶ ❆ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❞❡ ✉♠ ✜✲

❜r❛❞♦ ❞❡ ❋❡❧❧

  ◆❡st❛ s❡çã♦✱ ❞❡✜♥✐♠♦s ✜❜r❛❞♦s ❞❡ ❋❡❧❧ ❡ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❡①❡♠✲ ♣❧♦s✳ ❖❜t❡♠♦s ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛

  ✲á❧❣❡❜r❛✱ ❡ ❛ ♣❛rt✐r ❞✐ss♦ ❞❡✜♥✐♠♦s ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛✳ ❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❯♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ s♦❜r❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ G é g } g ✉♠❛ ❝♦❧❡çã♦ ❞❡ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ B = {B ∈G ♠✉♥✐❞❛ ❝♦♠ ✉♠❛ s × B t → B st ❢❛♠í❧✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s ❜✐❧✐♥❡❛r❡s · : B ❡ ✉♠❛ ❢❛♠í❧✐❛ t → B −1 , ❞❡ ❛♣❧✐❝❛çõ❡s ❝♦♥❥✉❣❛❞♦ ❧✐♥❡❛r❡s ∗ : B t ❝❤❛♠❛❞❛s ❞❡ ♠✉❧✲ t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s❛t✐s❢❛③❡♥❞♦✱ ♣❛r❛ q✉❛✐sq✉❡r b s ∈ B s , b t ∈ B t , b r ∈ B r

  ✱ ❡ s, t, r ∈ G✿ s b t )b r = b s (b t b r ) ✭✐✮ ❆s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ sã♦ ❛ss♦❝✐❛t✐✈❛s✿ (b ❀ t → B −1

  ✭✐✐✮ ∗ : B t é ✐♥✈♦❧✉t✐✈❛ ❡ ✐s♦♠étr✐❝❛❀ s b t ) = b b ∗ ∗ ∗ ✭✐✐✐✮ (b t s ❀ s b t k ≤ kb s kkb t k ✭✐✈✮ kb ❀ t b t k = kb t k 2

  ✭✈✮ kb ❀ t ∈ B t e b t = a a ∗ ∗ ✭✈✐✮ P❛r❛ t♦❞♦ b ✱ ❡①✐st❡ a ∈ B t❛❧ q✉❡ b t ✳ t

  P❛r❛ ❝❛❞❛ t ∈ G✱ ♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ B é ❞❡♥♦♠✐❞❛❞♦ ❛ ✜❜r❛ ❞❛ ❡♥tr❛❞❛ t ♦✉ ✜❜r❛ ❝♦♠ ❡♥tr❛❞❛ t ❞♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B✳ e ❖❜s❡r✈❛çã♦ ✷✳✶✳✷✳ ❆s ❝♦♥❞✐çõ❡s ✭✐✮✲✭✐✈✮ ♥♦s ❞✐③❡♠ q✉❡ B é ✉♠❛ ∗✲ e á❧❣❡❜r❛ ❞❡ ❇❛♥❛❝❤✳ ❆❞✐❝✐♦♥❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✈✮ t❡♠♦s q✉❡ B é✱ ❞❡ ❢❛t♦✱ ✉♠❛ C ✲á❧❣❡❜r❛✳ t B s

  ❆♥t❡s ❞❛ ♣ró①✐♠❛ ♦❜s❡r✈❛çã♦✱ ♣❛r❛ ✜①❛r ♥♦t❛çõ❡s✱ ❝♦♠ B q✉❡r❡✲ t b s : b t ∈ B t , b s ∈ B s } ♠♦s ❞✐③❡r ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❢❡❝❤❛❞♦ ❣❡r❛❞♦ ♣♦r {b ✳ −1 B g e ❖❜s❡r✈❛çã♦ ✷✳✶✳✸✳ P❛r❛ ❝❛❞❛ g ∈ G✱ B g é ✉♠ ✐❞❡❛❧ ❡♠ B ✳ −1 B g −1 a g ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ a ∈ B g s❡❥❛ ❞❛ ❢♦r♠❛ a g ✱ ❝♦♠ a −1 ∈ B −1 g ∈ B g e g g ❡ a ✳ ❙❡❥❛ b ∈ B ✳ ❊♥tã♦✱ ba = (ba g −1 )a g ∈ B e B g −1 B g ⊆ B g −1 B g

  ❡ ab = a g −1 (a g b) ∈ B g −1 B g B e ⊆ B g −1 B g . g a g g B g −1 −1

  ❯♠❛ ✈❡③ q✉❡ ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛ a ❣❡r❛♠ B ✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳ ❊①❡♠♣❧♦ ✷✳✶✳✹✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦✳ ❈♦❧♦❝❛♠♦s✱ ♣❛r❛ ❝❛❞❛ t ∈ G t = C × {t}

  ✱ B ✱ ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ♦❜t✐❞❛ ❛tr❛✈és t ❞❛ ❜✐❥❡çã♦ ❝❛♥ô♥✐❝❛ ❡♥tr❡ C × {t} ❡ C✳ ❊s❝r❡✈❡♠♦s λδ ♣❛r❛ (λ, t) ❡ t = Cδ t t } t ∈G ❛ss✐♠ B ✳ ❙❡❥❛ B = {B ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦ ❞❛❞❛s ♣♦r

  λδ s αδ t = λαδ st , t ) = ¯ λδ −1 , ❡ (λδ t t } t

  ♣❛r❛ s, t ∈ G ❡ λ, α ∈ C✳ ❊♥tã♦✱ B = {B ∈G é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✱ ❝❤❛♠❛❞♦ ✜❜r❛❞♦ tr✐✈✐❛❧✳

  (C) ❊①❡♠♣❧♦ ✷✳✶✳✺✳ ❙❡❥❛ A = M 3 ✳ ❙❡❥❛♠ B −1 ✱ B ❡ B 1 ♦s s✉❜❡s♣❛ç♦s ❞❡ A ❣❡r❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❡❧❛s ♠❛tr✐③❡s ❞❛ ❢♦r♠❛

        0 0 0 ∗ 0 0 ∗ ∗    ∗ 0 0  , ∗ ∗ 0 0 0  .

   ❡ ∗ 0 0 ∗ ∗ 0 0 0 n = {0} n } n

  P❛r❛ n ∈ Z\{−1, 0, 1} ❝♦❧♦❝❛♠♦s B ✳ ❊♥tã♦✱ B = {B ∈Z ♠✉✲ ♥✐❞♦ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦ ✉s✉❛✐s ❞❡ ♠❛tr✐③❡s é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ∗ ∗

  = B ❉❡♠♦♥str❛çã♦✿ ◆♦t❡♠♦s q✉❡ B é ✉♠❛ C ✲á❧❣❡❜r❛✱ B −1 ❜❡♠ 1

  = B ❝♦♠♦ B −1 ✳ ❆❧é♠ ❞✐ss♦✱ 1

      0 0 0 ∗ 0 0  

  B B =, 0 ∗ ∗  , B B =, 0 0 0  , −1 1 1 −1 0 ∗ ∗ 0 0 0 −1 B ⊆ B B ⊆ B B = 1 1 −1 1 ❞♦♥❞❡ B ❡ B ✳ ❖❜s❡r✈❛♠♦s t❛♠❜é♠ q✉❡ B B = B B B = B = B B B = {0} = B B = 1 1 ✱ B −1 −1 −1 ✱ B 1 1 2 ❡ B −1 −1

  {0} = B −2 ✳ ❙❡

    0 0   a = a 21 0 0 a 31 0 0

  é ✉♠❛ ♠❛tr✐③ ❡♠ B −1 ✱ ❡♥tã♦   2 2 |a 21 | + |a 31 | 0 0

   a a = 0 0  , 0 0 a ≥ 0

  é ✉♠ ❡❧❡♠❡♥t♦ ♣♦s✐t✐✈♦ ❡♠ B ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ a ❡♠ B ✱ ♣❛r❛ t♦❞❛ ♠❛tr✐③ a ∈ B 1 ✳ ❖s ♦✉tr♦s ❛①✐♦♠❛s ❞❛ ❉❡✜♥✐çã♦ s❡❣✉❡♠ ❞✐r❡t❛♠❡♥t❡ ❞♦ ❢❛t♦ n (C) n } n ∈Z q✉❡ M é ✉♠❛ C ✲á❧❣❡❜r❛✳ ❉♦♥❞❡ B = {B é ✉♠ ✜❜r❛❞♦ ❞❡

  ❋❡❧❧✳ ❊①❡♠♣❧♦ ✷✳✶✳✻✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s ❡ X ✉♠ A − B✲❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳ ❙❡❥❛ C ❛ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦ ❞❡ X✳ ❙❡❥❛♠ B −1 ✱ B

  ❡ B 1 ♦s s✉❜❡s♣❛ç♦s ❞❡ C ❞❡✜♥✐❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r A

  X , . ❡ e

  B

  X n = {0} P❛r❛ ❝❛❞❛ n ∈ Z \ {−1, 0, 1}✱ ❝♦❧♦❝❛♠♦s B ✳ ❊♥tã♦✱ ❝♦♠ ❛s n } n ∈Z ♦♣❡r❛çõ❡s ❤❡r❞❛❞❛s ❞❡ C✱ B = {B é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ♠♦str❛r s♦♠❡♥t❡ ❛ ❝♦♥❞✐çã♦ ✭✈✐✮ ❞❛ ❉❡✜♥✐çã♦

  ❙❡❥❛ x ∈ X✳ ❊♥tã♦✱ x = x ˜ hx, xi B 1 ! ! 1 = . 2 2 0 hx, xi hx, xi B B

  ❙✐♠✐❧❛r♠❡♥t❡✱ x A hx, xi = x ˜ A hx, xi A hx, xi 1 2 1 2 n } n ∈Z = .

  ▲♦❣♦✱ B = {B é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ 1 =

  ❊①❡♠♣❧♦ ✷✳✶✳✼✳ ❙❡❥❛ D ♦ ❞✐s❝♦ ✉♥✐tár✐♦ {z ∈ C : |z| ≤ 1} ❡ S {z ∈ C : |z| = 1}

  ♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦✳ ❙❡❥❛♠ A = C(D) ❛ C ✲á❧❣❡❜r❛ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❝♦♠ ✈❛❧♦r❡s ❝♦♠♣❧❡①♦s s♦❜r❡ D ❡ G ♦ ❣r✉♣♦ ♠✉❧✲ t✐♣❧✐❝❛t✐✈♦ ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s {−1, 1}✳ ❉❡✜♥✐♠♦s ♦s s❡❣✉✐♥t❡s ❡s♣❛ç♦s 1 −1

  ❢❡❝❤❛❞♦s B ❡ B ❞❡ A✿ 1 B = {f ∈ A : f (−z) = f (z), }, 1 ♣❛r❛ t♦❞♦ z ∈ S 1 B = {f ∈ A : f (−z) = −f (z), }. −1 ♣❛r❛ t♦❞♦ z ∈ S t } t ❊♥tã♦✱ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❤❡r❞❛❞❛s ❞❡ A✱ B = {B ∈G é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ ❛ ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ A = C(D) t B s ⊆ B ts é ❞❡✜♥✐❞❛ ♣♦♥t✉❛❧♠❡♥t❡✱ ✜❝❛ ❢á❝✐❧ ✈❡r q✉❡ B ✱ ♣❛r❛ t, s ∈ ∗ ∗ {−1, 1} = B = B

  ✳ ❚❛♠❜é♠ B 1 −1 1 ❡ B −1 ✳ ❱❛♠♦s ✈❡r✐✜❝❛r ❛ ❝♦♥❞✐çã♦ ✭✈✐✮ ❞❛ ❉❡✜♥✐çã♦ ❙❡❥❛ f ∈ B −1 ✳

  ❖❜s❡r✈❛♠♦s q✉❡ s❡ g ∈ A é ❞❛❞❛ ♣♦r D → C✱ z 7→ |f(z)|✱ ♣❛r❛ 1 z ∈ S t❡♠♦s g(−z) = |f (−z)| = | − f (z)| = |f (z)|. ∗ ∗ f = g g

  ❉♦♥❞❡ g ∈ B 1 ❡ f ✳ t } t ∈G P♦rt❛♥t♦✱ B = {B é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ t } t λ ) λ

  Pr♦♣♦s✐çã♦ ✷✳✶✳✽✳ ❙❡❥❛ B = {B ∈G ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❙❡ (u ∈Λ e é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ B ✱ ❡♥tã♦ lim b t u λ = lim u λ b t = b t , λ λ t ∈ B t

  ♣❛r❛ q✉❛✐sq✉❡r t ∈ G ❡ b ✳ ∗ ∗ b t ∈ B e b t = ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ✉♠❛ ✈❡③ q✉❡ b t ✱ ✈❛❧❡ q✉❡ b t ∗ ∗ lim λ b t b t u λ = lim λ u λ b t b t

  ✳ ❆ss✐♠✱ s❡❣✉❡ ❞♦ ❛①✐♦♠❛ ✭✈✮ ❞❛ ❉❡✜♥✐çã♦ q✉❡ kb t − b t u λ k = k(b t − b t u λ ) (b t − b t u λ )k ∗ ∗ ∗ ∗ λ b t u λ = b t t = lim λ u λ b t = kb b t − b b t u λ − u λ b b t + u λ b b t u λ k → 0. t t t t

  ▲♦❣♦✱ lim ✳ P❛r❛ ♣r♦✈❛r q✉❡ b ✱ ❜❛st❛ ❛♣❧✐❝❛r ♦ q✉❡ ❥á ❢♦✐ ❢❡✐t♦ ♣❛r❛ b t ❡ ✉s❛r q✉❡ ❛ ♦♣❡r❛çã♦ ❞❡ ✐♥✈♦❧✉çã♦ é ✐s♦♠étr✐❝❛✳ ❖❜s❡r✈❛çã♦ ✷✳✶✳✾✳ P♦❞❡♠♦s ♣❡r❝❡❜❡r ♥❛ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ q✉❡ b t = lim b t v λ λ

  ❡ λ ) λ ∈Λ ) λ ∈Λ t B t b t = lim v b t , λ λ −1 s❡♥❞♦ (v ❡ (v λ ✉♥✐❞❛❞❡s ❛♣r♦①✐♠❛❞❛s ♣❛r❛ ♦s ✐❞❡❛✐s B t B t −1 ❡ B ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  ◆♦ss♦ ♦❜❥❡t✐✈♦ ❛❣♦r❛ é✱ ♣❛rt✐♥❞♦ ❞❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✱ ❝♦♥str✉✐r ✉♠❛ C ✲á❧❣❡❜r❛✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ C ❞❛❞♦ ♣♦r

  ( )

  [ C c (B) = ξ : G → B t : ξ(t) ∈ B t , ∀t ∈ G t ∈G ❡ supp(ξ) é ✜♥✐t♦

  ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ❞❡✜♥✐❞❛s ♣♦♥t✉✲ c (B) g B g ❛❧♠❡♥t❡✳ ❖✉ s❡❥❛✱ C é s♦♠❛ ❞✐r❡t❛ ⊕ ∈G ✳ c (B)

  ◆❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦✱ ✈❛♠♦s ❞❡✜♥✐r ❡♠ C ✉♠❛ ❡str✉t✉r❛ ❞❡ ∗

  ✲á❧❣❡❜r❛✳ t } t Pr♦♣♦s✐çã♦ ✷✳✶✳✶✵✳ ❙❡❥❛ B = {B ∈G ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❉❡✜♥✐♠♦s c (B) ❡♠ C ❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

  ∗ : C c (B) × C c (B) → C c (B) (ξ, η) 7→ ξ ∗ η,

  P −1 ξ(t)η(t s)

  ❡♠ q✉❡ (ξ ∗ η)(s) = ✱ ♣❛r❛ t♦❞♦ s ∈ G✱ ❡ t ∈G ∗ : C c (B) → C c (B) ∗ −1 ∗ ξ 7→ ξ ,

  (t) = ξ(t ) c (B) ❡♠ q✉❡ ξ ✱ ♣❛r❛ t♦❞♦ t ∈ G✳ ❊♥tã♦✱ C é ✉♠❛ ∗✲ á❧❣❡❜r❛✳ ❉❡♠♦♥str❛çã♦✿ ❆ ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ∗ é ❜✐❧✐♥❡❛r✱ ♣♦✐s ❛ ♠✉❧✲ s × B t → B st t✐♣❧✐❝❛çã♦ · : B é ❜✐❧✐♥❡❛r✳ ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ ∗ é ❛ss♦❝✐❛t✐✈❛✳ c (B)

  ❉❡ ❢❛t♦✱ s❡❥❛♠ ξ✱ η ❡ ζ ❡♠ C ✳ ❙❡❥❛ s ∈ G✳ ❯s❛♥❞♦ ♦ ❛①✐♦♠❛ ❞❡ ❜✐❧✐♥❡❛r✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ B✱ t❡♠♦s

  X −1 ((ξ ∗ η) ∗ ζ)(s) = (ξ ∗ η)(t)ζ(t s) t ∈G

  !

  X X −1 −1 = ξ(r)η(r t) ζ(t s) t r ∈G ∈G

  X X −1 −1 = ξ(r)η(r t)ζ(t s) t ∈G r ∈G

  X X −1 −1 = ξ(r)η(r t)ζ(t s) r ∈G t ∈G

  !

  X X −1 −1 = ξ(r) η(r t)ζ(t s) r t ∈G ∈G

  !

  X X −1 −1 −1 = ξ(r) η(r t)ζ(t rr s) . r t ∈G ∈G

  ❋❛③❡♥❞♦ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ t = r −1 t

  =

  ✱ ξ 7→ P t ∈G kξ(t)k

  ❖❜s❡r✈❛çã♦ ✷✳✶✳✶✶✳ ❆ ❢✉♥çã♦ k·k 1 : C c (B) → R +

  C c (B) ✉♠❛ ∗✲á❧❣❡❜r❛✳

  ▼❛✐s ❛✐♥❞❛✱ ♦❜s❡r✈❛♥❞♦ q✉❡ ❛ ♦♣❡r❛çã♦ ❞❡ ✐♥✈♦❧✉çã♦ ❞❡ B é ✐♥✈♦❧✉✲ t✐✈❛✱ t❡♠♦s ♣❛r❛ t♦❞♦ t ∈ G (ξ ) (t) = (ξ (t −1 )) = (ξ(t) ) = ξ(t). P♦rt❛♥t♦✱ ❡st❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦ ❢❛③❡♠ ❞❡

  = η ∗ ξ

  = (η ∗ ξ )(s), ❝♦♥❝❧✉✐♥❞♦ q✉❡ (ξ ∗ η)

  X t ∈G η (t (t ′−1 s)

  ✱ ♦❜t❡♠♦s (ξ ∗ η) (s) =

  ❋❛③❡♥❞♦ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ t = st

  X t ∈G η (st)ξ ((st) −1 s).

  X t ∈G η (st)ξ (t −1 )

  ❡ s✉❜st✐t✉✐♥❞♦ ✈❡♠ q✉❡ ((ξ ∗ η) ∗ ζ)(s) =

  =

  X t ∈G η(t −1 s −1 ) ξ(t)

  ! ∗ =

  X t ∈G ξ(t)η(t −1 s −1 )

  (ξ ∗ η) (s) = (ξ ∗ η)(s −1 ) =

  = η ∗ ξ ✳ P❛r❛ s ∈ G✱

  ❉♦♥❞❡ ∗ é ❛ss♦❝✐❛t✐✈❛✳ ❱❡❥❛♠♦s q✉❡ (ξ ∗ η)

  X r ∈G ξ(r)(η ∗ ζ)(r −1 s) = (ξ ∗ (η ∗ ζ))(s).

  ! =

  X t ∈G η(t )ζ(t ′−1 r −1 s)

  X r ∈G ξ(r)

  ❢❛③ ❞❡ C c (B) ✉♠❛ ∗✲á❧❣❡❜r❛ ♥♦r♠❛❞❛✳ ▲❡♠❜r❡♠♦s q✉❡ s❡ P é ✉♠❛ s❡♥t❡♥ç❛ ❧ó❣✐❝❛✱ r❡♣r❡s❡♥t❛♠♦s ♣♦r [P ]

  ♦ ✈❛❧♦r 1 s❡ ❛ s❡♥t❡♥ç❛ P ❢♦r ✈❡r❞❛❞❡✐r❛✱ ❡ 0 s❡ ❛ s❡♥t❡♥ç❛ P ❢♦r ❢❛❧s❛✳ P♦r ❡①❡♠♣❧♦✱ ♦ sí♠❜♦❧♦ [s = t] t❡♠ ✈❛❧♦r 1 s❡ s = t✳ ❈❛s♦ ❝♦♥trár✐♦✱ [s = t] = 0

  ✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✷✳ P❛r❛ ❝❛❞❛ t ∈ G✱ ❝♦♥s✐❞❡r❡♠♦s ❛ tr❛♥s❢♦r♠❛çã♦ t : B t → C c (B) ❧✐♥❡❛r j ❞❛❞❛ ♣♦r c (B) = ⊕ t j t (B t ) j t (b t ) | s = [s = t]b t , ♣❛r❛ t♦❞♦ s ∈ G✳ ❊♥tã♦ C ∈G ❡✱ ♣❛r❛ q✉❛✐sq✉❡r s, t ∈ G t ∈ B t s ∈ B s ❡ b ✱ b ✱ ✈❛❧❡ q✉❡✿ t (b t )k = kb t k ✭✐✮ kj s (b s ) ∗ j t (b t ) = j st (b s b t ) 1 ❀ ✭✐✐✮ j ❀ t (b t ) = j t −1 (b ) ∗ ∗ ✭✐✐✐✮ j t ✳ c (B) t = ξ(t) ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ξ ∈ C ✳ P❛r❛ ❝❛❞❛ t ∈ G✱ s❡❥❛ b ✳ ❊♥tã♦✱ s❡ s ∈ G✱ t❡♠♦s q✉❡

  !

  X X t ∈G t ∈G j t (b t ) (s) = j t (b t ) | s = b s = ξ(s).

  P j t (b t ) = 0 ❆❧é♠ ❞✐ss♦✱ s❡ ❡ s ∈ G✱ s❡❣✉❡ q✉❡ t ∈G

  !

  X t (b t ) = 0 c (B) = ⊕ t j t (B t ) b s = j t (b t ) (s) = 0, t ∈G ❞♦♥❞❡ j ✱ ♣❛r❛ ❝❛❞❛ t✳ ▲♦❣♦✱ C ∈G ✳

  P❛r❛ ♦ ✐t❡♠ ✭✐✮✱ ♦❜s❡r✈❛♠♦s q✉❡

  X kj t (b t )k 1 = kj t (b t )(s)k = kj t (b t )(t)k = kb t k. s ∈G ❏á ♦ ✐t❡♠ ✭✐✐✮ s❡❣✉❡ ❞♦ s❡❣✉✐♥t❡ ❝á❧❝✉❧♦✿ −1

  (j s (b s ) ∗ j t (b t )) (r) = b s j t (b t )(s r) = [r = st]b s b t = j st (b s b t )(r), s (b s ) ∗ j t (b t ) = j st (b s b t ) ❡ ♣♦rt❛♥t♦ (j ✳ ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ ∗ −1 ∗ −1 ∗ t (b t ) = j −1 (b ) j t (b t ) (s) = j t (b t )(s ) = [s = t ]b ∗ ∗ t ❡ s❡❣✉❡ q✉❡ j t t ✳ c (B) → A

  Pr♦♣♦s✐çã♦ ✷✳✶✳✶✸✳ ❙❡❥❛♠ π : C ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ C c (B) c (B)

  ❡♠ ✉♠❛ C ✲á❧❣❡❜r❛ A ❡ ξ ∈ C ✳ ❊♥tã♦ kπ(ξ)k ≤ kξk c (B) 1 ✳ ❉❡♠♦♥str❛çã♦✿ ❙❡ π é ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ C ✱ ❛s ❛✜r♠❛çõ❡s e : B e → A ✭✐✐✮ ❡ ✭✐✐✐✮ ❞❛ Pr♦♣♦s✐çã♦ ✐♠♣❧✐❝❛♠ q✉❡ π ◦ j é ✉♠

  ✲❤♦♠♦♠♦r✜s♠♦ ❞❡ C ✲á❧❣❡❜r❛s✳ ❆ss✐♠✱ kπ(j e (b e ))k ≤ kb e k, e ∈ B e . t ∈ B t b t e ♣❛r❛ t♦❞♦ b P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ b ✱ b t é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ B ✳ ❉❡st❛

  ❢♦r♠❛✱ 2 ∗ ∗ ∗ 2 kπ(j t (b t ))k = kπ(j t (b t )) π(j t (b t ))k = kπ(j e (b b t ))k ≤ kb b t k = kb t k . t (b t ))k ≤ kb t k t t ❖✉ s❡❥❛✱ kπ(j ✳ c (B)

  ❆❣♦r❛✱ ♣❛r❛ ✉♠ ❡❧❡♠❡♥t♦ ❛r❜✐trár✐♦ ξ ❡♠ C ✱ ♥♦✈❛♠❡♥t❡ ✉s❛♥❞♦ P j t (ξ t ) t = ξ(t)

  ❛ Pr♦♣♦s✐çã♦ ❡s❝r❡✈❡♠♦s ξ = t ✱ ❡♠ q✉❡ ξ ✳ ∈G ❉❛í✱

  X X kπ(ξ)k ≤ kπ(j t (ξ t ))k ≤ kξ t k = kξk . t t ∈G ∈G c (B) 1 P♦rt❛♥t♦✱ kπ(ξ)k ≤ kξk 1 ✱ ♣❛r❛ t♦❞♦ ξ ∈ C ✳ c (B) ❈♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ s❡❣✉❡ q✉❡ C é

  ✉♠❛ ∗✲á❧❣❡❜r❛ ❛❞♠✐ssí✈❡❧✳ ❉❡✜♥✐çã♦ ✷✳✶✳✶✹✳ ❆ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❞❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ∗ ∗ B (B) c (B)

  ✱ ❞❡♥♦t❛❞❛ ♣♦r C ✱ é ❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❛ ∗✲á❧❣❡❜r❛ C ✳ ❊①❡♠♣❧♦ ✷✳✶✳✶✺✳ ❙❡❥❛ G = Z ❡ s❡❥❛ B ♦ ✜❜r❛❞♦ tr✐✈✐❛❧✳ ❖✉ s❡❥❛✱ 1 B n = C × {n} = Cδ n (B) ∼ = C(S )

  ✳ ❊♥tã♦✱ C ✳ (B)

  ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❛♠♦s q✉❡ C é ❛❜❡❧✐❛♥❛✱ ♣♦✐s Z é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ❜❡♠ ❝♦♠♦ ❛ C ✲á❧❣❡❜r❛ ❞♦s ♥ú♠❡r♦s

  (B) ❝♦♠♣❧❡①♦s✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❡❧❡♠❡♥t♦ δ n 1 é ✉♥✐tár✐♦ ❡ ❣❡r❛ C ✱ ♣♦✐s (δ ) = δ n 1 ✱ ♣❛r❛ ❝❛❞❛ n ∈ Z✳

  (B) ∼ )) ) ❉❡st❛ ❢♦r♠❛✱ s❛❜❡♠♦s q✉❡ C = C(σ(δ 1 ✱ ❡♠ q✉❡ σ(δ 1 ❞❡♥♦t❛

  ♦ ❡s♣❡❝tr♦ ❞♦ ❡❧❡♠❡♥t♦ δ 1 1 ✳ ❯♠❛ ✈❡③ q✉❡ δ 1 é ✉♥✐tár✐♦✱ t❡♠♦s q✉❡ 1 σ(δ ) ⊆ S ⊆ σ(δ ) 1 ❡✱ ♣♦rt❛♥t♦✱ ♣r❡❝✐s❛♠♦s ♠♦str❛r q✉❡ S 1 1 ✳

  ❉❡ ❢❛t♦✱ s❡❥❛ λ ∈ S ✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❛♣❧✐❝❛çã♦ C τ λ : ⊕ n δ n → C ∈Z

  X X i i i α i δ i 7→ α i λ . ❊♥tã♦✱ ♣❛r❛ α✱ β ∈ C ❡ n, m ∈ Z✱ t❡♠♦s n +m

  τ λ (αδ n βδ m ) = τ λ (αβδ n ) = αβλ n m +m = αλ βλ = τ λ (αδ n )τ λ (βδ m )

  ❡ ∗ ∗ −n ∗ n λ c (B) = ⊕ n δ n τ λ (αδ n ) = (αλ ) = ¯ αλ = τ λ ((αδ n ) ). C ▲♦❣♦✱ τ é ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ C ∈Z ❡✱ s❡❣✉❡ ❞❛

  (B) ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❡ C ✱ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ∗✲❤♦♠♦♠♦r✜s♠♦ τ λ : C (B) → C e t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛ ι

  C c (B) // C (B) τ λ τ $$ C f λ ❝♦♠✉t❛✳

  τ λ (B) ❉♦♥❞❡✱ e é ✉♠ ❝❛rát❡r s♦❜r❡ C ✱ ♦✉ s❡❥❛✱ ✉♠ ❤♦♠♦♠♦r✜s♠♦

  (B) ♥ã♦ ♥✉❧♦ ❞❡ C ❡♠ C✳ P♦rt❛♥t♦✱ λ = τ λ (δ τ λ (δ ) ∈ σ(δ ) 1 ) = e 1 1 1

  (B) ∼ = C(S ) ❡ ❝♦♥❝❧✉í♠♦s q✉❡ C ✳ t } t ❖❜s❡r✈❛çã♦ ✷✳✶✳✶✻✳ ❙❡ G é ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❡ B = {Cδ ∈G é ♦ ∗ ∗

  (B) ✜❜r❛❞♦ tr✐✈✐❛❧✱ C é ❝❤❛♠❛❞❛ C ✲á❧❣❡❜r❛ ❞♦ ❣r✉♣♦ G ❡ é ❞❡♥♦t❛❞❛

  (G) ♣♦r C ✳

  ✷✳✷ ❆ r❡♣r❡s❡♥t❛çã♦ r❡❣✉❧❛r

  ◆❡st❛ s❡çã♦✱ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❝♦♥str✉✐r ✉♠❛ r❡♣r❡s❡♥t❛çã♦ c (B) ✜❡❧ ❞❡ C ✱ q✉❡ s❡rá ❝❤❛♠❛❞❛ r❡♣r❡s❡♥t❛çã♦ r❡❣✉❧❛r✳ ❆ ♣❛rt✐r ❞❡❧❛✱ s❡rá ❞❡✜♥✐❞❛ ✉♠❛ ♦✉tr❛ C ✲á❧❣❡❜r❛ r❡❧❛❝✐♦♥❛❞❛ ❛ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❯♠❛ ✐♠♣♦rt❛♥t❡ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ r❡♣r❡s❡♥t❛çã♦ r❡❣✉❧❛r é q✉❡ ♣♦❞❡♠♦s c (B) (B) (B) ∗ ∗ ❝♦♥s✐❞❡r❛r C ❝♦♠♦ ✉♠❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡ C ❡✱ ❛❧é♠ ❞✐ss♦✱ C é ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛✱ ❝✉❥❛ ❞❡✜♥✐çã♦ ✈❡r❡♠♦s ♥♦ ❈❛♣ít✉❧♦

  ❈♦♠❡ç❛♠♦s ♦s ♣r❡♣❛r❛t✐✈♦s ♣❛r❛ ❝♦♥str✉✐r ❛ r❡♣r❡s❡♥t❛çã♦ r❡❣✉❧❛r e ❞❡✜♥✐♥❞♦ ✉♠❛ ❡str✉t✉r❛ ❞❡ B ✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ♥♦ ❡s♣❛ç♦ c (B)

  ✈❡t♦r✐❛❧ C ✳ c (B) e Pr♦♣♦s✐çã♦ ✷✳✷✳✶✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ C é ✉♠ B ✲♠ó❞✉❧♦ ❝♦♠ ♣r♦✲ ❞✉t♦ ✐♥t❡r♥♦ ❝♦♠ ❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❞❛❞❛ ♣♦r

  C c (B) × B e → C c (B) (ξ, a) 7→ ξa,

  ❡♠ q✉❡ (ξa)(t) = ξ(t)a✱ ♣❛r❛ t♦❞♦ t ∈ G✱ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h·, ·i B : C c (B) × C c (B) → B e e

  X c (B) (ξ, η) 7→ ξ(t) η(t), t ∈G ♣❛r❛ ξ, η ∈ C ✳ ❉❡♠♦♥str❛çã♦✿ ❆s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡ ❞❡ ❛çã♦ ❞❡ ♠ó✲ ❞✉❧♦ s❡❣✉❡♠ ❞✐r❡t❛♠❡♥t❡ ❞♦s ❛①✐♦♠❛s ❞❡ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡ ❞❛ ❡str✉t✉r❛ c (B) ❞❡ ∗✲á❧❣❡❜r❛ ❞❡ C ✳ e

  ❉❡♥♦t❛r❡♠♦s ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞♦ B ✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ C c (B) (B)

  ♣♦r l 2 ✳ c (B) ◆♦ss♦ ♦❜❥❡t✐✈♦ ❛❣♦r❛ é ❝♦♥str✉✐r ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ C ♥❛ C (B)

  ✲á❧❣❡❜r❛ ❞♦s ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s s♦❜r❡ l 2 ✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛✲ ♠♦s ❞♦ s❡❣✉✐♥t❡ ❧❡♠❛✿ s ∈ B s e ▲❡♠❛ ✷✳✷✳✷✳ ❙❡❥❛ b ❡ a ✉♠ ❡❧❡♠❡♥t♦ ♣♦s✐t✐✈♦ ❞❡ B ✳ ❊♥tã♦ ∗ ∗ b ab s ≤ kakb b s . s s

  B e ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ s❛❜❡♠♦s q✉❡ a ≤ kak ♥❛ ✉♥✐t✐③❛çã♦ f ✳ λ ) λ e ❙❡♥❞♦ ❛ss✐♠✱ s❡ (u ∈Λ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ B ❡ λ ∈ Λ✱ s❡❣✉❡ q✉❡ ∗ ∗ ∗ 0 ≤ u λ b (kak − a)b s u λ = kaku λ b b s u λ − u λ b ab s u λ . s s s ∗ + ∗ ab s ≤ kakb b s

  ❯♠❛ ✈❡③ q✉❡ B e é ❢❡❝❤❛❞♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ b s s ✳ ❆ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ♥♦t❛çã♦✱ ✈❛♠♦s ❞❡♥♦t❛r s✐♠✲ t t ∈ B t c (B)

  ♣❧❡s♠❡♥t❡ ♣♦r b ❛ ✐♠❛❣❡♠ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ b ❡♠ C ♣❡❧❛ t : B t → C c (B) tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r j ❞❡✜♥✐❞❛ ♥❛ Pr♦♣♦s✐çã♦ c (B) = ⊕ t B t ❆ss✐♠✱ C ∈G ✳ ❋✐❝❛rá ❝❧❛r♦ ♣❡❧♦ ❝♦♥t❡①t♦ q✉❛♥❞♦ ❡st❛r❡♠♦s t c (B) ♥♦s r❡❢❡r✐♥❞♦ ❛ b ❝♦♠♦ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ C ♦✉ ❝♦♠♦ ✉♠ ❡❧❡♠❡♥t♦ t ❞♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ B ✳ t ∈ B t

  P❛r❛ ❝❛❞❛ t ∈ G ❡ ♣❛r❛ ❝❛❞❛ b ✱ ❞❡✜♥✐♠♦s e T b : C c (B) → C c (B) t t ∗ ξ)(s) = b t ξ(t s) −1 ξ 7→ b t ∗ ξ.

  ◆♦t❡♠♦s q✉❡ (b ✱ ♣❛r❛ t♦❞♦ s ∈ G✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ e T b c (B) t é ❧✐♥❡❛r✱ ♣♦✐s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ∗ ❡♠ C é ❜✐❧✐♥❡❛r✳ ▼❛✐s ❛✐♥❞❛✱

  T b (ξ)k ≤ kb t kkξk B T b s❡❣✉❡ ❞♦ ▲❡♠❛ q✉❡ k e ✱ ❞♦♥❞❡ e s❡ ❡st❡♥❞❡ ❛ b : l (B) → l (B) b k ≤ kb t k t e t ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r T t t 2 2 t❛❧ q✉❡ kT ✳ b

  ❆ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♥♦s ❞✐③ q✉❡ T é ✉♠ ♦♣❡r❛❞♦r ❛❞❥✉♥tá✈❡❧ ❡♠ t l (B) 2 ✿ b b = T b Pr♦♣♦s✐çã♦ ✷✳✷✳✸✳ T é ❛❞❥✉♥tá✈❡❧ ❡ T ✳ t t t c (B) ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ❝♦♥s✐❞❡r❛♠♦s ξ✱ η ∈ C ✳ ❊♥tã♦✱

  X −1 ∗ ∗ −1 ∗ ∗ −1

  X hT b ξ, ηi B = ξ(t s) b η(s) = ξ(t s) b η(tt s). t e s ∈G s ∈G t t ′ −1 = t s

  ❋❛③❡♥❞♦ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ s ❡ s✉❜st✐t✉✐♥❞♦ ♥❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ✈❡♠ q✉❡

  X ′ ∗ ∗ ′ ′ ′

  X ∗ ∗ hT b ξ, ηi B = ξ(s ) b η(ts ) = ξ(s )T b (η)(s ) = hξ, T b ηi B . t e s ∈G s ∈G ′ ′ t t t e (B) b b

  ❆❣♦r❛✱ s❡ ξ✱ η ∈ l 2 ✱ ❞❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ T t ❡ T s❡❣✉❡ q✉❡ t hT b (ξ), ηi B = lim hT b (ξ m ), η m i B = lim hξ m , T b (η m )i B t e t e e m m t m ) m m ) m c (B) = hξ, T b (η)i B , t e ❡♠ q✉❡ (ξ ∈N ❡ (η ∈N sã♦ s❡q✉ê♥❝✐❛s ❡♠ C ❝♦♥✈❡r❣✐♥❞♦ ❛ ξ ❡ η✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ b = T b

  ▲♦❣♦✱ T ✳ t t t , b t ∈ B t s ∈ B s Pr♦♣♦s✐çã♦ ✷✳✷✳✹✳ ❙❡❥❛♠ a ✱ b ❡ λ ∈ C✳ ❊♥tã♦✿ a = T a + λT b ✭✐✮ T t +λb t t t ❀ b T b = T b b ✭✐✐✮ T ✳ t s t s c (B) ❉❡♠♦♥str❛çã♦✿ ✭✐✮ P❛r❛ ξ ∈ C ✱ ✈❛❧❡ q✉❡ T a (ξ) = (a t + λb t ) ∗ ξ = a t ∗ ξ + λb t ∗ ξ = T a (ξ) + λT b (ξ). t t t t +λb

  (B) P❛r❛ ♦ ❝❛s♦ ξ ∈ l 2 ✱ ❜❛st❛ ❛r❣✉♠❡♥t❛r ♣♦r ❝♦♥t✐♥✉✐❞❛❞❡✳ c (B)

  ✭✐✐✮ ◆♦✈❛♠❡♥t❡✱ s❡♥❞♦ ξ ∈ C ✱ t❡♠♦s q✉❡ T b (T b (ξ)) = b t ∗ (b s ∗ ξ) = (b t ∗ b s ) ∗ ξ = (b t b s ) ∗ ξ = T b b (ξ). t s t s (B)

  ❖ ❝❛s♦ ❣❡r❛❧ ❡♠ q✉❡ ξ ∈ l 2 s❡❣✉❡ ♣♦r ❝♦♥t✐♥✉✐❞❛❞❡✳ ❈♦r♦❧ár✐♦ ✷✳✷✳✺✳ ❆ ❛♣❧✐❝❛çã♦

  T : C c (B) → L(l (B)) 2 X ξ 7→ T , t ∈G ξ (t)

  é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦✳ ❉❡♠♦♥str❛çã♦✿ ❆s ♣r♦♣♦s✐çõ❡s ♥♦s ❞✐③❡♠ q✉❡ T é ✉♠ ∗

  ✲❤♦♠♦♠♦r✜s♠♦✳ ❆ss✐♠✱ r❡st❛ s♦♠❡♥t❡ ✈❡r✐✜❝❛r♠♦s q✉❡ T é ✐♥❥❡t♦r✳ c (B) λ ) λ ∈Λ ❙❡❥❛ ξ ∈ C t❛❧ q✉❡ T (ξ) = 0✳ ❙❡❥❛ (u ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦✲ e

  ①✐♠❛❞❛ ♣❛r❛ B ✳ ❊♥tã♦✱ ♣❛r❛ λ ∈ Λ ❡ s ∈ G✱

  X −1 0 = T (ξ)(j e (u λ )) | s = ξ(t)j e (u λ )(t s). t ∈G λ = 0 ❊s❝♦❧❤❡♥❞♦ s = t✱ ♦❜t❡♠♦s q✉❡ ξ(t)u ✳ ❈♦♠♦ λ ∈ Λ é ❛r❜✐trá✲ r✐♦✱ s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ q✉❡ ξ(t) = 0✳ P♦rt❛♥t♦✱ ξ = 0 ❡ T é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦✳ ❖ ∗✲❤♦♠♦♠♦r✜s♠♦ T ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛ é ❝❤❛♠❛❞♦ r❡♣r❡s❡♥t❛çã♦ c (B) r❡❣✉❧❛r à ❡sq✉❡r❞❛ ❞❡ C ✳

  ❉❡✜♥✐çã♦ ✷✳✷✳✻✳ ❆ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ r❡❞✉③✐❞❛ ❞♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B (B) c (B)) 2 (B))

  ✱ ❞❡♥♦t❛❞❛ ♣♦r C r ✱ é ♦ ❢❡❝❤♦ ❞❡ T (C ❡♠ L(l ✳ ❈♦r♦❧ár✐♦ ✷✳✷✳✼✳ ❆ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❞❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ + B c (B)

  é ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ C ♥❛ ♥♦r♠❛ |k · k| : C c (B) → R ξ 7→ sup{kπ(ξ)k : π c (B)}.

  é ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ C (B)

  ❉❡♠♦♥str❛çã♦✿ ❙❛❜❡♠♦s ❞♦ ❚❡♦r❡♠❛ q✉❡ C é ♦ ❝♦♠♣❧❡t❛✲ c (B)/N ♠❡♥t♦ ❞❡ C ♥❛ ♥♦r♠❛

  |kξ + N |k = sup{kπ(ξ)k : π c (B)}, é ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ C c (B) : sup kπ(η)k = 0}

  ❡♠ q✉❡ N = {η ∈ C π ✳ ◆❡st❡ ❝❛s♦✱ ♦ ❈♦r♦❧ár✐♦ ✐♠♣❧✐❝❛ q✉❡ N = {0}✱ ♦✉ s❡❥❛✱ ❛ ♣r♦❥❡✲ c (B) → C c (B)/N

  çã♦ ❝❛♥ô♥✐❝❛ ι : C é ✐♥❥❡t✐✈❛ ❡ ❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ c (B) c (B) ❞❡ C é s✐♠♣❧❡s♠❡♥t❡ ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ C ♥❛ ♥♦r♠❛ k|ξk| = sup{kπ(ξ)k : π c (B)}.

  é ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ C ❖❜s❡r✈❛çã♦ ✷✳✷✳✽✳ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✲ ∗ ∗

  T : C (B) → C (B) ✈❡♥t❡✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ∗✲❤♦♠♦♠♦r✜s♠♦ e r t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛ ι

  C c (B) // C (B) T $$ T e c (B) (B) C (B) r ❝♦♠✉t❛✱ ❡♠ q✉❡ ι é ❛ ✐♥❝❧✉sã♦ ❞❡ C ❡♠ C ✳

  T ❉❡✜♥✐çã♦ ✷✳✷✳✾✳ ❖ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B é ❞✐t♦ s❡r ❛♠❡♥❛❜❧❡ s❡ e é ✐♥❥❡t✐✈♦✳

  ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ❡♠ ♠❛✐s ❞❡t❛❧❤❡s s♦❜r❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧ ❛♠❡♥❛✲ ❜❧❡ ♣♦❞❡ ❡♥❝♦♥trá✲❧♦s ❡♠

  ❈❛♣ít✉❧♦ ✸

Pr♦❞✉t♦s ❈r✉③❛❞♦s

  ◆❡st❡ ❝❛♣ít✉❧♦ ❞❡✜♥✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❡♠ C ✲á❧❣❡❜r❛s✱ q✉❡ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ ❛çã♦ ❣❧♦❜❛❧✳ ❉❡✲ ♣♦✐s✱ ♠♦str❛♠♦s q✉❡ ❡①✐st❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✱ ❛tr❛✈és ❞♦ q✉❛❧ ❞❡✜♥✐♠♦s ♦s ♣r♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s ❝❤❡✐♦ ❡ r❡❞✉③✐❞♦✱ ❛♣❧✐❝❛♥❞♦ ♦ q✉❡ ❞❡s❡♥✈♦❧✈❡♠♦s ♥♦ ❈❛♣ít✉❧♦ ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s q✉❡ ✉s❛♠♦s sã♦

  ◆❛ ú❧t✐♠❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❜r❡✈❡♠❡♥t❡ ❛ t❡♦r✐❛ ❞❡ ♣r♦❞✉t♦s ❝r✉③❛❞♦s ♥♦ ❝❛s♦ ❞❡ ❛çõ❡s ❣❧♦❜❛✐s ❞❡ ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❛çã♦ é ❞❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❛❜❡❧✐❛♥♦✱ ♠♦str❛♠♦s q✉❡ é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠❛ ❛çã♦ ❞♦ s❡✉ ❣r✉♣♦ ❞✉❛❧ ♥♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ r❡✲ s✉❧t❛♥t❡✱ ♠✉✐t♦ ❡♠❜♦r❛ ✐st♦ s❡❥❛ ♣♦ssí✈❡❧ ♥♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧ ❞❡ ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s ❛❜❡❧✐❛♥♦s✳ ❆ t❡♦r✐❛ ❞❡ ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠✲ ♣❛❝t♦s é ❡♥❝♦♥tr❛❞❛ ❡♠ ❏á ✉♠❛ ót✐♠❛ r❡❢❡rê♥❝✐❛ ♣❛r❛ ♣r♦❞✉t♦s ❝r✉③❛❞♦s é é ♣r♦✈❛❞❛ ❛ ❡①✐stê♥❝✐❛ ❞❛ ❛çã♦ ❞✉❛❧ ♥♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧✱ ❛❧é♠ ❞❡ tr❛③❡r ✐♠♣♦rt❛♥t❡s r❡s✉❧t❛❞♦s r❡❧❛t✐✈♦s ❛♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♦❜t✐❞♦✳

  ✸✳✶ ❆çõ❡s ♣❛r❝✐❛✐s ❞❡ ❣r✉♣♦s

  ❉❡✜♥✐çã♦ ✸✳✶✳✶✳ ❯♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ G ❡♠ ✉♠❛ C

  ✲á❧❣❡❜r❛ A é ✉♠ ♣❛r α = ({D g } g , {α g } g ), g g ∈G ∈G

  ❡♠ q✉❡ ♣❛r❛ ❝❛❞❛ g ∈ G✱ D é ✉♠ ✐❞❡❛❧ ❢❡❝❤❛❞♦ ❡♠ A ❡ α é ✉♠ ∗ −1 g

  ✲✐s♦♠♦r✜s♠♦ ❞❡ D g ❡♠ D ✱ s❛t✐s❢❛③❡♥❞♦ ♣❛r❛ q✉❛✐sq✉❡r g, h ∈ G e = A

  ✭✐✮ D ✱ g (D g ∩ D h ) ⊆ D gh −1 ✭✐✐✮ α ✱ g (α h (a)) = α gh (a) h ∩ D h g −1 −1 −1 ✭✐✐✐✮ α ✱ ♣❛r❛ t♦❞♦ a ∈ D ✳

  ❉✐③❡♠♦s q✉❡ (A, G, α) é ✉♠ C ✲s✐st❡♠❛ ❞✐♥â♠✐❝♦ ♣❛r❝✐❛❧ ♦✉✱ s✐♠✲ g = A ♣❧❡s♠❡♥t❡✱ ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ♣❛r❝✐❛❧✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ D ✱ ♣❛r❛ t♦❞♦ g✱ α é ❞✐t❛ s❡r ✉♠❛ ❛çã♦ ❣❧♦❜❛❧✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ✉♠❛ ❛çã♦✱ ❞❡ G ❡♠ A ❡ (A, G, α) é ❝❤❛♠❛❞♦ s✐st❡♠❛ ❞✐♥â♠✐❝♦✳ ❖❜s❡r✈❛çã♦ ✸✳✶✳✷✳ ✭✐✐✮ ❝♦♥s✐st❡ ❞❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡ α g (α h (a))

  ❡♠ ✭✐✐✐✮ ❡st❡❥❛ ❜❡♠ ❞❡✜♥✐❞♦✳ −1 ∩ D −1 −1 ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ s❡ a ∈ D h h g ✱ ✭✐✐✮ ♥♦s ❞✐③ q✉❡ α h (a) ∈ D −1 −1 = D −1 h g g ✳ (h ) e ❖❜s❡r✈❛çã♦ ✸✳✶✳✸✳ α é ♦ ❛✉t♦♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡✳ e (α e (a)) = α e (a) ❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ a ∈ A✱ s❡❣✉❡ ❞❡ ✭✐✐✐✮ q✉❡ α ✳ e e (a) = a e ❯♠❛ ✈❡③ q✉❡ α é ✐♥❥❡t✐✈❛✱ ♦❜t❡♠♦s q✉❡ α ✳ ❉♦♥❞❡✱ α é ♦ ❛✉t♦♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡✳ ❖❜s❡r✈❛çã♦ ✸✳✶✳✹✳ ❙❡ θ é ✉♠❛ ❛çã♦ ❣❧♦❜❛❧ ❞❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ G g = θ g (I) ∩ I ❡♠ ✉♠❛ C ✲❛❧❣❡❜r❛ A ❡ I é ✉♠ ✐❞❡❛❧ ❞❡ A✱ ❝♦❧♦❝❛♥❞♦ D ❡ α g = θ g | D g } g , {α g } g ) g−1 ✱ ♣❛r❛ ❝❛❞❛ g ∈ G✱ t❡♠♦s q✉❡ α = ({D ∈G ∈G é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G ❡♠ I✳ g ❉❡♠♦♥str❛çã♦✿ ❈❛❞❛ D é ✉♠ ✐❞❡❛❧ ❡♠ I✱ ♣♦✐s é ✉♠❛ ✐♥t❡rs❡çã♦ ❞❡ g −1 g ✐❞❡❛✐s✳ P❛r❛ ✈❡r q✉❡ α é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ D g ❡ D ✱ ❝♦♥s✐❞❡r❡♠♦s −1 −1 (I)∩I g (x) = ✉♠ ❡❧❡♠❡♥t♦ x ❡♠ D g ✱ ♦✉ s❡❥❛✱ x ∈ θ g ✳ ❉❡st❛ ❢♦r♠❛✱ α θ g (x) ∈ I −1 (I) g (x) ∈ θ g (I) g (x) ∈ θ g (I) ∩ I = D g ♣♦✐s y ∈ θ g ❡ t❛♠❜é♠ α ✱ ✉♠❛ ✈❡③ q✉❡ x ∈ I✳ ▲♦❣♦✱ α ✳ g −1 (y) ∈ D −1

  ❉❛❞♦ y ∈ D ✱ ❜❛st❛ ❝♦❧♦❝❛r♠♦s x = α g g ✱ ❡ ❛ss✐♠ α g (x) = y g

  ✱ ❞♦♥❞❡ ❝❛❞❛ α é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❖s ♦✉tr♦s ❛①✐♦♠❛s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧ s❡❣✉❡♠ ❞✐r❡t❛♠❡♥t❡ ❞♦ ❢❛t♦ ❞❡ θ s❡r ✉♠❛ ❛çã♦ ❞❡ G ❡♠ A✳ Pr♦♣♦s✐çã♦ ✸✳✶✳✺✳ ❆ ❝♦♥❞✐çã♦ ✭✐✐✮ ♥❛ ❉❡✜♥✐çã♦ é ❡q✉✐✈❛❧❡♥t❡ ❛✿ g (D g ∩ D h ) = D g ∩ D gh −1 ✭✐✐✮✬ α ✱ ♣❛r❛ q✉❛✐sq✉❡r g, h ∈ G✳

  ❉❡♠♦♥str❛çã♦✿ ➱ ❢á❝✐❧ ✈❡r q✉❡ ✭✐✐✮✬ ✐♠♣❧✐❝❛ ✭✐✐✮✳ ❙✉♣♦♥❤❛ q✉❡ ✈❛❧❤❛ g ∩ D gh −1 (b) ∈ D −1 ∩ D h ✭✐✐✮ ❡ s❡❥❛ b ∈ D ✳ P♦r ✭✐✐✮✱ t❡♠♦s q✉❡ α g g ✳ g −1 (b) g −1 ∩ D h g (y) = ▲♦❣♦✱ ❝♦❧♦❝❛♥❞♦ y = α ✱ t❡♠♦s q✉❡ y ∈ D ❡ α α g (α g −1 (b)) = b

  ✳ ❊①❡♠♣❧♦ ✸✳✶✳✻✳ ❙❡❥❛ A ✉♠❛ C ✲❛❧❣❡❜r❛ ❡ s❡❥❛♠ I ❡ J ✐❞❡❛✐s ❞❡ A✳ ❙❡❥❛ α : I → J ✉♠ ∗✲✐s♦♠♦r✜s♠♦✳

  = A = J = I n ❉❡✜♥❛ D ✱ D 1 ❡ D −1 ✳ P❛r❛ ❝❛❞❛ n > 1✱ s❡❥❛ D

  ❞❡✜♥✐❞♦ ✐♥❞✉t✐✈❛♠❡♥t❡ ❝♦♠♦ −(n−1) D n = {x ∈ D n : α (x) ∈ J} −1

  ❡ ♣❛r❛ n < −1 −(n+1) n := α n D n = {x ∈ D n : α (x) ∈ I}. +1 ❙❡❥❛ α ✱ ❡♠ q✉❡ α é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ❛✉t♦♠♦r✜s♠♦ n } n , {α n } n )

  ✐❞❡♥t✐❞❛❞❡ ❞❡ A✳ ❊♥tã♦ α = ({D ∈Z ∈Z é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❡ (A, Z, α) é ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ♣❛r❝✐❛❧✳ n ❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❛♠♦s q✉❡ ❝❛❞❛ D é✱ ❞❡ ❢❛t♦✱ ♦ ❞♦♠í♥✐♦ ❞❡ −n α n : D → D n

  ❡ α −n é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ n > 0 ❡ n (x) = α (x) ∈ J n s❡❥❛ x ∈ D −n ✳ ❊♥tã♦ α ❡ t❡♠♦s✱ ♣❛r❛ n ≥ k ≥ 1 q✉❡ −k+1 n n −k+1 α (α (x)) = α (x) ∈ J

  ♣♦✐s 1 ≤ n − k + 1 ≤ n✱ ♦✉ s❡❥❛✱ ❛s ♣♦tê♥❝✐❛s ❞❡ α sã♦ ♣♦s✐t✐✈❛s✳ ❆❧é♠ n (y) ∈ D (x) = y −n n ❞✐ss♦✱ s❡ y ∈ D ✱ t♦♠❛♥❞♦ x = α −n s❡❣✉❡ q✉❡ α ✳ ❖ ❝❛s♦ n < 0 é ❛♥á❧♦❣♦✳ n

  ❱❛♠♦s ❛❣♦r❛ ♣r♦✈❛r ♣♦r ✐♥❞✉çã♦ ❡♠ n ≥ 0 q✉❡ ❝❛❞❛ D é ✉♠ ✐❞❡❛❧ ❡♠ A✳ n n

  P❛r❛ ✐ss♦✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ D é ✉♠ ✐❞❡❛❧ ❡♠ D −1 ✱ ♣❛r❛ n > 0 n n ❡ D é ✉♠ ✐❞❡❛❧ ❡♠ D +1 ✱ ♣❛r❛ n < 0✳ ❈♦♠♦ I ❡ J sã♦ ✐❞❡❛✐s ❡♠ A✱ n t❡r❡♠♦s q✉❡ D é ✐❞❡❛❧ ❞❡ A✱ ♣❛r❛ t♦❞♦ n✳

  = J P❛r❛ n = 0 ❡ n = 1 ♥ã♦ ❤á ♥❛❞❛ ❛ ❢❛③❡r✱ ♣♦✐s D 1 ✳ ❙✉♣♦♥❤❛ n > 1 n n n

  ❡ q✉❡ D −1 é ✉♠ ✐❞❡❛❧ ❡♠ D −2 ✳ ❱❛♠♦s ♠♦str❛r q✉❡ D é ✉♠ n ✐❞❡❛❧ ❡♠ D −1 ✳ n n n

  ❙❡❥❛♠ b ∈ D ❡ y ∈ D −1 ✳ ❯♠❛ ✈❡③ q✉❡ D é ❢❡❝❤❛❞♦ ♣♦r ✐♥✈♦❧✉çã♦✱ n ❜❛st❛ ♣r♦✈❛r♠♦s q✉❡ é ✉♠ ✐❞❡❛❧ à ❡sq✉❡r❞❛✱ ♦✉ s❡❥❛✱ yb ∈ D ✳ ❈♦♠♦ D n n −1 é ✉♠ ✐❞❡❛❧ ❡♠ J✱ s❡❣✉❡ q✉❡ yb ∈ D −1 ✳ ❚❡♠♦s q✉❡ −(n−1) −1 −(n−2) −1 −(n−2) −(n−2)

  α (yb) = α [α (yb)] = α [α (y)α (b)],

  n −(n−2)

  ♣♦✐s y, b ∈ D −2 ✱ q✉❡ é ♦ ❞♦♠í♥✐♦ ❞❡ α ✳ P♦r ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡ −(n−2) −(n−2) α (y), α (y) ∈ J −(n−1) −1 −(n−2) −1 −(n−2) ✱ ❞♦♥❞❡ −(n−1) −1 −(n−2) α (yb) = α [α (y)]α [α (b)] ∈ J

  (b) = α [α (b)] ∈ J ♣♦✐s α ✳ n n n n

  ▲♦❣♦✱ yb ∈ D ❡ ❛ss✐♠ D é ✐❞❡❛❧ ❡♠ D −1 ✱ ♦✉ s❡❥❛✱ D é ✐❞❡❛❧ ❡♠ A

  ✳ P❛r❛ n < 0 ❛ ♣r♦✈❛ é ❛♥á❧♦❣❛✳ m (D −m ∩ D n ) ⊆ D m +n ❙❡❥❛♠ m, n ∈ Z✳ Pr❡❝✐s❛♠♦s ♣r♦✈❛r q✉❡ α ✱

  ❛ ❝♦♥❞✐çã♦ ✭✐✐✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧✳ ❖ ❝❛s♦ m = 0 é ❞✐r❡t♦ ❡ ♦ ❝❛s♦ n = 0 ❥á ❛r❣✉♠❡♥t❛♠♦s✳ ❙❡ m, n > 0 −m ∩ D n

  ❡ x ∈ D ✱ t❡♠♦s q✉❡ −k+1 −k+1 m m α (α (x)) = α (x) ∈ J

  ♣❛r❛ ❝❛❞❛ 1 ≤ k ≤ m < m + n✱ ♣♦✐s m − k + 1 > 0✳ ❙❡ k = m + r✱ ❝♦♠ 1 ≤ r ≤ n ❡♥tã♦ −k+1 −m−r+1 −r+1 m m

  α (α (x)) = α (α (x)) = α (x) ∈ J, n ✉♠❛ ✈❡③ q✉❡ x ∈ D ✳ −m ∩ D n

  ❙✉♣♦♥❤❛ ❛❣♦r❛ m > 0 ❡ n < 0 ❡ s❡❥❛ x ∈ D ✳ ❙✉♣♦♥❤❛ m m > m + n > 0 (x) ∈ D m ⊆ ✳ ❖ r❡s✉❧t❛❞♦ s❡❣✉❡ ❞♦ ❢❛t♦ q✉❡ α

  D m +n ✳ ❆❣♦r❛✱ ♣❛r❛ n < m + n < 0✱ s❡❥❛ n < m + n ≤ k ≤ −1✳ m

  (x) ∈ I ❖❜s❡r✈❛♠♦s q✉❡ α ✱ ♣♦✐s ♥❡st❡ ❝❛s♦ −m − 1 ≥ n ❡ ❛ss✐♠ m −((−m−1)+1) (α (x)) = α (x) ∈ I −k−1 −k−1+m m ✳ ▼❛✐s ❛✐♥❞❛✱ r α (α (x)) = α (x) ∈ I,

  (x) ∈ I ♣♦✐s α ♣❛r❛ t♦❞♦ 0 ≤ r ≤ −n − 1✳

  ❖s ❝❛s♦s ❡♠ q✉❡ m < 0 ❡ n > 0✱ ❡ m < 0 ❡ n < 0 sã♦ ❛♥á❧♦❣♦s✳ n +m (x) = α n (α m (x)) −m ∩ D −m−n ❖ ❢❛t♦ q✉❡ α ✱ ♣❛r❛ x ∈ D ✱ s❡❣✉❡ n ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ α ❝♦♠♦ ❝♦♠♣♦s✐çã♦ ❞❡ ✐s♦♠♦r✜s♠♦s✳ n } n , {α n } n )

  ▲♦❣♦✱ α = ({D ∈Z ∈Z é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✳

  ✸✳✷ Pr♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s

  ◆♦ss♦ ♦❜❥❡t✐✈♦ ♥❡st❛ s❡çã♦ é ❝♦♥str✉✐r ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ ♣❛rt✐r ❞❡ ✉♠ ❛çã♦ ♣❛r❝✐❛❧ α ❞❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ G ❡♠ ✉♠❛ C ✲á❧❣❡❜r❛ A✳ ■st♦ é ❢❡✐t♦ ❡♠ ♥♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧✱ ❡ ♠❛✐s ❝♦♠♣❧✐❝❛❞♦✱ ❞❡ ❛çõ❡s ♣❛r❝✐❛✐s t♦r❝✐❞❛s✳ ◆♦ ❝❛s♦ ❞❡ ❛çõ❡s ♣❛r❝✐❛✐s ♥ã♦ t♦r❝✐❞❛s✱ ❞❡✜♥✐♥❞♦ ❛ ❝♦❧❡çã♦ ❞❡

  ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ❡ ❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦✱ é ✉♠ t❛♥t♦ ♠❛✐s s✐♠♣❧❡s ♣r♦✈❛r ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞♦ ♣r♦❞✉t♦✳ ❈♦♠❡ç❛♠♦s ♣♦r ❞❡✜♥✐r ❛ ❝♦❧❡çã♦ ❞❡ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ✐♥❞❡①❛❞♦s

  ❡♠ G✳ g } g , {α g } g ) ❙❡❥❛ α = ({D ∈G ∈G ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G ❡♠ ✉♠❛ C ✲

  á❧❣❡❜r❛ A✳ ❙❡❥❛ e B = {(a, g) ∈ A × G : a ∈ D g }, t B

  ❡ ♣❛r❛ ❝❛❞❛ t ∈ G✱ s❡❥❛ B ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ e ❢♦r♠❛❞♦ ♣♦r t♦❞♦s t ♦s ♣❛r❡s (a, g) ❝♦♠ g = t✳ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦ aδ ♣❛r❛ (a, t)✱ ❞♦♥❞❡ B t = D t δ t

  ✳ t ❉♦t❛♠♦s B ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ♦❜t✐❞❛ ❛tr❛✈és t t

  ❞❛ ❜✐❥❡çã♦ ❝❛♥ô♥✐❝❛ ❡♥tr❡ B ❡ D ✳ g } g ❈♦♥s✐❞❡r❡ ❛ ❢❛♠í❧✐❛ B = {B ∈G ❞❡ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳ ❉❡✜♥✐♠♦s

  ❡♠ B ❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦ ♣♦r (a s δ s ) ∗ (b t δ t ) = α s (α −1 (a s )b t )δ st s

  ❡ ∗ ∗ s s t t (a s δ s ) = α −1 (a )δ −1 , s s s ♣❛r❛ a ❡♠ D ❡ b ❡♠ D ✳ Pr♦♣♦s✐çã♦ ✸✳✷✳✶✳ ❈♦♠ ❛s ♦♣❡r❛çõ❡s ❛❝✐♠❛✱ B é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❉❡♠♦♥str❛çã♦✿ s → B t

  ❆ ❜✐❧✐♥❡❛r✐❞❛❞❡ ❞❛ ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ∗ : B s❡❣✉❡ ❞♦ g ❢❛t♦ q✉❡ ❝❛❞❛ α é ✉♠ ✐s♦♠♦r✜s♠♦ ❡✱ ♣♦rt❛♥t♦✱ ❧✐♥❡❛r✳ ❱❛♠♦s ♠♦str❛r s δ s ) ∗ (b t δ t ) ∈ D st δ st q✉❡ ❛ ♦♣❡r❛çã♦ ∗ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♦✉ s❡❥❛✱ (a ✱ s, t ∈ G

  ✳ ❉❡ ❢❛t♦✱ −1 (a s δ s ) ∗ (b t δ t ) = α s (α s (a s )b t )δ st , s (a s )b t ∈ D s ∩ D t −1 −1

  ❡ ♦❜s❡r✈❛♥❞♦ q✉❡ α ✱ ❝♦♥❝❧✉í♠♦s ❞♦ ✐t❡♠ ✭✐✐✮ ❞❛ s (α s (a s )b t ) ∈ D st −1 ❉❡✜♥✐çã♦ q✉❡ α ✱ ❞♦♥❞❡ ❛ ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐✲ ♣❧✐❝❛çã♦ ∗ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ g ∈ D g

  ❱❛♠♦s ♠♦str❛r q✉❡ ∗ é ❛ss♦❝✐❛t✐✈❛✳ ❙❡❥❛ c ✳ ❚❡♠♦s (a s δ s ∗ b t δ t ) ∗ (c g δ g ) = α s (α −1 (a s )b t )δ st ∗ c g δ g s = α st (α −1 −1 (α s (α −1 (a s )b t ))c g )δ stg . t s s

  P♦r ♦✉tr♦ ❧❛❞♦✱ (a s δ s ) ∗ (b t δ t ∗ c g δ g ) = a s δ s ∗ α t (α t −1 (b t )c g )δ tg −1 −1 = α s (α s (a s )α t (α t (b t )c g ))δ stg . st (α t s (α s (α s (a s )b t )c g ) = −1 −1 −1

  ❖✉ s❡❥❛✱ ♣r❡❝✐s❛♠♦s ♣r♦✈❛r q✉❡ α −1 −1 α s (α s (a s )α t (α t (b t )c g )) −1 (a s )α t (α −1 (b t )c g ) ∈ D −1 ∩ D t

  ❯♠❛ ✈❡③ q✉❡ α s t s ✱ ♣❡❧♦ ✐t❡♠ ✭✐✐✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧ ❝♦♥❝❧✉í♠♦s q✉❡ α s (α −1 (a s )α t (α −1 (b t )c g )) ∈ D s ∩ D st . s t

  ▲♦❣♦✱ α s (α s −1 (a s )α t (α t −1 (b t )c g )) = α st (α t −1 s −1 (α s (α s −1 (a s )α t (α t −1 (b t )c g )), st

  ❡ ♣❡❧❛ ✐♥❥❡t✐✈✐❞❛❞❡ ❞❡ α é s✉✜❝✐❡♥t❡ ♠♦str❛r♠♦s ❛ s❡❣✉✐♥t❡ ✐❣✉❛❧❞❛❞❡✿ −1 −1 −1 −1 −1 −1 −1 α t s (α s (α s (a s )b t ))c g = α t s [α s (α s (a s )α t (α t (b t )c g ))]. λ ) λ ∈Λ t s ∩ −1 −1 ✭†✮ −1 µ µ −1 P❛r❛ ✐ss♦✱ s❡❥❛ (u ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ D D t ) ∈L t

  ❡ s❡❥❛ (w ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ D ✳ ❯s❛♥❞♦ ♦ −1 −1 α s = α −1 −1 ∩ D t ❢❛t♦ q✉❡ α t s t ❡♠ D s ✱ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❡ s❡ t♦r♥❛

  α −1 (α −1 (a s )b t )c g = lim α −1 (α −1 (a s )b t )u λ c g t s t s λ = lim α −1 (α −1 (a s )b t )α −1 (α t (u λ c g ) λ t s t = lim α t −1 [α s −1 (a s )b t α t (u λ c g )] λ = lim lim α t −1 [α s −1 (a s )b t α t (u λ c g w µ )] λ µ −1 −1 λ ) λ −1 −1 ∩ D −1 t = lim lim α t [α s (a s )b t α t (u λ )α t (c g w µ )]. λ µ

  ❈♦♠♦ (u ∈Λ é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ D t s t ❡ α −1 −1 ∩ D −1 t ∩ D −1 é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ D t s t ❡♠ D s ✭Pr♦♣♦s✐çã♦ t (u λ )} λ t ∩ D −1 s❡❣✉❡ q✉❡ {α ∈Λ é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ D s ✳ ❆ss✐♠✱ s❡ s✉♣✉s❡r♠♦s ♣♦r ✉♠ ✐♥st❛♥t❡ q✉❡ ♣♦❞❡♠♦s tr♦❝❛r ❛ ♦r❞❡♠ ❞♦ ❧✐♠✐t❡✱ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ s❡ t♦r♥❛ lim α t −1 [α s −1 (a s )b t α t (c g w µ )] = lim α t −1 [α s −1 (a s )α t (α t −1 (b t )c g w µ )] µ µ −1 −1 −1 t = α t s α s s ∩ D t −1 −1 −1 −1 = α t [α s (a s )α t (α t (b t )c g )], ❡ ♥♦✈❛♠❡♥t❡ ✉s❛♥❞♦ q✉❡ α ❡♠ D ✱ ❛ ❡①♣r❡ssã♦ ♦❜t✐❞❛ ❛❝✐♠❛ é ✐❞ê♥t✐❝❛ ❛♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡

  ❆ss✐♠✱ r❡st❛ ♣r♦✈❛r♠♦s q✉❡ ♣♦❞❡♠♦s tr♦❝❛r ❛ ♦r❞❡♠ ❞♦ ❧✐♠✐t❡✳ ▼❛s✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ α −1 [α −1 (a s )α t (α −1 (b t )c g )] = lim lim α −1 [α −1 (a s )b t α t (u λ )α t (c g w µ )] t s t t s µ λ−1 −1 −1 −1

  α t (α s (a s )b t )c g = lim lim α t [α s (a s )b t α t (u λ )α t (c g w µ )], λ µ t❡♠♦s −1 −1 −1 −1 −1 kα t [α s (a s )α t (α t (b t )c g )] − α t (α s (a s )b t )c g k ≤ (△), ❡♠ q✉❡ −1 −1 −1 −1 −1 (△) = kα t [α s (a s )α t (α t (b t )c g )] − α t [α s (a s )b t α t (c g w µ )]k kα t [α s (a s )b t α t (c g w µ )] − α t [α s (a s )b t α t (u λ )α t (c g w µ )]k −1 −1 −1 −1

  • kα t [α s (a )b α (u )α (c w )] − α t (α s (a )b )c k. −1 −1 s t t λ t g µ −1 −1 s t g t (c g w µ )k ≤ kc g k
  • ❖❜s❡r✈❛♥❞♦ q✉❡ kα ✱ ♣❛r❛ t♦❞♦ µ ∈ L✱ ✈❡♠ q✉❡

  (△) ≤ kα t −1 [α s −1 (a s )α t (α t −1 (b t )c g )] − α t −1 [α s −1 (a s )b t α t (c g w µ )]k

  • kα t −1 [α s −1 (a s )b t ] − α t −1 [α s −1 (a s )b t α t (u λ )]kkc g k + kα t −1 [α s −1 (a s )b t α t (u λ )α t (c g w µ ) − α t −1 (α s −1 (a s )b t )c g k.

  ❆♣❧✐❝❛♥❞♦ ♦ ❧✐♠✐t❡ s♦❜r❡ µ ❡ ❡♠ s❡❣✉✐❞❛ s♦❜r❡ λ ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♦❜t✐❞❛✱ ❝♦♥❝❧✉í♠♦s q✉❡

  α −1 [α −1 (a s )α t (α −1 (b t )c g )] = α −1 (α −1 (a s )b t )c g , t s t t s ♦✉ s❡❥❛✱ ♣♦❞❡♠♦s tr♦❝❛r ❛ ♦r❞❡♠ ❞♦ ❧✐♠✐t❡✳ P♦rt❛♥t♦✱ ✜❝❛ ♣r♦✈❛❞❛ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ B✳

  ▼❛✐s ❛✐♥❞❛✱ t❡♠♦s ∗ ∗ ∗ (a g δ g ) ∗ (a g δ g ) = α g −1 (a )δ g −1 ∗ a g δ g = α g −1 (a a g )δ e , g g e = Aδ e q✉❡ é ✉♠ ❡❧❡♠❡♥t♦ ♣♦s✐t✐✈♦ ❡♠ B ✳ t −1 s −1 α s = α t −1 s −1 ∩

  ❆❣♦r❛✱ ♠❛✐s ✉♠❛ ✈❡③ ♦❜s❡r✈❛♥❞♦ q✉❡ α ❡♠ D D t

  ✱ t❡♠♦s (a s δ s ∗ b t δ t ) = (α s (α s −1 (a s )b t )δ st ) −1 −1 −1 −1 −1 ∗ ∗

  = α t s [α s (b α s (a ))]δ t s −1 −1 −1 −1 ∗ ∗ t s = α t [b α s (a )]δ t s −1 −1 −1 −1 t s ∗ ∗ = (α t (b )δ t ) ∗ (α s (a )δ s ) ∗ ∗ t s g = (b t δ t ) ∗ (a s δ s ) .

  ❈♦♠♦ ❝❛❞❛ α é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ C ✲á❧❣❡❜r❛s✱ t❡♠♦s q✉❡ ❛ ♦♣❡✲ r❛çã♦ ❞❡ ✐♥✈♦❧✉çã♦ ❡♠ B é ✐s♦♠étr✐❝❛✳ ▲♦❣♦✱ B é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❉❡✜♥✐çã♦ ✸✳✷✳✷✳ ❖ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❞❡ A ❡ G ♣♦r α✱ ❞❡♥♦t❛❞♦ α G ♣♦r A ⋊ ✱ é ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❞♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B = {B g } g ∈G

  ✳ ❉❡✜♥✐çã♦ ✸✳✷✳✸✳ ❖ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ r❡❞✉③✐❞♦ ❞❡ A ❡ G ♣♦r α α,r G

  ✱ ❞❡♥♦t❛❞♦ ♣♦r A ⋊ ✱ é ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ r❡❞✉③✐❞❛ ❞♦ ✜❜r❛❞♦ g } g ❞❡ ❋❡❧❧ B = {B ∈G ✳ ❖❜s❡r✈❛çã♦ ✸✳✷✳✹✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ α é ✉♠❛ ❛çã♦ ❣❧♦❜❛❧ ❞❡ G ❡♠ A✱ A ⋊ α G

  é ❝❤❛♠❛❞♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ❞❡ A ❡ G ♣♦r α✳ ❙✐♠✐❧❛r♠❡♥t❡✱ A ⋊ α,r G

  é ❞✐t♦ s❡r ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ r❡❞✉③✐❞♦ ❞❡ A ❡ G ♣♦r α✳ n n , x , . . . , x n , x n ) ∈ C :

  ❊①❡♠♣❧♦ ✸✳✷✳✺✳ ❙❡❥❛ A = C ✱ I = {(x n 1 2 −1 x n = 0} , x , . . . , x n , x n ) ∈ C : x = 0} ❡ J = {(x 1 2 −1 1 ✳ ❙❡❥❛ α : I → J✱

  (x , . . . , x n , 0) 7→ (0, x , . . . , x n ) n } n , {α n } n ) 1 −1 1 −1 ✳ ❊ s❡❥❛ α = ({D ∈Z ∈Z n Z ⋊ α =

  ❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡✜♥✐❞❛ ❝♦♠♦ ♥♦ ❊①❡♠♣❧♦ ❊♥tã♦✱ C M n (C)

  ✳ ❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❛♠♦s q✉❡

  D = J, 1 D = {(0, x , x , . . . , x n ) ∈ J : x = 0}, 2 2 3 2 ✳✳✳ D n = {(0, x , x , . . . , x n ) ∈ J : x = x = · · · x n = 0}, −1 2 3 2 3 −1 D m = {0},

  ♣❛r❛ t♦❞♦ m ≥ n ❡ s✐♠✐❧❛r♠❡♥t❡

  D = I, −1 D = {(x , x , . . . , x n , 0) ∈ I : x n = 0}, −2 1 2 −1 −1

  ✳✳✳ D = {(x , x , . . . , x n , 0) ∈ I : x = x = · · · x n = 0}, −n+1 1 2 −1 2 3 −1

  D m = {0}, i,j := e i δ i i = (0, 0, . . . , ♣❛r❛ t♦❞♦ m ≤ −n.

  1 , . . . , 0) ❙❡❥❛ e −j ✱ ❡♠ q✉❡ e ✱ ♣❛r❛ 1 ≤

  |{z}

  ❡♥tr❛❞❛ i

  i ≤ n ✱ 1 ≤ j ≤ n✳ i,j ∈ B i = D i δ i

  ◆♦t❡♠♦s q✉❡✱ ♣❛r❛ i − j < 0✱ e −j −j −j ✱ ♣♦✐s −i+j−1 α (e i ) = (0, 0, . . . , 1 , . . . , 0) ∈ I

  |{z}

  ❡♥tr❛❞❛ j−1

  ❡ ♣❛r❛ i − j > 0✱ −i+j+1 α (e i ) = (0, 0, . . . , 1 , . . . , 0) ∈ J.

  |{z}

  ❡♥tr❛❞❛ j+1

  ❙❡❥❛♠ 1 ≤ k, l ≤ n✳ ❊♥tã♦✱ e i,j e k,l = e i δ i −j ∗ e k δ l −k = α i −j (α j −i (e i )e k )δ (i−j)+(k−l) i,j : 1 ≤ i, j ≤ n} = [j = k]e i δ i −l = [j = k]e i δ i −l = e i,l , ❞♦♥❞❡ {e é ✉♠ s✐st❡♠❛ ❞❡ ✉♥✐❞❛❞❡s ♠❛tr✐❝✐❛✐s✳ ❈♦♠♦ α α = M n (C) Z Z ❣❡r❛♠ C ⋊ ✱ s❡❣✉❡ q✉❡ C ⋊ ✳ n 1 (C) ⊕ M n 2 (C) ⊕ · · · ⊕ M n (C) ❊①❡♠♣❧♦ ✸✳✷✳✻✳ ❙❡❥❛ A = M k ✉♠❛ C 1 + n 2 + · · · + n k

  ✲á❧❣❡❜r❛ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❙❡❥❛ N = n ❡ ♣❛r❛ ❝❛❞❛ n l 1 ≤ l ≤ k l l l ✱ s❡❥❛ α ♦ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ✐❞❡❛✐s I ❡ J ❞❡ C ❝♦♠♦ ♥♦ N n n 1 k

  = C ⊕ · · · ⊕ C ⊕ I ⊕ · · · ⊕ I k ❡①❡♠♣❧♦ ❛❝✐♠❛✳ ❈♦❧♦❝❛♠♦s C ✱ I = I 1 2

  ⊕ J ⊕ · · · ⊕ J k ❡ J = J 1 2 ✳ ❙❡❥❛ α : I → J ♦ ✐s♦♠♦r✜s♠♦ ❞❛❞♦ ♣♦r N

  α ⊕ · · · ⊕ α k 1 ❡ α ❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ Z ❡♠ C ❞❡✜♥✐❞❛ ♥♦ ❊①❡♠♣❧♦ N α Z ❊♥tã♦ A = C ⋊ ✳ l C ❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❛♠♦s q✉❡✱ ♥❡st❡ ❝❛s♦✱ s❡ D i ❞❡♥♦t❛ ♦ ✐❞❡❛❧ ❡♠ n l l

  ❝♦rr❡s♣♦♥❞❡♥t❡ à ❛çã♦ ♣❛r❝✐❛❧ α ❞♦ ❊①❡♠♣❧♦ t❡♠♦s q✉❡ D = J ⊕ J ⊕ · · · ⊕ J k , 1 1 1 2 2 k

  D = D ⊕ D ⊕ · · · ⊕ D , 2 2 2 2 ✳✳✳ 1 2 k

  D n = D ⊕ D ⊕ · · · ⊕ D , −1 n n n 1 −1 2 −1 −1 k D m = {0},

  ♣❛r❛ t♦❞♦ m ≥ n, , n , . . . , n k }

  ❡♠ q✉❡ n = max{n 1 2 ✳ ❙✐♠✐❧❛r♠❡♥t❡✱

  D = −1 I ⊕ I ⊕ · · · ⊕ I k , 1 1 2 2 k D = D ⊕ D ⊕ · · · ⊕ D , −2 −2 −2 −2

  ✳✳✳ 1 2 k D −n+1 = D ⊕ D ⊕ · · · ⊕ D , −n 1 +1 −n 2 +1 −n +1 k

  D m = {0}, ♣❛r❛ t♦❞♦ m ≤ −n.

  := 0 ❋✐①❛♠♦s l ∈ {1, 2, . . . , k} ❡✱ ❞❡✜♥✐♥✐♥❞♦ n ✱ s❡❥❛ l n +i = (0, 0, . . . , e := e n +i δ i −j , i,j l−1 1 , . . . , 0) l

  ❡♠ q✉❡ e l−1 ❡ ♣❛r❛ 1 ≤ i ≤ n ✱ |{z} +i

  

❡♥tr❛❞❛ n l−1

  1 ≤ j ≤ n l ✳

  ❆ss✐♠✱ ♦s ♠❡s♠♦s ❛r❣✉♠❡♥t♦s ✉s❛❞♦s ♥♦ ❊①❡♠♣❧♦ ❣❛r❛♥t❡♠ l n ∈ D i q✉❡ e i,j ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ♦✉ s❡❥❛✱ e l−1 +i −j ✳ l l l e = [j = h]e ❏á ✈✐♠♦s ♥♦ ❊①❡♠♣❧♦ q✉❡ e ✱ ❞♦♥❞❡ é l m i,j h,k i,k e = 0 s✉✜❝✐❡♥t❡ ♠♦str❛r♠♦s q✉❡ e i,j h,k ✱ ♣❛r❛ l 6= m✳ ❈♦♠ ❡❢❡✐t♦✱ l m e e = e n δ i ∗ e n δ h i,j h,k +i −j +h −k l−1 m−1

  = α i (α j (e n )e n )δ −j −i +i +h (i−j)+(h−k) l−1 m−1 = 0. l

  : 1 ≤ i, j ≤ n l , 1 ≤ l ≤ k} P♦rt❛♥t♦✱ ✉♠❛ ✈❡③ q✉❡ {e i,j é ✉♠❛ ❜❛s❡ N α Z

  ♣❛r❛ C ⋊ ✱ ❛ ❛♣❧✐❝❛çã♦ l e 7→ (0, 0, . . . , e i,j , . . . , 0) i,j |{z} N Z

  ❡♥tr❛❞❛ l

  ⋊ α é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ C ❡ A✳

  

✸✳✸ ❆çõ❡s ❞❡ ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s

  ❆ ✜♠ ❞❡ ❢♦r♥❡❝❡r ❡①❡♠♣❧♦s ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ✉♠❛ ♠❡❧❤♦r ❝♦♠✲ ♣r❡❡♥sã♦ ❞❛ ❞❡✜♥✐çã♦ ❞♦ ♣r♦❞✉t♦ ❙♠❛s❤✱ q✉❡ s❡rá ❛♣r❡s❡♥t❛❞❛ ♥♦ ❈❛♣í✲ t✉❧♦ ❞❡✜♥✐♥✐♠♦s ♥❡st❛ s❡çã♦ ♦ q✉❡ é ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ❞❡ ✉♠ ❣r✉♣♦

  ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ G ❡♠ ✉♠❛ C ✲á❧❣❡❜r❛ A✳ ❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥✲ t❛♠♦s ❜r❡✈❡♠❡♥t❡ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ❞❡ A ♣♦r ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ❞❡ G

  ❡✱ ♣❛r❛ ♦ ❝❛s♦ ❞✐s❝r❡t♦✱ ♠♦str❛♠♦s q✉❡ s❡ G é ❛❜❡❧✐❛♥♦✱ ♦ ♣r♦❞✉t♦ α G G ❝r✉③❛❞♦ A ⋊ ❛❞♠✐t❡ ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ❞♦ ❣r✉♣♦ ❞✉❛❧ b ✳

  Pr✐♠❡✐r❛♠❡♥t❡✱ ❧❡♠❜r❡♠♦s q✉❡ ✉♠ ❣r✉♣♦ t♦♣♦❧ó❣✐❝♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ G ❛❞♠✐t❡ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❇♦r❡❧ µ r❡❣✉❧❛r ❡ ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛✱ ♦✉ s❡❥❛✱ s❡ E é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ❇♦r❡❧ ❞❡ G ❡ s ∈ G✱ ❡♥✲ tã♦ µ(sE) = µ(E)✳ ❚❛❧ ♠❡❞✐❞❛ é ú♥✐❝❛ ❛ ♠❡♥♦s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ❡ é ❝❤❛♠❛❞❛ ♠❡❞✐❞❛ ❞❡ ❍❛❛r✳ ❯♠❛ ♣r♦✈❛ ❞❡ s✉❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ◗✉❛♥❞♦ G é ❝♦♠♣❛❝t♦✱ µ é ✜♥✐t❛✱ ❡ ♥❡st❡ ❝❛s♦ ✉s❛♠♦s ❛ ♠❡❞✐❞❛ ♥♦r♠❛❧✐③❛❞❛ µ(G) = 1✳ ❙❡ G é ❞✐s❝r❡t♦✱ µ é ❛ ♠❡❞✐❞❛ ❞❡ ❝♦♥t❛❣❡♠✳

  ❊♠ ❣❡r❛❧✱ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❍❛❛r ♣♦❞❡ ♥ã♦ s❡r ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s❧❛✲ R + çã♦ à ❞✐r❡✐t❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡①✐st❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❝♦♥tí♥✉♦ ∆ : G →

  ✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❢✉♥çã♦ ♠♦❞✉❧❛r✱ t❛❧ q✉❡ µ(Es) = ∆(s)µ(E)✳ ❙❡ G

  é ❝♦♠♣❛❝t♦ ♦✉ ❛❜❡❧✐❛♥♦✱ ∆ é ♦ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧✳ ❯s❛r❡♠♦s t❛♠❜é♠ ♥❡st❛ s❡çã♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥t❡❣r❛çã♦ s♦❜r❡ ❣r✉♣♦s

  ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s ❝♦♠ ✈❛❧♦r❡s ❡♠ ✉♠❛ C ✲á❧❣❡❜r❛✳ ▼❛✐s ❞❡t❛❧❤❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡st❡ ❝♦♥❝❡✐t♦ ❞❡ ✐♥t❡❣r❛çã♦ ❜❡♠ ❝♦♠♦ ❛ t❡♦r✐❛ ❣❡r❛❧ ❞❡ ♣r♦❞✉t♦s ❝r✉③❛❞♦s ♣♦r ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s sã♦ ❡♥❝♦♥tr❛❞♦s ❡♠

  ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛✳ ❱❛♠♦s ❞❡♥♦t❛r ♣♦r Aut(A) ♦ ❣r✉♣♦ ❞❡ t♦❞♦s ♦s ∗✲❛✉t♦♠♦r✜s♠♦s ❞❡ A✳ ❉❡✜♥✐çã♦ ✸✳✸✳✶✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦✳ ❯♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ❞❡ G ❡♠ A é ✉♠ ❤♦♠♦♠♦r✜s♠♦ α : G → Aut(A) q✉❡ é ❝♦♥tí♥✉♦ ♥❛ t♦♣♦❧♦❣✐❛ ❢♦rt❡✱ ♦✉ s❡❥❛✱ ♣❛r❛ ❝❛❞❛ a ∈ A✱ ❛ ❛♣❧✐❝❛çã♦ g 7→ α g (a)

  é ❝♦♥tí♥✉❛✳ c (G, A)

  ❙❡❥❛♠ α : G → Aut(A) ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ❡ C ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ❞❡ G ❡♠ A c (G, A)

  ✳ P❛r❛ f, g ∈ C ❡ s ∈ G✱ ❞❡✜♥✐♠♦s Z −1 (f ∗ g)(s) = f (r)α r (g(r s))dµ(r). G

  ❖ ❈♦r♦❧ár✐♦ ✶✳✶✵✹ ❞❡ ❣❛r❛♥t❡ q✉❡ s 7→ (f ∗g)(s) ❞❡✜♥❡ ✉♠ ❡❧❡♠❡♥t♦ c (G, A) ❞❡ C ✱ ❝❤❛♠❛❞♦ ❝♦♥✈♦❧✉çã♦ ❞❡ f ❡ g✳ c (G, A)

  ❉❡✜♥✐♠♦s ❡♠ C ✉♠❛ ♦♣❡r❛çã♦ ❞❡ ✐♥✈♦❧✉çã♦ ♣♦r ∗ −1 −1 ∗ f (s) = ∆(s) α s (f (s ) ). c (G, A) ❈♦♠ ❡st❛s ♦♣❡r❛çõ❡s✱ C é ✉♠❛ ∗✲á❧❣❡❜r❛✳ c (G, a)

  ❆ ∗✲á❧❣❡❜r❛ C ♣♦ss✉✐ ✉♠❛ ♥♦r♠❛✱ ❛ s❛❜❡r Z kf k = kf (t)kdµ(t) 1 G

  ❡ q✉❡ s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦ ❞❡ C c (G, a)

  ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ c (G, A) Pr♦♣♦s✐çã♦ ✸✳✸✳✷✳ ❙❡❥❛♠ f, g ∈ C ✳ ❊♥tã♦✱ kf k 1 = kf k

  ❡ kf ∗ gk 1 ≤ kf k 1 kgk 1 .

  ❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡ ∗ ∗ −1 −1 ∗ Z Z kf k = kf (s)kdµ(s) = ∆(s )kα s (f (s ) )kdµ(s) 1 G G Z −1 −1 = ∆(s )kf (s )kdµ(s) G

  Z = kf (s)kdµ(s) = kf k G 1 .

  ❆❝✐♠❛✱ ✉s❛♠♦s ♦ ▲❡♠❛ ✶✳✻✼ ❞❡ q✉❡ ♥♦s ❞✐③ q✉❡ Z Z −1 −1 G G c (G) ∆(s )f (s )dµ(s) = f (s)dµ(s), s❡♠♣r❡ q✉❡ f ∈ C ✳

  ❆❧é♠ ❞✐ss♦✱ Z Z −1 kf ∗ gk 1 = k f (r)α r (g(r s))dµ(r)kdµ(s) G G

  Z Z −1 ≤ kf (r)α r (g(r s))kdµ(r)dµ(s) G G

  Z Z −1 ≤ kf (r)kkα r (g(r s))kdµ(r)dµ(s) G G

  Z Z −1 = kf (r)k kα r (g(r s))kdµ(s) dµ(r) G G = kf k kgk . 1 1 c (G, A)

  ❈♦♠♦ ✉♠ r❡s✉❧t❛❞♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✱ t❡♠♦s q✉❡ C é ✉♠❛ ∗

  ✲á❧❣❡❜r❛ ♥♦r♠❛❞❛✳ c (G, A) ❉❡✜♥✐çã♦ ✸✳✸✳✸✳ ❙❡❥❛ π ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ C ✳ ❉✐③❡✲ ♠♦s q✉❡ π é L 1 ✲♥♦r♠❛ ❞❡❝r❡s❝❡♥t❡ s❡ kπ(f)k ≤ kfk 1 ✱ ♣❛r❛ t♦❞♦ f ∈ C c (G, A)

  ✳ c (G, A)) ❱❛♠♦s ❞❡♥♦t❛r ♣♦r Rep(C ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ∗✲r❡♣r❡s❡♥✲ c (G, A) t❛çõ❡s L 1 ✲♥♦r♠❛ ❞❡❝r❡s❝❡♥t❡s ❞❡ C ✳

  ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ❝♦♥s✐st❡ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠❛ C ✲♥♦r♠❛ ❡♠ C c (G, A)

  ✳ ❊♠❜♦r❛ ♥ã♦ ❞❡♠♦♥str❛r❡♠♦s ❛q✉✐✱ ❡❧❛ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ Pr♦♣♦s✐çã♦ ✷✳✷✸ ❡ ❞♦ ▲❡♠❛ ✷✳✷✻ ❞❡ Pr♦♣♦s✐çã♦ ✸✳✸✳✹✳ ❙❡❥❛ (A, G, α) ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ❡ ♣❛r❛ ❝❛❞❛ f ∈ C c (G, A)

  ✱ ❝♦❧♦❝❛♠♦s kf k := sup{kπ(f )k : π ∈ Rep(C c (G, A))}. c (G, A) ❊♥tã♦✱ k · k é ✉♠❛ C ✲♥♦r♠❛ ❡♠ C ✳ ❉❡✜♥✐çã♦ ✸✳✸✳✺✳ ❙❡❥❛ (A, G, α) ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦✳ ❉❡✜♥✐♠♦s ♦ α G ♣r♦❞✉t♦ ❝r✉③❛❞♦ ❞❡ A ❡ G ♣♦r α✱ ❞❡♥♦t❛❞♦ ♣♦r A ⋊ ✱ ❝♦♠♦ s❡♥❞♦ ♦ c (G, A) ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ C ♥❛ ♥♦r♠❛ ❞❡✜♥✐❞❛ ♥❛ Pr♦♣♦s✐çã♦

  ❖ ♣r♦❞✉t♦ ❝r✉③❛❞♦ t❛♠❜é♠ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❝♦♠♣❧❡t❛✲ 1 (G, A)

  ♠❡♥t♦ ❞❛ ∗✲á❧❣❡❜r❛ ❞❡ ❇❛♥❛❝❤ L ❝♦♠ ❛s ♠❡s♠❛s ♦♣❡r❛çõ❡s ❞❡ ❝♦♥✈♦❧✉çã♦ ❡ ✐♥✈♦❧✉çã♦ ❡ ❛ ♠❡s♠❛ ❞❡✜♥✐çã♦ ❞❡ ♥♦r♠❛✳ ❈❧❛r♦✱ ♥❡st❡ ❝❛s♦ ♦ s✉♣r❡♠♦ é t♦♠❛❞♦ s♦❜ t♦❞❛s ❛s r❡♣r❡s❡♥t❛çõ❡s✱ q✉❡ ❥á sã♦ ❛✉t♦✲ 1

  ♠❛t✐❝❛♠❡♥t❡ L ✲♥♦r♠❛ ❞❡❝r❡s❝❡♥t❡s✳ ❖ ❆♣ê♥❞✐❝❡ ❇ ❞❡ ❛♣r❡s❡♥t❛ ✉♠ ♣♦✉❝♦ ❞❛ t❡♦r✐❛ ❞❡✜♥✐❞❛ ❞❡st❛ ❢♦r♠❛✳

  ∗

  ❯♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ r❡❞✉③✐❞♦ ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ A ❡ ✉♠ ❣r✉♣♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ G ♣♦r ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ α é ❞❡✜♥✐❞♦ ♥❛ ❙❡çã♦ c (G, A) ✼ ❞❡ ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ C ❡♠ ✉♠❛ ♥♦r♠❛ ❞❛❞❛ ♣♦r ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ❡s♣❡❝✐❛❧✳ ◆♦ ❡♥t❛♥t♦✱ ✈❛❧❡ r❡ss❛❧t❛r q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ♣r♦❞✉t♦ ❝r✉③❛❞♦ r❡❞✉③✐❞♦ ♣♦r ❣r✉♣♦s ❞✐s❝r❡t♦s q✉❡ ❛♣r❡✲ s❡♥t❛♠♦s ❡♠ ❡ ✐ss♦ s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✽ ❞❡

  ❉❛❞❛ ✉♠❛ ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❛❜❡❧✐❛♥♦ G✱ é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ❞♦ ✉♠ ❝❡rt♦ ❣r✉♣♦ ❝♦♠♣❛❝t♦ ❛❜❡❧✐❛♥♦ ♥♦ ♣r♦❞✉t♦ α G ❝r✉③❛❞♦ A ⋊ ✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ ❛❣♦r❛ é ❝♦♥str✉✐r ❡st❛ ❛çã♦✳ ❉❡✜♥✐çã♦ ✸✳✸✳✻✳ ❙❡❥❛ G é ✉♠ ❣r✉♣♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❛❜❡❧✐❛♥♦✳ 1

  ❯♠ ❝❛rát❡r s♦❜r❡ G é ✉♠ ❤♦♠♦♠♦r✜s♠♦ χ : G → S ✳ G

  ❖ ❝♦♥❥✉♥t♦ b ❞❡ t♦❞♦s ♦s ❝❛rát❡r❡s ❝♦♥tí♥✉♦s ❞❡ G é ✉♠ ❣r✉♣♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛ ♣♦♥t✉✲ ❛❧♠❡♥t❡ ❡ t♦♣♦❧♦❣✐❛ ♣❛r❛ ❛ q✉❛❧ ♦s ❝♦♥❥✉♥t♦s

  P (F, ε) = {χ ∈ b G : |χ(x) − 1| < ε, ♣❛r❛ t♦❞♦ x ∈ F },

  ♣❛r❛ F ⊆ G ❝♦♠♣❛❝t♦ ❡ ε > 0✱ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♥❛ ✐❞❡♥t✐❞❛❞❡ 1 ✭1(x) = 1✱ ♣❛r❛ t♦❞♦ x ∈ G✮✳

  G ❖ ❣r✉♣♦ b é ❝❤❛♠❛❞♦ ❣r✉♣♦ ❞✉❛❧ ❞❡ G✳ ❖ ❚❡♦r❡♠❛ ✶✳✷✳✺ ❞❡

  G ❛✜r♠❛ q✉❡ s❡ G é ❞✐s❝r❡t♦✱ ❡♥tã♦ b é ❝♦♠♣❛❝t♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ G

  G é ❝♦♠♣❛❝t♦✱ ❡♥tã♦ b é ❞✐s❝r❡t♦✳ Pr♦♣♦s✐çã♦ ✸✳✸✳✼✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❛❜❡❧✐❛♥♦ ❡ (A, G, α) ✉♠

  G γ : s✐st❡♠❛ ❞✐♥â♠✐❝♦✳ P❛r❛ ❝❛❞❛ γ ∈ b ✱ ❞❡✜♥✐♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦ bα C c (G, A) → A ⋊ α G

  ♣♦r !

  X α γ a g δ g = γ(g)a g δ g . γ α G b g ❊♥tã♦ bα s❡ ❡st❡♥❞❡ ❛ ✉♠ ∗✲❛✉t♦♠♦r✜s♠♦ ❞❡ A ⋊ ❡ ❛ ❛♣❧✐❝❛çã♦ α : b G → Aut(A ⋊ α

  G) γ b ✱ γ 7→ bα é ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛✳ G

  ❉❡♠♦♥str❛çã♦✿ ❊s❝r❡✈❡♠♦s Γ ♣❛r❛ b ❡ s❡❥❛ γ ∈ Γ✳ ◆♦t❡♠♦s q✉❡ α γ (aδ g α γ (bδ h ) = γ(g)γ(h)aα g (b)δ gh b )b

  = γ(gh)aα g (b)δ gh = α γ (aδ g bδ h ) b

  ❡ ∗ ∗ α(aδ g ) = γ(g)α g −1 (a )δ g −1 −1 ∗

  = γ(g )α g −1 (a )δ g −1 = α γ ((aδ g ) ). γ c (G, A) b ▲♦❣♦✱ bα é ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ C ✱ ❡ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ α G γ

  ✉♥✐✈❡rs❛❧ ❞♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ A ⋊ ✱ s❡❣✉❡ q✉❡ bα s❡ ❡st❡♥❞❡ ❛ ✉♠ ∗ α G γ

  ✲❡♥❞♦♠♦r✜s♠♦ ❞❡ A ⋊ ✱ q✉❡ ✈❛♠♦s ❝♦♥t✐♥✉❛r ❞❡♥♦t❛♥❞♦ ♣♦r bα ✳ γ P❛r❛ ✈❡r q✉❡ bα é ❞❡ ❢❛t♦ ✉♠ ∗✲❛✉t♦♠♦r✜s♠♦✱ ✈❛♠♦s ♠♦str❛r q✉❡

  α γ 1 α γ 2 α γ 1 γ 2 1 2 ∈ Γ b b = b ✱ ♣❛r❛ q✉❛✐sq✉❡r γ ✱ γ ✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❝♦♠♦ α 1 α G γ b é ♦ ❛✉t♦♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ A ⋊ ✱ ❝♦♥❝❧✉✐r❡♠♦s q✉❡ bα é ✉♠ −1 −1

  ∗ α γ ✲❛✉t♦♠♦r✜s♠♦ ❝♦♠ bα γ = b ✳

  ❉❡ ❢❛t♦✱ α γ 1 α γ 2 (aδ g )) = α γ 1 (γ 2 (g)aδ g ) b (b b

  = γ (g)γ (g)aδ g 1 2 = (γ γ )(g)aδ g 1 2 = α γ γ (aδ g ). b 1 2 γ α γ α γ γ

  ❆r❣✉♠❡♥t❛♥t♦ ♣♦r ❝♦♥t✐♥✉✐❞❛❞❡✱ ♦❜t❡♠♦s q✉❡ bα 1 b 2 = b 1 2 ✳ γ P❛r❛ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ♥❛ t♦♣♦❧♦❣✐❛ ❢♦rt❡ ❞❛ ❛♣❧✐❝❛çã♦ γ 7→ bα ✱ ❜❛st❛ γ (b) g

  ✈❡r q✉❡ ❛ ❛♣❧✐❝❛çã♦ γ 7→ bα é ❝♦♥tí♥✉❛ s❡♠♣r❡ q✉❡ b é ❞❛ ❢♦r♠❛ aδ ✱ γ k = 1 ❡ ❝♦♠♦ kbα ✱ ♣❛r❛ t♦❞♦ γ ∈ Γ✱ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ t❛♠❜é♠ ♣❛r❛ α G q✉❛❧q✉❡r b ∈ A ⋊ ✳

  ❆ ❛çã♦ bα ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r é ❝❤❛♠❛❞❛ ❛çã♦ ❞✉❛❧ ❞❡ α✳ ➱ ♣♦s✲ G α G sí✈❡❧ ❞❡✜♥✐r ✉♠❛ ❛çã♦ ❞✉❛❧ ❞❡ b ❡♠ A ⋊ ♣❛r❛ ✉♠ ❣r✉♣♦ ❧♦❝❛❧♠❡♥t❡

  ❝♦♠♣❛❝t♦ ❛❜❡❧✐❛♥♦ q✉❛❧q✉❡r✳ ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ♣♦❞❡ ❡♥❝♦♥tr❛r ♠❛✐s α b

  G) ⋊ α G ❞❡t❛❧❤❡s ❡♠ ✈❛♠♦s ✈❡r q✉❡ (A ⋊ é ✐s♦✲ 2 b

  (G)) ♠♦r❢♦ ❛ A ⊗ K(l ✳ ❊st❡ r❡s✉❧t❛❞♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❞✉❛❧✐❞❛❞❡ ❞❡ ❚❛❦❛✐✳

  ❈❛♣ít✉❧♦ ✹ ❚❡♦r❡♠❛ ❞❡ ❇r♦✇♥✲●r❡❡♥✲❘✐❡✛❡❧

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ♥♦ss♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ é ♦ t❡♦r❡♠❛ ❞❡ ❇r♦✇♥✲ ●r❡❡♥✲❘✐❡✛❡❧✱ q✉❡ ❛✜r♠❛ q✉❡ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ C ✲á❧❣❡❜r❛s ♣♦ss✉✐♥❞♦ ❡❧❡♠❡♥t♦s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦s ✭❉❡✜♥✐çã♦ ❡q✉✐✈❛❧❡ ❛ ✐s♦♠♦r✜s♠♦ ❡stá✈❡❧✳ ❊st❡ t❡♦r❡♠❛ t❡rá ✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ❝r✉❝✐❛❧ ♥♦ ♥♦ss♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦✱ q✉❡ s❡rá ❛♣r❡s❡♥t❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ❡♠❜♦r❛ s❡❥❛ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❡♠ ❖✉ s❡❥❛✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ♦ r❡s✉❧t❛❞♦ é ♦❜t✐❞♦ ♣❛r❛ ✉♠ ❝❛♥t♦ ❝❤❡✐♦ ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ ❡ ✉s❛♥❞♦ ❛ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦✱ é ❡st❡♥❞✐❞♦ ♣❛r❛ ♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧ ❞❡ C ✲á❧❣❡❜r❛s ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✳

  ❉❡♣♦✐s ❞❡ ❞❡✜♥✐r C ✲á❧❣❡❜r❛s ❡stá✈❡✐s✱ ❞❡✜♥✐♠♦s ✜❜r❛❞♦s ❞❡ ❋❡❧❧ ❡stá✈❡✐s ❡ ♠♦str❛♠♦s q✉❡✱ ♥❡st❡ ❝❛s♦✱ s✉❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ é ❡stá✈❡❧✳ ❆ ♣r♦✈❛ ❞✐st♦ ❢♦✐ ♦❜t✐❞❛ ❞❡ ❊♠❜♦r❛ ♥ã♦ t❡♥❤❛♠♦s ❢❡✐t♦ ✐ss♦ ❛q✉✐✱ ✉♠❛ ❞❡♠♦♥str❛çã♦ ✐❞ê♥t✐❝❛ ❣❛r❛♥t❡ q✉❡ ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ r❡❞✉③✐❞❛ ❞❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡stá✈❡❧ é t❛♠❜é♠ ❡stá✈❡❧✳

  ❯s❛r❡♠♦s ❛♠♣❧❛♠❡♥t❡ ❛ t❡♦r✐❛ ❞❡ ♣r♦❞✉t♦s t❡♥s♦r✐❛✐s ❞❡ C ✲á❧❣❡✲ ❜r❛s✱ q✉❡ é ❛❜♦r❞❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✻ ❞❡ ❆❧é♠ ❞✐ss♦✱ ✈❛♠♦s ✉s❛r s❡♠ ∗ M (B) ∗

  B) ♠❡♥❝✐♦♥❛r q✉❡ ❡①✐st❡ ✉♠❛ ✐♥❝❧✉sã♦ ❞❡ M(A) ⊗ ❡♠ M(A ⊗ ✱ ∗ B ❡♠ q✉❡ A⊗ ❞❡♥♦t❛ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❡s♣❛❝✐❛❧ ❞❡ A ❡ B ✭Pr♦♣♦s✐çã♦

  P❛r❛ ✜①❛r ♥♦t❛çõ❡s✱ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ ❝❛♣ít✉❧♦✱ K ❞❡♥♦t❛ ❛ C ✲ á❧❣❡❜r❛ ❞❡ t♦❞♦s ♦s ♦♣❡r❛❞♦r❡s ❝♦♠♣❛❝t♦s s♦❜r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt i,j : i, j ∈ N} s❡♣❛rá✈❡❧ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ❡ {e é ✉♠ s✐st❡♠❛ ❞❡ i } i

  ✉♥✐❞❛❞❡s ♠❛tr✐❝✐❛✐s q✉❡ ❣❡r❛♠ K✳ ❖✉ s❡❥❛✱ s❡♥❞♦ {e ∈N ✉♠❛ ❜❛s❡ i,j

  ♦rt♦♥♦r♠❛❧ ♣❛r❛ t❛❧ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡ s❡♥❞♦ e ♦ ♦♣❡r❛❞♦r h 7→ he j , hie i i,j : i, j ∈ N} ✱ t❡♠♦s q✉❡ span{e é ✉♠❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡♥s❛ i,j e k,l = [j = k]e i,l

  ❞❡ K ❡ ❛ ✐❞❡♥t✐❞❛❞❡ e é s❛t✐s❢❡✐t❛✱ ♣❛r❛ q✉❛✐sq✉❡r i, j, k, l ∈ N ✳

  ∗

  ✹✳✶ C ✲á❧❣❡❜r❛s ❡stá✈❡✐s

  ❉❡✜♥✐çã♦ ✹✳✶✳✶✳ ❯♠❛ C ✲á❧❣❡❜r❛ A é ❞✐t❛ s❡r ❡stá✈❡❧ s❡ A é ✐s♦♠♦r❢❛ ❛ A ⊗ K✳ ❊①❡♠♣❧♦ ✹✳✶✳✷✳ K é ❡stá✈❡❧✳ ❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡ ♠♦str❛r q✉❡ K ❡ K ⊗ K sã♦ ✐s♦♠♦r❢❛s✳ 1 , i 2 ) P❛r❛ ✐ss♦✱ s❡❥❛ f : N → N × N ✉♠❛ ❜✐❥❡çã♦✳ ❉❡t♦♥❛r❡♠♦s ♣♦r (i ❛ ✐♠❛❣❡♠ f(i) ❞❡ ✉♠ ❡❧❡♠❡♥t♦ i ∈ N✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❛♠♦s ❞❡✜♥✐r ✉♠ ❤♦♠♦♠♦r✜s♠♦ ♥❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡♥s❛ S ❞❡ K ❣❡r❛❞❛ ♣❡❧♦ s✐st❡♠❛ ij : i, j ∈ N} ❞❡ ✉♥✐❞❛❞❡s ♠❛tr✐❝✐❛✐s {e ✳

  ϕ : S → K ⊗ K ϕ(e ij ) = ❈♦♥s✐❞❡r❡♠♦s ❛ ❛♣❧✐❝❛çã♦ ˜ ❞❛❞❛ ♣♦r ˜ e i 1 j 1 ⊗ e i 2 j 2

  ✳ ❊♥tã♦✱ ♣❛r❛ k, l ∈ N ϕ(e ˜ ij e k,l ) = [j = k] ˜ ϕ(e il ) = [j = k](e i 1 l 1 ⊗ e i 2 l 2 ). P♦r ♦✉tr♦ ❧❛❞♦✱

  ϕ(e ˜ i,j ) ˜ ϕ(e k,l ) = (e i 1 j 1 ⊗e i 2 j 2 )(e k 1 l 1 ⊗e k 2 l 2 ) = [j 1 = k 1 ][j 2 = k 2 ](e i 1 l 1 ⊗e i 2 l 2 ), = k = k ϕ(e ij ) ˜ ϕ(e kl ) =

  ❡ ❝♦♠♦ j = k s❡✱ ❡ s♦♠❡♥t❡ s❡✱ j 1 1 ❡ j 2 2 ✱ s❡❣✉❡ q✉❡ ˜ ϕ(e ˜ ij e kl )

  ✳ ❆❧é♠ ❞✐ss♦✱ ∗ ∗ ∗ ϕ((e ˜ ij ) ) = ˜ ϕ(e ji ) = e j i ⊗ e j i = (e i j ⊗ e i j ) = ˜ ϕ(e ij ) . 1 1 2 2 1 1 2 2

  ϕ ▲♦❣♦✱ ✜❝❛ ♣r♦✈❛❞♦ q✉❡ ˜ é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦✳

  ϕ Pr♦✈❡♠♦s ❛❣♦r❛ q✉❡ ˜ é ✐♥❥❡t✐✈❛✳ ❖❜s❡r✈❛♠♦s q✉❡ s❡ ❞♦✐s ♣❛r❡s

  (i, j) ❡ (k, l) sã♦ ❞✐st✐♥t♦s✱ ❞✐❣❛♠♦s✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ i 6= k 1 6= k 1 2 6= k 2 ϕ(e ij ) = e i 1 j 1 ⊗ e i 2 j 2

  ✱ ❡♥tã♦ i ♦✉ i ✱ ❞♦♥❞❡ ˜ ❡ ϕ(e ˜ kl ) = e k l ⊗ e k l 1 1 2 2 sã♦ ❞✐st✐♥t♦s ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❧✐♥❡❛r♠❡♥t❡

  ✐♥❞❡♣❡♥❞❡♥t❡s✳ ▼❛✐s ❣❡r❛❧♠❡♥t❡✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ✐♠❛❣❡♠ ❞❛ ❜❛s❡ {e ij : i, j ∈ N} ϕ

  ❞❡ S ♣❡❧♦ ❤♦♠♦♠♦r✜s♠♦ ˜ é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ ij ⊗ e kl : i, j, k, l ∈ N} ❥á q✉❡ ♦ ❝♦♥❥✉♥t♦ {e é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡

  ϕ ❡♠ K ⊗ K✳ P♦rt❛♥t♦✱ ˜ é ✐♥❥❡t✐✈❛✳

  ∗ F ⊆ S

  P❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ ✜♥✐t♦ F ⊆ N✱ t❡♠♦s ✉♠❛ C ✲á❧❣❡❜r❛ C F = { λ i,j e ij : i, j ∈ F } ϕ P ❛ss♦❝✐❛❞❛✳ ◆❛ ✈❡r❞❛❞❡✱ C ✳ ❈❧❛r♦ q✉❡ ˜ F i,j r❡str✐t❛ à C ✲á❧❣❡❜r❛ C é ✐♥❥❡t✐✈❛ ❡✱ ♣♦rt❛♥t♦✱ ✐s♦♠étr✐❝❛✳ ❈♦♠♦ S é

  ϕ ❛ ✉♥✐ã♦ ❞❡st❛s C ✲á❧❣❡❜r❛s✱ s❡❣✉❡ q✉❡ ˜ é ✐s♦♠étr✐❝♦✳

  ϕ ❉❡st❛ ❢♦r♠❛✱ ˜ s❡ ❡st❡♥❞❡ ❛ ✉♠ ❤♦♠♦r✜s♠♦ ✐s♦♠étr✐❝♦ ϕ : K →

  K ⊗ K ✳ ❯♠❛ ✈❡③ q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ϕ ❝♦♥té♠ ❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡♥s❛ ij ⊗ e kl : (i, j), (k, l) ∈ N × N}

  ❣❡r❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ {e ✱ s❡❣✉❡ q✉❡ ϕ é ✉♠ ✐s♦♠♦r✜s♠♦✳ Pr♦♣♦s✐çã♦ ✹✳✶✳✸✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛✳ ❊♥tã♦✱ A é ❡stá✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ C ✲á❧❣❡❜r❛ B t❛❧ q✉❡ A é ✐s♦♠♦r❢❛ ❛ B ⊗ K✳ ❉❡♠♦♥str❛çã♦✿ ❙❡ A é ❡stá✈❡❧✱ ♣♦r ❞❡✜♥✐çã♦ t❡♠♦s q✉❡ A é ✐s♦♠♦r❢❛ ❛ A⊗K✱ ❞♦♥❞❡ ❜❛st❛ t♦♠❛r♠♦s B = A✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ A é ✐s♦♠♦r❢❛ ❛ B ⊗ K ♣❛r❛ ❛❧❣✉♠❛ C ✲á❧❣❡❜r❛ B✱ ❡♥tã♦

  A ⊗ K ∼ = (B ⊗ K) ⊗ K ∼ = B ⊗ (K ⊗ K). ❉♦ ❊①❡♠♣❧♦ s❡❣✉❡ q✉❡ A ⊗ K ∼ = B ⊗ K ∼ = A ❡✱ ♣♦rt❛♥t♦✱ A é ❡stá✈❡❧✳

  ✹✳✷ ❋✐❜r❛❞♦s ❞❡ ❋❡❧❧ ❡stá✈❡✐s

  ◆❡st❛ s❡çã♦✱ ✈❛♠♦s ❞❡✜♥✐r ✜❜r❛❞♦s ❞❡ ❋❡❧❧ ❡stá✈❡✐s ❡ ❞❡✜♥✐r ✉♠❛ ❡s♣é❝✐❡ ❞❡ ❡st❛❜✐❧✐③❛çã♦ ❞❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧✳ ▼♦str❛r❡♠♦s q✉❡ ❛ C ✲ á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ♦❜t✐❞❛ ❛tr❛✈és ❞❡st❡ ♥♦✈♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ é✱ ♥❛ ✈❡r❞❛❞❡✱ ❛ ❡st❛❜✐❧✐③❛çã♦ ❞❛ ❛♥t✐❣❛✳

  P❛r❛ ♠♦str❛r ❛ ❜♦❛ ❞❡✜♥✐çã♦ ❞❛ ❡st❛❜✐❧✐③❛çã♦✱ ♣r❡❝✐s❛♠♦s ❞♦ ❝♦♥✲ ❝❡✐t♦ ❞❡ ✐s♦♠♦r✜s♠♦s ❡♥tr❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧✱ q✉❡ é ♥♦ss❛ ♣r✐♠❡✐r❛ ❞❡✜✲ ♥✐çã♦✳ g } g

  ❉❡✜♥✐çã♦ ✹✳✷✳✶✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❡ A = {A ∈G ✱ B = {B g } g ∈G

  ✜❜r❛❞♦s ❞❡ ❋❡❧❧✳ ❯♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ A ❡ B é ✉♠❛ ❝♦❧❡çã♦ t } t ∈G t : A t → B t ❞❡ ❛♣❧✐❝❛çõ❡s {φ t❛❧ q✉❡✱ ♣❛r❛ ❝❛❞❛ t ∈ G✱ φ é ❧✐♥❡❛r t ∈ A t s ∈ A s ❡ ✐♥❥❡t✐✈❛✱ ❡ ♣❛r❛ q✉❛✐sq✉❡r s, t ∈ G✱ a ❡ a ✱ t (A t ) = B t ✭✐✮ φ ❀ st (a s a t ) = φ s (a s )φ t (a t ) ✭✐✐✮ φ ❀ t (a ) = (φ t (a t )) −1 ∗ ∗ ✭✐✐✐✮ φ t ✳ e : A e → B e

  ❖❜s❡r✈❛çã♦ ✹✳✷✳✷✳ ❯♠❛ ✈❡③ q✉❡ φ é ✉♠ ✐s♦♠♦r✜s♠♦ t (a t )k = ka t k ❡♥tr❡ C ✲á❧❣❡❜r❛s✱ s❡❣✉❡ q✉❡ kφ ✱ ♣❛r❛ ❝❛❞❛ t ∈ G ❡ ❝❛❞❛ a t ∈ B t

  ✳ ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ❛ ❛✜r♠❛çã♦ s❡❣✉❡ ❞♦s s❡❣✉✐♥t❡s ❝á❧❝✉❧♦s✿ 2 ∗ ∗ ∗ 2 kφ t (a t )k = kφ t (a t ) φ t (a t )k = kφ e (a a t )k = ka a t k = ka t k . t t

  ❊①❡♠♣❧♦ ✹✳✷✳✸✳ ❙❡❥❛ (A, G, α) ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ♣❛r❝✐❛❧ ❡ s✉♣♦♥❤❛ q✉❡ ϕ : A → B s❡❥❛ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ C ✲á❧❣❡❜r❛s✳ ❙❡❥❛ θ = ({D } g ∈G , {θ g } g ∈G ) g ❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G ❡♠ B ♦❜t✐❞❛ ❛tr❛✈és ❞❡ ϕ ❝♦❧♦❝❛♥❞♦ ′ −1

  D g = ϕ(D g ) g = ϕ ◦ α g ◦ ϕ , g = D g δ g g = D δ g g } g ❡ θ ♣❛r❛ ❝❛❞❛ g ∈ G. ❙❡♥❞♦ A ✱ B g ✱ s❡❣✉❡ q✉❡ A = {A ∈G ❡ B = {B g } g ∈G sã♦ ✜❜r❛❞♦s ❞❡ ❋❡❧❧ ✐s♦♠♦r❢♦s✳ g : A g → B g g 7→ ϕ(a)δ g . ❉❡♠♦♥str❛çã♦✿ ❉❡✜♥✐♠♦s φ ♣♦r aδ ❖❜✲ g g g s❡r✈❛♠♦s q✉❡✱ ❝♦♠♦ a ∈ D ✱ ϕ(a) ∈ D ✱ ♣♦r ❞❡✜♥✐çã♦✱ ❞♦♥❞❡ φ ❡stá g (A g ) ⊆ B g g

  ❜❡♠ ❞❡✜♥✐❞❛ ❡ φ ✳ ❚❛♠❜é♠ é ❢á❝✐❧ ✈❡r q✉❡ φ é ❧✐♥❡❛r✳ g ∈ B g = ϕ(D g ) ❆❧é♠ ❞✐ss♦✱ ❞❛❞♦ bδ ✱ t❡♠♦s q✉❡ b ∈ D g ✳ ▲♦❣♦✱ t♦✲ −1

  (b) g g (aδ g ) = bδ g ♠❛♥❞♦ a = ϕ ✱ t❡♠♦s q✉❡ a ∈ D ❡ φ ✱ ❡ ♣♦rt❛♥t♦ φ g (A g ) = B g s δ s t δ t

  ❉❛❞♦s a ❡ a ❡♠ A✱ t❡♠♦s q✉❡ φ s (a s δ s )φ t (a t δ t ) = ϕ(a s )δ s ϕ(a t )δ t −1

  = θ s (θ s (ϕ(a s ))ϕ(a t ))δ st −1 = θ s (ϕ(α s (a s ))ϕ(a t ))δ st = θ s (ϕ(α −1 (a s )a t ))δ st s = ϕ(α s (α −1 (a s )a t ))δ st s = φ st (a s δ s a t δ t ).

  ▼❛✐s ❛✐♥❞❛✱ ∗ ∗ ∗ −1 −1 φ t (aδ t ) = (ϕ(a)δ t ) = θ t (ϕ(a) ))δ t −1 −1 −1 ∗ ∗ g } g = ϕ(α t (a ))δ t = φ t ((aδ t ) ).

  P♦rt❛♥t♦✱ {φ ∈G é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ A ❡ B✳ g } g g } g

  ❖❜s❡r✈❛çã♦ ✹✳✷✳✹✳ ❙❡❥❛♠ A = {A ∈G ❡ B = {B ∈G ✜❜r❛❞♦s ❞❡ ❋❡❧❧ ✐s♦♠♦r❢♦s✳ ❊♥tã♦ ∗ ∗ ∗ ∗ C (A) ∼ (B) (A) ∼ (B).

  = C ❡ C r = C r g } g c (A) ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ {φ ∈G ✉♠ ✐s♦♠♦r✜s♠♦ ❡ ♣❛r❛ f ∈ C ✱

  S φ(f ) : G → B t φ(f )(t) = φ t (f (t)) φ(f )

  ❝♦❧♦❝❛♠♦s e ♣♦r e ✳ ❊♥tã♦ e é c (B) c (B) c (A) t ∈G ✉♠ ❡❧❡♠❡♥t♦ ❞❡ C ✳ ❆❧é♠ ❞✐ss♦✱ s❡ g ∈ C ✱ ❡s❝♦❧❤❡♥❞♦ f ∈ C −1

  (g(t)) φ(f ) = g ❞❛❞❛ ♣♦r t 7→ φ t ✱ s❡❣✉❡ q✉❡ e ✳ t } t

  ❯s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❢❛♠í❧✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s {φ ∈G ✱ é ❢á❝✐❧ ✈❡r φ c (A) c (B) q✉❡ e é ✉♠ ∗✲✐s♦♠♦r✜s♠♦ ❡♥tr❡ ❛s ∗✲á❧❣❡❜r❛s C ❡ C ✳ ❆ss✐♠✱ ∗ ∗ e

  φ (A) (B) s❡ ❡st❡♥❞❡ ❛ ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ❞❡ C ❡♠ C q✉❡ t❡♠ ❝♦♠♦ −1 ∗ ∗ φ (B)

  ✲❤♦♠♦♠♦r✜s♠♦ ✐♥✈❡rs♦ ❛ ❡①t❡♥sã♦ ❞❡ e ❛ C ✳ ∗ ∗ ∗ (A) (B)

  ❱❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ C r ❡ C r sã♦ C ✲á❧❣❡❜r❛s ✐s♦♠♦r❢❛s✳ c (A) e ▲❡♠❜r❡♠♦s ❞❛ Pr♦♣♦s✐çã♦ q✉❡ C é ✉♠ A ✲♠ó❞✉❧♦ ❝♦♠ ♣r♦✲ ❞✉t♦ ✐♥t❡r♥♦ ❝♦♠ ❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❞❛❞❛ ♣♦r

  (ξ, a) 7→ ξa, ❡♠ q✉❡ (ξa)(t) = ξ(t)a✱ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦

  X c (B) e (ξ, η) 7→ ξ(t) η(t). t ∈G ❙✐♠✐❧❛r♠❡♥t❡✱ C é ✉♠ B ✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳ c (A)

  ❖❜s❡r✈❛♠♦s q✉❡✱ s❡ ξ ∈ C ✱

  X hξ, ξi A = ξ(t) ξ(t) e t ∈G

  !

  X ∗ ∗

  X h e φ(ξ), e φ(ξ)i B = φ t (ξ(t)) φ t (ξ(t)) = φ e ξ(t) ξ(t) , e t t ∈G ∈G ❞♦♥❞❡ s❡❣✉❡ q✉❡ kξk A = k e φ(ξ)k B . e e c (A)

  ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ ♣❛r❛ η ∈ C ✱ ✈❛❧❡ q✉❡ A : C c (A) → L(l (A)) B : C c (B) → L(l (B)) k e φ(ξ) ∗ e φ(η)k B = kξ ∗ ηk A . e e ❙❡❥❛♠ T 2 ❡ T 2 ❛s r❡♣r❡✲ c (A) c (B)

  s❡♥t❛çõ❡s r❡❣✉❧❛r❡s ❞❡ C ❡ C ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✱ ♥♦✈❛✲ φ : C c (A) → C c (B)

  ♠❡♥t❡ ✉s❛♥❞♦ q✉❡ e é ✐s♦♠♦r✜s♠♦✱ kT B ( e φ(ξ))k = sup k e φ(ξ) ∗ η k B kη k ≤1 Be e = sup k e φ(ξ) ∗ e φ(η)k B kηk ≤1 Ae e = sup kξ ∗ ηk A kηk ≤1 Ae e A (C c (A)) B (C c (B)) = kT A (ξ)k.

  P♦rt❛♥t♦✱ t❡♠♦s q✉❡ T ❡ T sã♦ ∗✲á❧❣❡❜r❛s ✐s♦♠❡✲ tr✐❝❛♠❡♥t❡ ✐s♦♠♦r❢❛s✱ ❞♦♥❞❡ t❛❧ ✐s♦♠♦r✜s♠♦ s❡ ❡st❡♥❞❡ ❛ ✉♠ ✐s♦♠♦r✲ ∗ ∗ (A) (B)

  ✜s♠♦ ❡♥tr❡ C r ❡ C r ✳ ▲❡♠❜r❡♠♦s ❞❛ Pr♦♣♦s✐çã♦ q✉❡ ♣❛r❛ ❝❛❞❛ t ∈ G✱ ❛ tr❛♥s❢♦r✲ t : B t → C c (B)

  ♠❛çã♦ ❧✐♥❡❛r j ❞❛❞❛ ♣♦r j t (b t ) | s = [s = t]b t , c (B) = ⊕ t j t (B t ) t (b t )k = ∈G 1 ♣❛r❛ t♦❞♦ s ∈ G é t❛❧ q✉❡ C ❡ ✈❛❧❡ q✉❡ kj ∗ ∗ kb t k s (b s ) ∗ j t (b t ) = j st (b s b t ) t (b t ) = j −1 (b )

  ✱ j ❡ j t t ♣❛r❛ q✉❛✐sq✉❡r s, t ∈ G t ∈ B t s ∈ B s c (B)

  ❡ b ✱ b ✳ ❈♦♠♦ ❛ ✐♥❝❧✉sã♦ ❞❡ C é ✐♥❥❡t✐✈❛✱ ❡s❝r❡✈❡♥❞♦ t t (B t ) (B) (B) = ∗ ∗ s✐♠♣❧❡s♠❡♥t❡ B ♣❛r❛ ❛ ✐♠❛❣❡♠ ❞❡ j ❡♠ C t❡♠♦s q✉❡ C ⊕ g B g ∈G ✳ g } g

  ❉❡✜♥✐çã♦ ✹✳✷✳✺✳ ❉✐③❡♠♦s q✉❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B = {B ∈G é e ❡stá✈❡❧ s❡ ❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ B é ❡stá✈❡❧✳ g } g g ⊗

  ❙❡❥❛ B = {B ∈G ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❈♦❧♦❝❛♠♦s B ⊗ K = {B K} g g ⊗ K =: span{b g ⊗ u : b g ∈ B g , u ∈ K} ⊆ C (B) ⊗ K ∈G ✱ ❡♠ q✉❡ B

  ✱ ♣❛r❛ ❝❛❞❛ g ∈ G✳ ❊♥tã♦ B⊗K ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❝❛♥ô♥✐❝❛ ❞❡ ✜❜r❛❞♦

  (B) ⊗ K ❞❡ ❋❡❧❧ ❝♦♠ ❛ ♥♦r♠❛ ❡ ❛s ♦♣❡r❛çõ❡s ❤❡r❞❛❞❛s ❞❡ C ✳ ❱❛♠♦s ♥♦s r❡❢❡r✐r ❛♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B ⊗ K ❝♦♠♦ ❛ ❡st❛❜✐❧✐③❛çã♦ ❞❡ B✳ c (B) ❖❜s❡r✈❛çã♦ ✹✳✷✳✻✳ ❙❡❥❛ ϕ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ✜❡❧ ❞❡ C ❡♠ ✉♠❛ C B = { f B g } g

  ✲á❧❣❡❜r❛ A ❡ ❞❡✜♥✐♠♦s e ∈G ✱ ❡♠ q✉❡ f B g = ϕ(B g ) ⊗ K =: span{ϕ(b g ) ⊗ u : b g ∈ B g , u ∈ K}.

  B ❉♦t❛♠♦s e ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ♦❜t✐❞❛ ❞❡ A⊗K✳ ❊♥tã♦ ♦s

  B ϕ⊗1 ✜❜r❛❞♦s ❞❡ ❋❡❧❧ B⊗K ❡ e sã♦ ✐s♦♠♦r❢♦s✱ ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ e ✱ ❡♠

  ϕ : C (B) → A q✉❡ e é ❛ ❡①t❡♥sã♦ ❞❡ ϕ ♦❜t✐❞❛ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧

  ∗

  ϕ ⊗ 1 : C (B) ⊗ K → A ⊗ K ϕ ⊗ 1)(b ⊗ u) = ϕ(b) ⊗ u ❡ e é t❛❧ q✉❡ ( e ✱ ♣❛r❛

  (B) q✉❛✐sq✉❡r b ∈ C ❡ u ∈ K✳ P n a g ⊗

  ❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡✱ ♣❛r❛ ✉♠❛ s♦♠❛ ✜♥✐t❛ i i =1 u i ✈❛❧❡ q✉❡ n n

  X X g ∈ B g i ∈ K k a g ⊗ u i k = k ϕ(a g ) ⊗ u i k, 1 i i 1 ❡♠ q✉❡ a i ❡ u ✱ ♣❛r❛ i = 1, · · · , n✳ g B g

  ❙❡ ϕ é ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ✐♥❥❡t✐✈❛ ❞❡ ⊕ ∈G ✱ ❡♠ ♣❛rt✐❝✉❧❛r t❡✲ e e ⊗ K ♠♦s q✉❡ ϕ é ✜❡❧ s♦❜r❡ ❛ C ✲á❧❣❡❜r❛ B ✳ ▲♦❣♦✱ t❡♠♦s q✉❡ B ❡ ϕ(B e ) ⊗ K sã♦ ✐s♦♠♦r❢♦s ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ ϕ ⊗ 1✳ ❆❣♦r❛✱ ❞❛❞♦ 2 ∗ a ∈ B g ⊗ K = ka ak ✱ t❡♠♦s ♣❡❧♦ C ✲❛①✐♦♠❛ q✉❡ kak ❡✱ ❛✐♥❞❛✱ a a ∈ B e ⊗ K n n n ✳ P♦rt❛♥t♦✱ 2 ! ∗

  !

  X X

  X i i i =1 =1 =1 a g ⊗ u i = a g ⊗ u i a g ⊗ u i i i i n n ! ∗ !!

  X X = (ϕ ⊗ 1) a g ⊗ u i a g ⊗ u i n i i =1 =1 2 i i

  X = ϕ(a g ) ⊗ u i , g ⊗ K B g i =1 i

  ❞♦♥❞❡ s❡❣✉❡ q✉❡ B ❡ f sã♦ ✐s♦♠♦r❢♦s ✐s♦♠❡tr✐❝❛♠❡♥t❡ ❝♦♠♦ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✱ ♣❛r❛ ❝❛❞❛ g ∈ G✳

  ❆❧é♠ ❞✐ss♦✱ é ❢á❝✐❧ ✈❡r q✉❡ ❛s ♦♣❡r❛çõ❡s ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦ ❞♦s B ϕ⊗1

  ✜❜r❛❞♦s ❞❡ ❋❡❧❧ B⊗K ❡ e sã♦ ❡q✉✐✈❛❧❡♥t❡s✱ ♣♦✐s e é ∗✲❤♦♠♦♠♦r✜s♠♦✳ B

  ❉❡st❛ ❢♦r♠❛✱ B ⊗ K ❡ e sã♦ ✐s♦♠♦r❢♦s✳ ❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ♥♦s ❞✐③ q✉❡ ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❞❛

  ❡st❛❜✐❧✐③❛çã♦ ❞❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ é ❛ ❡st❛❜✐❧✐③❛çã♦ ❞❡ s✉❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛✳ g } g Pr♦♣♦s✐çã♦ ✹✳✷✳✼✳ ❙❡❥❛ B = {B ∈G ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡ s❡❥❛ B ⊗K s✉❛ ❡st❛❜✐❧✐③❛çã♦✳ ❊♥tã♦✱ ∗ ∗ C (B) ⊗ K ∼ (B ⊗ K).

  = C (B g ⊗

  ❉❡♠♦♥str❛çã♦✿ P♦r ✉♠ ❧❛❞♦✱ t❡♠♦s ❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ı ❞❡ ⊕ g ∈G K) (B) ⊗ K

  ❡♠ C ❞❛❞❛ ♣❡❧❛ ✐♥❝❧✉sã♦✱ ❞♦♥❞❡ s❡❣✉❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡

  ∗ ∗

  (B ⊗K) (B)⊗K ✉♥✐✈❡rs❛❧ q✉❡ ❡①✐st❡ ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ eı❞❡ C ❡♠ C g B g ⊗ K q✉❡ ❝♦✐♥❝✐❞❡ ❝♦♠ ı ❡♠ ⊕ ∈G ✳

  P n a g ⊗ u i ∈ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ ❝❛❞❛ h ∈ G ❡ ✉♠❛ s♦♠❛ ✜♥✐t❛ i 1 e e

  B g ⊗ K h ⊗ L 1 h ⊗ R

  1 ✱ ❞❡✜♥✐♠♦s b ❡ b ♣♦r n n !

  X X e (b h ⊗ L 1) a g ⊗ u i = b h a g ⊗ u i 1 i i 1

  ❡ n n !

  X X g ∈ B g i ∈ K 1 a g ⊗ u i (b h ⊗ e R 1) = a g b h ⊗ u i , i i 1 ❡♠ q✉❡ a i ❡ u ✱ ♣❛r❛ i = 1, 2, . . . , n✳ λ ) λ

  ❙❡❥❛ (v ∈Λ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ K✳ ◆♦t❡♠♦s q✉❡ n n !

  X X (b h ⊗ e L 1) a g ⊗ u i = b h a g ⊗ u i 1 i i 1 n

  !

  X = lim (b h ⊗ v λ ) a g ⊗ u i λ n 1 i

  X = ≤ kb h kk a g ⊗ u i k. 1 i ❙✐♠✐❧❛r♠❡♥t❡✱ n n !

  X X e 1 a g ⊗ u i (b h ⊗ R 1) ≤ kb h kk a g ⊗ u i k. i i 1 h ⊗ L e e 1 h ⊗ R 1 h ⊗ 1 =

  ❆ss✐♠✱ b ❡ b s❡ ❡st❡♥❞❡♠ ❛ ✉♠ ♠✉❧t✐♣❧✐❝❛❞♦r b (b h ⊗ L 1, b h ⊗ R 1) g B g ⊗ K

  ❞❡ ⊕ ∈G ✳ ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ ♣r♦✈❛✲s❡ q✉❡ b h ⊗ 1 = (b h ⊗ L 1, b h ⊗ R 1) ❞❡ ❢❛t♦ é ❝♦♥tí♥✉♦ ❝♦♠ r❡❧❛çã♦ à ♥♦r♠❛

  (B ⊗ K) ✉♥✐✈❡rs❛❧✱ ❞♦♥❞❡ s❡ ❡st❡♥❞❡ ❛ ✉♠ ♠✉❧t✐♣❧✐❝❛❞♦r ❞❡ C ✱ q✉❡ ❝♦♥t✐♥✉❛r❡♠♦s ❞❡♥♦t❛♥❞♦ ❞❛ ♠❡s♠❛ ❢♦r♠❛✳ g B g

  P♦r ✜♠✱ ❝♦♥s✐❞❡r❡♠♦s ❛ r❡♣r❡s❡♥t❛çã♦  ❞❡ ⊕ ∈G ♥❛ á❧❣❡❜r❛ ❞❡ (B ⊗ K))

  ♠✉❧t✐♣❧✐❝❛❞♦r❡s M(C ❞❡✜♥✐❞❛ ♣♦r

  X X h h ∈G ∈G b h 7→ b h ⊗ 1. (B)

  ◆♦✈❛♠❡♥t❡✱  s❡ ❡st❡♥❞❡ ❛ ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ e ❞❡ C ❡♠ M (C (B ⊗ K))

  ✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ρ ❞❛ á❧❣❡❜r❛

  ∗

  (B ⊗ K)) ❞❡ ♦♣❡r❛❞♦r❡s ❝♦♠♣❛❝t♦s K ❡♠ M(C ❞❛❞❛ ♣♦r u 7→ 1 ⊗ u✳

  (B)) ❈♦♠♦ e(C ❡ ρ(K) ❝♦♠✉t❛♠✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ∗✲❤♦♠♦♠♦r✜s♠♦ π −1

  (B) ⊗ K ❞❡ C t❛❧ q✉❡ π(a ⊗ b) = e(a) ⊗ ρ(b)✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ π = eı ✳

  (B) Pr♦♣♦s✐çã♦ ✹✳✷✳✽✳ ❙❡❥❛ B ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡stá✈❡❧✳ ❊♥tã♦ C é ✉♠❛ C ✲á❧❣❡❜r❛ ❡stá✈❡❧✳ ▼❛✐s ❛✐♥❞❛✱ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ϕ : ∗ ∗ C (B) → C (B) ⊗ K t❛❧ q✉❡

  ϕ(B g ) = B g ⊗ K, ♣❛r❛ t♦❞♦ g ∈ G✳ j = 1⊗e jj ∈ M (B e ⊗ ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ❝♦♥s✐❞❡r❡♠♦s E

  P K) j E j = 1 ⊗ 1

  ✳ ❊♥tã♦✱ E é ✉♠❛ ♣r♦❥❡çã♦✱ ♣❛r❛ ❝❛❞❛ j ∈ N✱ ❡ e ⊗ K) j ❝♦♠ ❝♦♥✈❡r❣ê♥❝✐❛ ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ ❞❡ M(B ✳ ▼❛✐s ❛✐♥❞❛✱ ❛s j ♣r♦❥❡çõ❡s E sã♦ ♠✉t✉❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s ❡ ♦rt♦❣♦♥❛✐s✱ ✉♠❛ ✈❡③ q✉❡ (1 ⊗ e ij )(1 ⊗ e ji ) = E i ji )(1 ⊗ e ij ) = E j

  ❡ (1 ⊗ e ✳ e e e ⊗K ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ B s❡❥❛ ❡stá✈❡❧✳ ❆ss✐♠✱ B é ✐s♦♠♦r❢♦ ❛ B ❡✱ e ⊗K) e )

  ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ M(B ❡ M(B sã♦ ✐s♦♠♦r❢♦s✳ ❚❛❧ ✐s♦♠♦r✜s♠♦ e ⊗ K) é t❛♠❜é♠ ❝♦♥tí♥✉♦ ♥❛s r❡s♣❡❝t✐✈❛s t♦♣♦❧♦❣✐❛s ❡str✐t❛s ❞❡ M(B e ) ❡ M(B ✭Pr♦♣♦s✐çã♦ ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s ❞♦ q✉❡ ❢♦✐ ❢❡✐t♦ ❛❝✐♠❛ j ) j e ) q✉❡ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣r♦❥❡çõ❡s (E ∈N ❡♠ M(B ✱ ♠✉t✉❛♠❡♥t❡

  P E j = 1

  ❡q✉✐✈❛❧❡♥t❡s ❡ ♦rt♦❣♦♥❛✐s✱ ❡ t❛✐s q✉❡ j ❝♦♠ ❝♦♥✈❡r❣ê♥❝✐❛ ♥❛ t♦✲ e ) e (B) ♣♦❧♦❣✐❛ ❡str✐t❛ ❞❡ M(B ✳ ❖❜s❡r✈❛♥❞♦ q✉❡ ❛ ✐♥❝❧✉sã♦ ❞❡ B ❡♠ C é ♥ã♦ ❞❡❣❡♥❡r❛❞❛ ❡ ♥♦✈❛♠❡♥t❡ ✉s❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ♣♦❞❡♠♦s ❝♦♥✲ j ) j ∈N (B)) s✐❞❡r❛r ✉♠❛ s❡q✉ê♥❝✐❛ (E ❡♠ M(C s❛t✐s❢❛③❡♥❞♦ ❛s ♠❡s♠❛s ♣r♦♣r✐❡❞❛❞❡s✳ ∗ ∗

  (B) ∼ = A 1 ⊗ K 1 = E 1 C (B)E 1 ❱❛♠♦s ♠♦str❛r q✉❡ C ✱ ❡♠ q✉❡ A ✳ n = E j n = p n C (B)p n P n ❙❡❥❛ p ❡ ❝♦❧♦❝❛♠♦s A ✳ ◆♦t❡♠♦s q✉❡ 1 A n ⊆ A n +1 n

  ✱ ♣♦✐s s❡ b ∈ A ✱ t❡♠♦s q✉❡ n : A n → A n p n bp n = p n p n bp n p n = p n bp n = b. +1 +1 +1 +1 ❙❡❥❛ ϕ +1 ❛ ✐♥❝❧✉sã♦ ❡ s❡❥❛ n

  ψ : A n → A ⊗ K 1 X b 7→ v bv j ⊗ e ij , 1≤i,j≤n i , v , . . . (B)) = E

  ❡♠ q✉❡ v ∗ ∗ 1 2 sã♦ ✐s♦♠❡tr✐❛s ♣❛r❝✐❛✐s ❡♠ M(C ❝♦♠ v 1 n 1 ✱ v v j = E j v = E j j 1 ❡ v j ✱ ♣❛r❛ j ≥ 2✳ ❱❛♠♦s ♠♦str❛r q✉❡ ψ é ✉♠

  ✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t♦r✳

  j = v j v v j = v j E = v v j v = E v ∗ ∗ ∗ ∗ ∗

  ❈♦♠♦ v j n n 1 ❡ v j j j 1 j ✱ ♣❛r❛ ❝❛❞❛ j✱ t❡♠♦s q✉❡ ψ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ P❛r❛ ✈❡r q✉❡ ψ é ∗✲❤♦♠♦♠♦r✜s♠♦✱ n s❡❥❛♠ a, b ∈ A ✳ ❊♥tã♦✱     n n

  X ∗ ∗

  X    

  ψ (a)ψ (b) = v av j ⊗ e ij v bv l ⊗ e kl 1≤i,j≤n 1≤k,l≤n i k n !

  X ∗ ∗

  X = v a v k v bv l ⊗ e il 1≤i,l≤n =1 i k k

  X = v ap n bv l ⊗ e il 1≤i,l≤n i

  X n = v abv l ⊗ e il = ψ (ab). 1≤i,l≤n i ▼❛✐s ❛✐♥❞❛✱ n n ∗ ∗ ∗ ∗

  X ψ (a) = v a v i ⊗ e ji = ψ (a ). n 1≤i,j≤n j n ▼♦str❡♠♦s q✉❡ ψ é ✐♥❥❡t♦r✳ ❙❡❥❛ a ∈ A t❛❧ q✉❡ n ∗

  X ψ (a) = v av j ⊗ e ij = 0. 1≤i,j≤n i av j = 0, ■ss♦ ♥♦s ❞✐③ q✉❡ v i ♣❛r❛ 1 ≤ i, j ≤ n✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ i i aE j = 0 ♣♦r v ♣❡❧❛ ❡sq✉❡r❞❛ ❡ ♣♦r v j ♣❡❧❛ ❞✐r❡✐t❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ E ✱ n ap n = ( E j ) a ( E j ) = 0 P n P n ♣❛r❛ 1 ≤ i, j ≤ n✳ ▲♦❣♦✱ a = p ✳ n 1 1

  ❖❜s❡r✈❛♠♦s q✉❡ s❡ a ∈ A ❡ j = n + 1✱ ❡♥tã♦ ∗ ∗ ∗ av j = av j v v j = aE n v n = 0 j +1 +1 a = v v j v a = 0 ❡ s✐♠✐❧❛r♠❡♥t❡✱ v j j j ✳ P♦rt❛♥t♦✱ ♦ ❞✐❛❣r❛♠❛ ϕ n

  A n // A n ψ n ## +1 ψ n+1 A ⊗ K 1

  ❝♦♠✉t❛✳ n , ϕ n ) ❉❡st❛ ❢♦r♠❛✱ s❡♥❞♦ A ♦ ❧✐♠✐t❡ ❞✐r❡t♦ ❞❛ s❡q✉ê♥❝✐❛ (A n ✱ =1

  ⊗ K ❡①✐st❡ ✉♠ ú♥✐❝♦ ∗✲❤♦♠♦♠♦r✜s♠♦ ψ : A → A 1 t❛❧ q✉❡✱ ♣❛r❛ ❝❛❞❛ n ∈ N

  ✱ ♦ ❞✐❛❣r❛♠❛ ϕ n A n // A ψ n ## ψ n A ⊗ K 1 n

  ❝♦♠✉t❛✱ ❡♠ q✉❡ ϕ é ❛ ✐♥❝❧✉sã♦ ♥❛t✉r❛❧ ❞❡ A ❡♠ A ✭❚❡♦r❡♠❛ ✻✳✶✳✷✱

  A n ◆♦ ❡♥t❛♥t♦✱ ♥❡st❡ ❝❛s♦ A é ❡①❛t❛♠❡♥t❡ ∪ ✭❖❜s❡r✈❛çã♦ ✻✳✶✳✸✱ n = p n C (B)p n n ) n n =1

  ❯♠❛ ✈❡③ q✉❡ A ❡ ❛ s❡q✉ê♥❝✐❛ (p ∈N ❝♦♥✈❡r❣❡ ∗ ∗ (B)) (B)

  ❛ 1 ❡str✐t❛♠❡♥t❡ ❡♠ M(C ✱ s❡❣✉❡ q✉❡ A = C ✳ ❆ss✐♠✱ r❡st❛ (B)

  ♣r♦✈❛r♠♦s q✉❡ ψ é ✉♠ ✐s♦♠♦r✜s♠♦ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡ C é ❡stá✈❡❧✳ n ❉❡ ❢❛t♦✱ ψ é ✐♥❥❡t✐✈❛ ♣♦✐s ❝❛❞❛ ψ é ✐s♦♠étr✐❝❛✳ ❱❛♠♦s ♣r♦✈❛r q✉❡

  ψ é s♦❜r❡❥❡t♦r❛✳ kl ∈ A ⊗ K bE = ∗ ∗ ∗ ❙❡❥❛♠ k, l ∈ N ❡ b ⊗ e 1 ✳ ❚❡♠♦s q✉❡ b = E 1 1 v v k bv v l k bv n k l ✳ ❚♦♠❛♠♦s a = v l ❡ t❡♠♦s q✉❡ a ∈ A ✱ ❡♠ q✉❡ n = max{k, l} v j = 0

  ✳ ❯♠❛ ✈❡③ q✉❡ v i ♣❛r❛ i 6= j✱ t❡♠♦s n

  X ∗ ∗ ψ(a) = ψ (a) = v v k bv v j ⊗ e ij = b ⊗ e kl . 1≤i,j≤n 1 ⊗K i l ❈♦♠♦ ❡st❡s ❡❧❡♠❡♥t♦s ❣❡r❛♠ A ✱ s❡❣✉❡ q✉❡ ψ é ✉♠ ∗✲✐s♦♠♦r✜s♠♦

  (B) ❡✱ ♣♦rt❛♥t♦✱ C é ❡stá✈❡❧✳ j ∈ M (B e ) ⊆ M (C (B))

  P♦r ✜♠✱ s❡❥❛ g ∈ G✳ ❈♦♠♦ v ✱ ♣❛r❛ ❝❛❞❛ j ∈ N g ) = E B g E ⊗ K s ✱ t❡♠♦s ψ(B 1 1 ✱ ♣♦✐s ♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s v j ♥ã♦ 1 ⊗K 1 ⊗K⊗K

  ❛❧t❡r❛♠ ❛ ✜❜r❛ g✳ ❆❧é♠ ❞✐ss♦✱ ♦ ✐s♦♠♦r✜s♠♦ ϕ ❡♥tr❡ A ❡ A ❝♦♥str✉í❞♦ ♥❛ Pr♦♣♦s✐çã♦ é t❛❧ q✉❡

  ϕ (E 1 B g E 1 ⊗ K) = E 1 B g E 1 ⊗ K ⊗ K.

  (B) ⊗ K g B g E ⊗ K ❆ss✐♠✱ ✐❞❡♥t✐✜❝❛♥❞♦ C ❝♦♠ A ∗ ∗ 1 ❡ B ❝♦♠ E 1 1 ✱ ♦

  (B) (B) ⊗ K g ) = B g ⊗ K ✐s♦♠♦r✜s♠♦ ϕ ❡♥tr❡ C ❡ C é t❛❧ q✉❡ ϕ (B ✳

  ✹✳✸ Pr♦❥❡çõ❡s ❝❤❡✐❛s

  ◆❡st❛ s❡çã♦✱ ✈❛♠♦s ✈❡r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s s❛t✐s❢❡✐t❛s ♣♦r ♣r♦❥❡✲ çõ❡s ❝❤❡✐❛s ❡ ♠♦str❛r q✉❡ s❡ p ∈ M(A) é ❝❤❡✐❛✱ ❡♠ ❝❛s♦ ❞❡ ✉♠❛ C ✲ á❧❣❡❜r❛ ❝♦♠ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦ ♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❝♦♠ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ s❡q✉❡♥❝✐❛❧✱ ♦❜t❡♠♦s ✉♠❛ ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ v

  

∗ ∗

  v = 1 = p ⊗ 1 ❡♠ M(A⊗K) s❛t✐s❢❛③❡♥❞♦ v ❡ vv ✳ ❚❛❧ ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ ❢♦r♥❡❝❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡stá✈❡❧ ❡♥tr❡ A ❡ ♦ ❝❛♥t♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r p✳ ❊st❡ r❡s✉❧t❛❞♦✱ ❛ss✐♠ ❝♦♠♦ ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡st❛ s❡çã♦✱ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❆ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ ❞❡ ❇r♦✇♥✲●r❡❡♥✲❘✐❡✛❡❧ é ♦❜t✐❞❛ ❞❡

  ❉❡✜♥✐çã♦ ✹✳✸✳✶✳ ❯♠ ❝❛♥t♦ ❞❡ A é ✉♠❛ C ✲á❧❣❡❜r❛ ❞❛ ❢♦r♠❛ pAp✱ ♣❛r❛ ❛❧❣✉♠❛ ♣r♦❥❡çã♦ p ❡♠ M(A)✳ jj )(A ⊗ K)(1 ⊗ e jj ) = A ⊗ e jj ❊①❡♠♣❧♦ ✹✳✸✳✷✳ ❯♠❛ ✈❡③ q✉❡ (1 ⊗ e ✱ ♣❛r❛ q✉❛❧q✉❡r j ∈ N✱ ♣♦❞❡♠♦s ✈❡r A ❝♦♠♦ ✉♠ ❝❛♥t♦ ❞❡ A ⊗ K✳ ∗ ∗

  ❯♠❛ C ✲s✉❜á❧❣❡❜r❛ ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ A é ❞✐t❛ s❡r ❝❤❡✐❛ s❡ ♥ã♦ ❡stá ❝♦♥t✐❞❛ ❡♠ ♥❡♥❤✉♠ ✐❞❡❛❧ ♣ró♣r✐♦ ❞❡ A✳ ▲❡♠❛ ✹✳✸✳✸✳ ❙❡❥❛ p ✉♠❛ ♣r♦❥❡çã♦ ❡♠ M(A)✳ ❊♥tã♦ sã♦ ❡q✉✐✈❛❧❡♥t❡s✿ ✭✐✮ P❛r❛ q✉❛❧q✉❡r r❡♣r❡s❡♥t❛çã♦ ♥ã♦ ❞❡❣❡♥❡r❛❞❛ π ❞❡ A✱ t❡♠✲s❡ ˜π(p) 6=

  ✱ ❡♠ q✉❡ ˜π ❞❡♥♦t❛ ❛ ú♥✐❝❛ ❡①t❡♥sã♦ ✉♥✐t❛❧ ❞❡ π ❛ M(A)❀ ✭✐✐✮ P❛r❛ q✉❛❧q✉❡r r❡♣r❡s❡♥t❛çã♦ ✐rr❡❞✉tí✈❡❧ π ❞❡ A✱ t❡♠✲s❡ ˜π(p) 6= 0❀ ✭✐✐✐✮ pAp é ❝❤❡✐❛❀ ✭✐✈✮ ApA = A❀ ✭✈✮ pA ♥ã♦ ❡stá ❝♦♥t✐❞♦ ❡♠ ♥❡♥❤✉♠ ✐❞❡❛❧ ♣ró♣r✐♦ ❢❡❝❤❛❞♦ ❞❡ A❀ ✭✈✐✮ Ap ♥ã♦ ❡stá ❝♦♥t✐❞♦ ❡♠ ♥❡♥❤✉♠ ✐❞❡❛❧ ♣ró♣r✐♦ ❢❡❝❤❛❞♦ ❞❡ A❀ ✭✈✐✐✮ ❖ ✐❞❡❛❧ ❢❡❝❤❛❞♦ ❞❡ M(A) ❣❡r❛❞♦ ♣♦r p ❝♦♥té♠ A❀ ✭✈✐✐✐✮ p ♥ã♦ ♣❡rt❡♥❝❡ ❛ ♥❡♥❤✉♠ ✐❞❡❛❧ ♣ró♣r✐♦ ❡str✐t❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❞❡ M (A)

  ✳ ❉❡♠♦♥str❛çã♦✿ P❛r❛ ✭✐✮⇒✭✐✐✮ ♥ã♦ ❤á ♥❛❞❛ ❛ ❢❛③❡r✳ ❙❡ pAp ♥ã♦ é ❝❤❡✐❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✺✳✸✳✸ ❞❡ ❡①✐st❡ ✉♠ ❡st❛❞♦ ♣✉r♦ ρ ❞❡ A t❛❧ q✉❡ pAp ⊆ I ⊆ N ρ ρ

  ✱ ❡♠ q✉❡ I ❞❡♥♦t❛ ♦ ✐❞❡❛❧ ❢❡❝❤❛❞♦ ❣❡r❛❞♦ ♣♦r pAp ❡ N é ♦ ♥ú❝❧❡♦ ❞♦ ❡st❛❞♦ ♣✉r♦ ρ✳ ❈♦♠♦ ρ é ✉♠ ❡st❛❞♦ ♣✉r♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✺✳✶✳✻ ρ , ϕ ρ ) ❞❡ ❛ r❡♣r❡s❡♥t❛çã♦ GNS (H ❛ss♦❝✐❛❞❛ ❛ ρ é ✐rr❡❞✉tí✈❡❧✳ ❊st❛ r❡♣r❡s❡♥t❛çã♦ s❡ ❛♥✉❧❛ ❡♠ I ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❡♠ pAp ❡ t❡r❡♠♦s ϕ ρ (p) = 0 f ✱ ❝♦♥tr❛❞✐③❡♥❞♦ ❛ ❤✐♣ót❡s❡✳ ❉♦♥❞❡ s❡❣✉❡ ✭✐✐✮⇒✭✐✐✐✮✳

  P❛r❛ ✭✐✐✐✮⇒✭✐✈✮✱ é s✉✜❝✐❡♥t❡ ♦❜s❡r✈❛♠♦s q✉❡ pAp = pA ∩ Ap ❡✱ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ A✱ s❡❣✉❡ q✉❡ ApA ⊇ pA ⊇ pAp ✳ ❯t✐❧✐③❛♥❞♦ ♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦✱ ♦❜t❡♠♦s ❛ ✐♠♣❧✐❝❛çã♦ ✭✐✈✮⇒✭✈✮✳

  = Ap ✭✈✮⇒✭✈✐✮ s❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞♦ ❢❛t♦ ❞❡ (pA) ❡ q✉❛❧q✉❡r ✐❞❡❛❧ ❢❡❝❤❛❞♦ ❞❡ A é ❛✉t♦❛❞❥✉♥t♦✳ ❈♦♠♦ ♦ ✐❞❡❛❧ ❢❡❝❤❛❞♦ ❣❡r❛❞♦ ♣♦r p ❝♦♥té♠ Ap

  ❡ pA✱ t❡♠♦s ✭✈✐✮⇒✭✈✐✐✮ ❡ ✉♠❛ ✈❡③ q✉❡ A é ❞❡♥s♦ ❡♠ M(A) ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛✱ t❡♠♦s ✭✈✐✐✮⇒✭✈✐✐✐✮✳ ❘❡st❛ ♣r♦✈❛r♠♦s ❛ ✐♠♣❧✐❝❛çã♦ ✭✈✐✐✐✮⇒✭✐✮✳ ❖❜s❡r✈❛♠♦s q✉❡✱ s❡ π é

  ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♥ã♦ ❞❡❣❡♥❡r❛❞❛ ❞❡ A s♦❜r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H

  ✱ ❡♥tã♦ ker(˜π) é ❢❡❝❤❛❞♦ ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ ❞❡ M(A)✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ π λ ) λ λ → x

  é ♥ã♦ ❞❡❣❡♥❡r❛❞❛✱ s❡ ✉♠ ♥❡t (x ∈Λ ❡♠ ker(˜π) é t❛❧ q✉❡ x ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ ❞❡ M(A)✱ ❡♠ q✉❡ x ∈ M(A)✱ s❡❣✉❡ q✉❡ 0 = ˜ π(x λ )π(a)h = ˜ π(x λ a)h → ˜ π(xa)h = ˜ π(x)π(a)h,

  ♣❛r❛ a ∈ A ❡ h ∈ H✳ ▲♦❣♦✱ x ∈ ker(˜π) ❡ ker(˜π) é ❢❡❝❤❛❞♦ ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ ❞❡ M(A)✳ P♦rt❛♥t♦ s❡❣✉❡ ✭✈✐✐✮⇒✭✐✮✱ ❝♦♥❝❧✉✐♥❞♦ ❛ ♣r♦✈❛✳ ❉❡✜♥✐çã♦ ✹✳✸✳✹✳ ❯♠❛ ♣r♦❥❡çã♦ p ❡♠ M(A) é ❞✐t❛ s❡r ❝❤❡✐❛ s❡ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❞♦ ▲❡♠❛ ❊①❡♠♣❧♦ ✹✳✸✳✺✳ ❆ ♣r♦❥❡çã♦ ✐❞❡♥t✐❞❛❞❡ é ❝❤❡✐❛✱ ♣❛r❛ q✉❛❧q✉❡r C ✲

  P e ii ∈ K á❧❣❡❜r❛ ❡ s❡ F ⊆ N é ✜♥✐t♦ ❡ ♥ã♦ ✈❛③✐♦✱ ❛ ♣r♦❥❡çã♦ ✱ é ✉♠❛ i ∈F ♣r♦❥❡çã♦ ❝❤❡✐❛✳

  P e ii ∈ K ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ s❡ p = ✱ ♦❜s❡r✈❛♠♦s q✉❡ q✉❛❧q✉❡r il i ∈F ❡❧❡♠❡♥t♦ ❞❛ ❢♦r♠❛ e ✱ ❝♦♠ i ∈ F ❡ j ∈ N✱ ♣❡rt❡♥❝❡ ❛ pK✳ ❆ss✐♠✱ s❡ I é ✉♠ ✐❞❡❛❧ ❢❡❝❤❛❞♦ ❞❡ A ❝♦♥t❡♥❞♦ pK s❡❣✉❡ q✉❡ q✉❛❧q✉❡r ♠❛tr✐③ ✉♥✐❞❛❞❡ e kl ∈ I

  ✳ ❈♦♠♦ ❛s ♠❛tr✐③❡s ✉♥✐❞❛❞❡ ❣❡r❛♠ K✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳ Pr♦♣♦s✐çã♦ ✹✳✸✳✻✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛ ❡ I ✉♠ ✐❞❡❛❧ ❞❡ A q✉❡ é ❞❡♥s♦ ❡♠ A✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ❝r❡s❝❡♥t❡ ❞❡ A ❝♦♥s✐st✐♥❞♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ I✳

  A ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ˜ ❛ ✉♥✐t✐③❛çã♦ ❞❡ A ❡ s❡❥❛ Λ ♦ ❝♦♥❥✉♥t♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ✜♥✐t♦s ❞❡ I ♦r❞❡♥❛❞♦s ♣♦r ✐♥❝❧✉sã♦✳ 1 , x , . . . , x n } ∈ Λ 2 P❛r❛ λ = {x ✱ ❝♦❧♦❝❛♠♦s ∗ ∗ v λ = x x + · · · + x n x ∈ I 1 n 1

  ❡ −1

  1 u λ = v λ + v λ . n

  λ + v λ A

  1

  ❯♠❛ ✈❡③ q✉❡ v é ♣♦s✐t✐✈♦ ❡✱ ♣♦rt❛♥t♦✱ é ✐♥✈❡rtí✈❡❧ ❡♠ ˜ ✱ λ n t❡♠♦s q✉❡ u ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ I é ✉♠ ✐❞❡❛❧ ❡♠ ˜

  A λ ∈ I ✱ s❡❣✉❡ q✉❡ u ✳ ❯♠❛ ✈❡③ q✉❡ ❛ ❢✉♥çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧ 1 −1 t 7→ t + t n só ❛♣r❡s❡♥t❛ ✈❛❧♦r❡s ❡♥tr❡ 0 ❡ 1 ♣❛r❛ t ≥ 0✱ t❡♠♦s q✉❡ 0 ≤ u λ ≤ 1 λ ) λ

  ✳ ❱❛♠♦s ♠♦str❛r q✉❡ (u ∈Λ é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ A✳ n ❚❡♠♦s✱ ♣❛r❛ λ ∈ Λ✱

  X i =1 ((u λ − 1)x i )((u λ − 1)x i ) = (u λ − 1)v λ (u λ − 1) −1 !

  1 = v λ + v λ − 1 v λ n −1 !

  1 v λ + v λ − 1 n −2 ! 2

  1 = v λ v + v λ λ n −1 !

  1

  • v λ −2v λ + v λ + 1 n 2 −2

  ! 2

  1

  1

  1 λ + = v λ v − 2v λ + v λ + v λ + v λ −2 n n n v λ 1 = + v λ . 2 n n 1 −2

  • t ❆❣♦r❛✱ ❛ ❢✉♥çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧ t 7→ t é s❡♠♣r❡ n n
  • 1 2 n t n + 2 + t ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ ✳ ❉❡ ❢❛t♦✱ ❞❡r✐✈❛♥❞♦ ❛ ❢✉♥çã♦ t 7→ ❝♦♠

      4 1 r❡❧❛çã♦ ❛ t ♦❜t❡♠♦s q✉❡ t = é ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ♥♦ ✐♥t❡r✈❛❧♦ (0, +∞)✱ n

      1 1 −2

    • t ♦✉ s❡❥❛✱ t = é ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ♣❛r❛ ❛ ❢✉♥çã♦ t 7→ t ❡♠ n n

      [0, +∞) ✳ ❉♦♥❞❡ −2 −2

      1

      1

      1 1 n t + + t ≤ = . n n n n

      4

      ▲♦❣♦✱ n

      X

      1 ((u λ − 1)x i )((u λ − 1)x i ) ≤ . i =1 4n λ − 1)x i )((u λ − 1)x i ) ≤

      1 P❛r❛ i = 1, 2, . . . , n ❞❡❞✉③✐♠♦s q✉❡ ((u 4n

      ❡ s❡❣✉❡ q✉❡ 2

      1 k(u λ − 1)x i k ≤ . 4n

      (u λ x) = x λ k ≤ 1 ❆ss✐♠✱ ♣❛r❛ x ∈ I ❝♦♥❝❧✉í♠♦s q✉❡ lim ✳ ❈♦♠♦ ku ✱ λ λ ) λ ∈Λ

      ♣❛r❛ t♦❞♦ λ ∈ Λ✱ ❡ I é ❞❡♥s♦ ❡♠ A✱ t❡♠♦s q✉❡ (u é ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ A✳ λ ) λ

      ❋✐♥❛❧♠❡♥t❡✱ ♣r♦✈❡♠♦s q✉❡ (u ∈Λ é ❝r❡s❝❡♥t❡✳ ❙❡❥❛♠ λ, µ ∈ Λ t❛✐s , x , . . . , x n } , x , . . . , x p } q✉❡ λ ≤ µ✳ ❚❡♠♦s λ = {x λ ≤ v µ 1 2 ❡ µ = {x 1 2 ✱ n ≤ p ❡

      ❛ss✐♠✱ v ❡ −1 −1

      1

      1 + v λ ≥ + v µ . n n

      P❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ t ≥ 0 t❡♠♦s −1 −1

      1

      1

      1

      1

    • t ≥ + t , n n p p

      ❞♦♥❞❡ −1 −1

      1

      1

      1

      1

    • v µ ≥ + v µ , n n p p

      ♦ q✉❡ ✐♠♣❧✐❝❛ −1 −1 −1

      1

      1

      1

      1

      1

      1 1 − + v λ ≤ 1 − + v µ ≤ 1 − + v µ .

      ✭✶✮ n n n n p p ◆♦ ❡♥t❛♥t♦✱ −1 !

      1

      1

      1 v λ + v + 1 − λ = v λ , n n n

      ♦✉ s❡❥❛✱

      −1 −1 ! −1

      1

      1

      1

      1

      1

    • v λ + v λ = v λ 1 − + v + λ v λ n n n n n −1

      1

      1 = 1 − + v λ . n n

      ❉❡st❛ ❢♦r♠❛✱ ❝♦♥❝❧✉í♠♦s ❞❡ q✉❡ −1 −1

      1

      1 u λ = v λ + v λ ≤ v µ + v µ = u µ λ ) λ n n ❡ (u ∈Λ é ❝r❡s❝❡♥t❡✳

      ❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❣❡♥❡r❛❧✐③❛ ♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ♣r♦✈❛r✿ ❚❡♦r❡♠❛ ✹✳✸✳✼✳ ❙❡ R é ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❞❡ A ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❢❡❝❤❛❞♦✮ q✉❡ ❣❡r❛ ✉♠ ✐❞❡❛❧ ❞❡♥s♦ ❡♠ A✱ ❡♥tã♦ A ♣♦ss✉✐ ✉♠❛ ✉♥✐❞❛❞❡

      P r r j ❛♣r♦①✐♠❛❞❛ ❝r❡s❝❡♥t❡ ❝♦♥s✐st✐♥❞♦ ❞❡ s♦♠❛s ✜♥✐t❛s ❞❛ ❢♦r♠❛ j ✱ r j ∈ R

      ✳ ∗ ∗ ∗ R R =: span{r s :

      ❉❡♠♦♥str❛çã♦✿ ❖ ✐❞❡❛❧ ❢❡❝❤❛❞♦ R ✭❡♠ q✉❡ R r, s ∈ R} λ ) λ ✮ ❝♦♥té♠ R✱ ♣♦✐s s❡ (u ∈Λ é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ 2 ∗ ∗ R R λ k = k(r − u λ r )(r − ru λ )k → 0

      ❡ r ∈ R✱ t❡♠♦s kr − ru ✳ ▲♦❣♦✱ R

      ♦ ✐❞❡❛❧ I = R é ❞❡♥s♦ ❡♠ A✳ P❡❧❛ Pr♦♣♦s✐çã♦ A ♣♦ss✉✐ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ❝r❡s❝❡♥t❡ ❝♦♥s✐st✐♥❞♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ I✳ ❱❛♠♦s ♠♦str❛r q✉❡ ♣♦❞❡♠♦s t♦♠❛r ❡st❡s ❡❧❡♠❡♥t♦s ❝♦♠♦ ❞❡s❡❥❛❞♦✳

      A ❙❡❥❛✱ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à Pr♦♣♦s✐çã♦ ˜ ❛ ✉♥✐t✐③❛çã♦ ❞❡ A ❡ Λ

      ♦ ❝♦♥❥✉♥t♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ✜♥✐t♦s ❞❡ R ♦r❞❡♥❛❞♦s ♣♦r ✐♥❝❧✉sã♦✳ , x , . . . , x n } ∈ Λ

      P❛r❛ λ = {x 1 2 ✱ ❝♦❧♦❝❛♠♦s ∗ ∗ v λ = x x + · · · + x x n ∈ I 1 1 n ❡✱ ♥♦✈❛♠❡♥t❡✱ −1

      1 u λ = v λ + v λ . n n 2 1 λ = r r j j = x j + v λ P

      1 ◆♦t❡♠♦s q✉❡ u j ✱ ❡♠ q✉❡ r ✳ P❡❧♦ j n =1 λ k ≤ 1 q✉❡ ✜③❡♠♦s ♥❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ ku ♣❛r❛ t♦❞♦ λ ∈ Λ λ ) λ λ ) λ

      ❡ (u ∈Λ é ❝r❡s❝❡♥t❡✳ ❉❡st❛ ❢♦r♠❛✱ r❡st❛ ♣r♦✈❛r♠♦s q✉❡ (u ∈Λ é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ A✱ ✉♠❛ ✈❡③ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ í♥❞✐❝❡s Λ

      ❝♦♥s✐❞❡r❛❞♦ ❛q✉✐ é ❢♦r♠❛❞♦ ♣♦r ❝♦♥❥✉♥t♦s ✜♥✐t♦s ❞❡ R✱ ❡ ♥ã♦ ♠❛✐s ❞♦ R

      ✐❞❡❛❧ R ✳ (a − b) = ∗ ∗ ∗ ∗ ❖❜s❡r✈❛♠♦s q✉❡✱ ❞❛❞♦s a, b ∈ A✱ ❡♥tã♦ 0 ≤ (a − b) a a + b b − a b − b a

      ♦ q✉❡ ✐♠♣❧✐❝❛ ∗ ∗ ∗ ∗ a b + b a ≤ a a + b b. ✭†✮ s

      ❙❡ x, y ∈ I sã♦ ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛ r ❝♦♠ r, s ∈ R✱ ❞✐❣❛♠♦s ∗ ′∗ ′ ′∗ ′ x = r s s s ❡ y = r ✱ ❛♣❧✐❝❛♥❞♦ ♣❛r❛ a = s ❡ b = rr ✱ ❝♦♥❝❧✉í♠♦s q✉❡ ∗ ∗ ∗ ∗ x y + y x ≤ a a + b b, ′∗ ′ ✭‡✮ s

      ❡♠ q✉❡ a = s ❡ b = rr ♣❡rt❡♥❝❡♠ ❛ R✳ ▼❛✐s ❛✐♥❞❛✱ ❞❛❞♦ ✉♠ x ❡❧❡♠❡♥t♦ ❛r❜✐trár✐♦ x ❡♠ I✱ ❛♣❧✐❝❛♥❞♦ ♥❛s ♣❛r❝❡❧❛s ❞❡ x ✱ ♦❜t❡♠♦s n j x ≤ c c j ∗ ∗ P q✉❡ ❡①✐st❡♠ c ✬s ❡♠ R✱ j = 1, . . . , n ✱ t❛✐s q✉❡ x j ✳ n j =1

      P = {c , . . . , c n } λ = c c j

      ▲♦❣♦✱ t♦♠❛♥❞♦ λ 2 ∗ 1 ✱ t❡♠♦s v j ❡ j =1 kx(1 − u λ )k = k(1 − u λ )x x(1 − u λ )k n

      X ≤ k(1 − u λ ) c c j (1 − u λ )k j =1 j

      1 = k(1 − u λ )v λ (1 − u λ )k ≤ ,

      4n ❝♦♠♦ ❥á ❥✉st✐✜❝❛❞♦ ♥❛ Pr♦♣♦s✐çã♦ ❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ λ ≥ λ ✱ u λ x = x ❞♦♥❞❡ s❡❣✉❡ q✉❡ lim ✱ ❝♦♠ x ∈ I✳ ❈♦♠♦ I é ❞❡♥s♦ ❡♠ A✱ λ ) λ λ ❝♦♥❝❧✉í♠♦s q✉❡ (u ∈Λ é ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ A✳ ▲❡♠❛ ✹✳✸✳✽✳ ❙❡ p é ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛ ❡♠ M(A)✱ e ∈ A ❡ ε > 0✱ n

      P , a , . . . , a n ∈ A a pa i ≤ 1

      ❡♥tã♦ ❡①✐st❡♠ a 1 2 t❛✐s q✉❡ ❡ n i =1 i

      X k(1 − a pa i )ek < ε. i =1 i ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ p ∈ M(A) é ♣r♦❥❡çã♦ ❝❤❡✐❛✱ R = pA é ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❞❡ A q✉❡ ❣❡r❛ ✉♠ ✐❞❡❛❧ ❞❡♥s♦ ❡♠ A✳ ❙❡ e ∈ A ❡ ε > 0✱ ♣❡❧♦ ❚❡♦r❡♠❛ A ♣♦ss✉✐ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ❢♦r♠❛❞❛ ♣♦r P a pa i , . . . , a n ∈ A ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛ i ❡✱ ♣♦rt❛♥t♦✱ ♣♦❞❡♠♦s ♦❜t❡r a n 1 P a pa j ≤ 1 t❛✐s q✉❡ j ❡ j =1

        n

      X 1 − a pa j  e < ε. j =1 j ▲❡♠❛ ✹✳✸✳✾✳ ❙❡❥❛ p ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛ ❡♠ M(A) ❡ s✉♣♦♥❤❛ q✉❡ A ♣♦ss✉❛ ✉♠ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦ e✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥✲ n ) n ∈N a pa i = 1 P ❝✐❛ (a ❡♠ A t❛❧ q✉❡ i ✱ ❝♦♠ ❝♦♥✈❡r❣ê♥❝✐❛ ♥❛ t♦♣♦❧♦❣✐❛ i =1 ❡str✐t❛ ❞❡ M(A)✳

      < n < . . . < n k ❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ❝♦♥str✉✐r r❡❝✉rs✐✈❛♠❡♥t❡ n i k 1 2

      ❡ a ✱ 1 ≤ i ≤ n ✱ t❛✐s q✉❡ n k

      X s k = a pa i ≤ 1 i =1 i

      1 k(1 − s k )ek ≤ . 1 k P❛r❛ k = 1✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ s s❛t✐s❢❛③❡♥❞♦ t❛✐s ❝♦♥❞✐çõ❡s s❡❣✉❡ ❞♦

      ▲❡♠❛ 1 , b 2 , . . . b m = b pb j ≤ 1 ′ ∗ P ♣♦❞❡♠♦s ❡s❝♦❧❤❡r b t❛✐s q✉❡ s j ❡ 1 1 2 2

      1 k(1 − s k ) (1 − s )(1 − s k ) ek < . k = n k + m n = b j (1 − s k ) 2 1 k + 1 ❙❡❥❛ n +1 ❡ a k +j ✱ ♣❛r❛ 1 ≤ j ≤ m✳ ❆ss✐♠✱ m

      X 2 1 2 1 1 − s k +1 = (1 − s k ) − (1 − s k ) b j pb j (1 − s k ) j =1 1   m 1 2 X 2 k ≤ 1 = (1 − s k ) 1 − b pb j  (1 − s k ) ≥ 0, j =1 j ❞♦♥❞❡ s +1 ✳

      ❆❧é♠ ❞✐ss♦✱ m

      X 2 1 2 1 k(1 − s k +1 )ek = k(1 − s k )e − (1 − s k ) b j pb j (1 − s k ) ek 1 j =1 1 2 2

      1 = k(1 − s k ) (1 − s )(1 − s k ) ek < . k + 1 s k = 1

      ❆❣♦r❛✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ lim ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ ❞❡ M(A)✳ k →∞ ❉❡ ❢❛t♦✱ ❝♦♠♦ e é ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✱ t❡♠♦s q✉❡ eA é ❞❡♥s♦ ❡♠ A

      ✭Pr♦♣♦s✐çã♦ ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ Ae é ❞❡♥s♦ ❡♠ A✳ ❉❡st❛ ❢♦r♠❛✱ ❞❛❞♦ b ∈ A ❡ ε > 0✱ s❡❥❛ a ∈ A t❛❧ q✉❡ kb − eak < ε✳ ❚❡♠♦s kb − s k bk ≤ kb − eak + kea − s k eak + ks k ea − s k bk

      ≤ kb − eak + ke − s k ekkak + ks k kkea − bk < 3ε, k k ≤ 1 ♣❛r❛ k s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ ✉♠❛ ✈❡③ q✉❡ ks ✱ ♣❛r❛ t♦❞♦ k✳ k k < 3ε ❆♥❛❧♦❣❛♠❡♥t❡ s❡❣✉❡ kb − bs ✳

      P a pa i = 1 ❘❡st❛ ♣r♦✈❛r♠♦s q✉❡ i ✱ ❝♦♠ ❝♦♥✈❡r❣ê♥❝✐❛ ♥❛ t♦♣♦❧♦❣✐❛

      ❡str✐t❛ ❞❡ M(A)✳ ❉❛❞♦ x ∈ A✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦s ♣♦s✐t✐✈♦s n ∗ ∗ P x (1 − a pa i )x i =1 i é ♠♦♥ót♦♥❛ ❞❡❝r❡s❝❡♥t❡ ❡ ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ n n ∗ ∗ ∗ P P (1 − a pa i )x → 0 a pa i k ≤

      ❝♦♥✈❡r❣✐♥❞♦ ❛ 0✳ ▲♦❣♦✱ x i ✳ ❈♦♠♦ k1 − i i =1 i =1

      1 ✱ ♣❛r❛ t♦❞♦ n ∈ N ❡ n n n

      X ∗ ∗ ∗

      X 2 1 X 2 1 k(1 − a pa i )xk ≤ k(1 − a pa i ) kk(1 − a pa i ) xk, i =1 i =1 i =1 i i i ♦ ❧❡♠❛ é ♣r♦✈❛❞♦✳ ▲❡♠❛ ✹✳✸✳✶✵✳ ❙❡❥❛ p ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛ ❡♠ M(A) ❡ s✉♣♦♥❤❛ q✉❡ A ♣♦ss✉❛ ✉♠ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✳ ❊♥tã♦✱ ♣❛r❛ ❝❛❞❛ j ∈ N✱ j ∈ M (A ⊗ K) u j = 1 ⊗ e jj ❡①✐st❡ ✉♠❛ ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ u t❛❧ q✉❡ u j ❡ u j u ≤ p ⊗ 1 j ✳

      ❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ❝♦♥str✉✐r t❛❧ ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ ♣❛r❛ j = 1✱ n ) n ✉♠❛ ✈❡③ q✉❡ ♣❛r❛ j ∈ N ❛r❜✐trár✐♦ ❛ ❝♦♥str✉çã♦ é ❛♥á❧♦❣❛✳ ❙❡❥❛ (a ∈N

      P a pa i = 1 ❝♦♠♦ ♥♦ ▲❡♠❛ ❆ss✐♠✱ i ✱ ❝♦♠ ❝♦♥✈❡r❣ê♥❝✐❛ ♥❛ t♦♣♦✲ i ❧♦❣✐❛ ❡str✐t❛ ❞❡ M(A)✳ P pa i ⊗ e i ❉❡✜♥✐♠♦s u = i 1 ❡ ♠♦str❡♠♦s q✉❡ u ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱

      ♦✉ s❡❥❛✱ q✉❡ ❛ sér✐❡ ❝♦♥✈❡r❣❡ ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ ❞❡ M(A)✳ n = pa i ⊗ e i P n ◆♦t❡♠♦s q✉❡ s❡ u 1 n 1 ✱ ❡♥tã♦ 2 ∗ ∗

      X ku n k = ku u n k = a pa i ⊗ e n i 1 11 ≤ 1.

      P b jk ⊗ e jk ❉❡st❛ ❢♦r♠❛✱ ❝♦♠♦ s♦♠❛s ✜♥✐t❛s ❞❛ ❢♦r♠❛ sã♦ ❞❡♥s❛s n k ≤ 1

      ❡♠ A ⊗ K ❡ ku ♣❛r❛ t♦❞♦ n✱ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r♠♦s q✉❡ ❛s n (b ⊗ e jk )) jk )u n ) s❡q✉ê♥❝✐❛s (u n ❡ ((b ⊗ e n ❝♦♥✈❡r❣❡♠✱ ♣❛r❛ t♦❞♦ ∈N ∈N b ∈ A ❡ j, k ∈ N✳ ▼❛s✱ jk n k j

      (b ⊗ e )u = [n ≥ k]bpa ⊗ e 1 (b ⊗ e jk )u n = bpa k ⊗ e j

      ❞♦♥❞❡ lim n 2 ∗ ∗ ∗ 1 ✳ P❛r❛ ♦ ♦✉tr♦ ❝❛s♦✱ t❡♠♦s k(u m − u n )b ⊗ e jk k = k(b ⊗ e kj )(u − u )(u m − u n )(b ⊗ e jk )k m n m ! ∗ ∗

      X = (b ⊗ e kj ) a pa i ⊗ e (b ⊗ e jk ) m n +1 i 11 X ∗ ∗

      = [j = 1] b a pa i b ⊗ e kk n m +1 i

      X ∗ ∗ = [j = 1] b a pa i b → 0, n +1 i

      P a pa i ✉♠❛ ✈❡③ q✉❡ i i ❝♦♥✈❡r❣❡ ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ ❞❡ M(A)✳ ∗ ∗ u = 1 ⊗ e ≤ p ⊗ 1

      ❘❡st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ u 11 ❡ uu ✳ ❚❡♠♦s q✉❡ ∗ ∗

      X ∗ ∗ u u = a pa i ⊗ e = 1 ⊗ e . i i 11 11 (p ⊗ 1) = uu

      ❆❧é♠ ❞✐ss♦✱ uu ❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✸✳✷ ❞❡ s❡❣✉❡ q✉❡ uu ≤ p ⊗ 1 ✱ ❝♦♥❝❧✉✐♥❞♦ ❛ ♣r♦✈❛✳

      ❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ t❡rá ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ✉♠ ✐s♦♠♦r✜s♠♦ ❡s✲ tá✈❡❧ ❡♥tr❡ ✉♠❛ C ✲á❧❣❡❜r❛ ♣♦ss✉✐♥❞♦ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦ ❡ ✉♠ ❝❛♥t♦ ❝❤❡✐♦✳ ❈♦♠ ✐ss♦ ❡♠ ♠ã♦s✱ ♣❛r❛ ❞✉❛s C ✲á❧❣❡❜r❛s ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✱ ❛♠❜❛s ♣♦ss✉✐♥❞♦ ❡❧❡♠❡♥t♦s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦s✱ s❡rá s✉✜❝✐❡♥t❡ t♦♠❛r ❛ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦ ❞❡ ✉♠ ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐✲ ❞❛❞❡ ♣❛r❛ ♦❜t❡r ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❡ ❝❛♣ít✉❧♦✱ ❛ s❛❜❡r ♦ t❡♦r❡♠❛

      ❞❡ ❇r♦✇♥✲●r❡❡♥✲❘✐❡✛❡❧✳ P❛r❛ ✜♥s ❞❛ ♣ró①✐♠❛ ❞❡♠♦♥str❛çã♦✱ ❛❧❡rt❛♠♦s ♦ ❧❡✐t♦r q✉❡ ✉s❛r❡♠♦s

      ❛♠♣❧❛♠❡♥t❡ ♦ ❚❡♦r❡♠❛ ✷✳✸✳✷ ❞❡ q✉❡✱ ❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ ❞✐③ q✉❡ s❡ p ❡ q sã♦ ♣r♦❥❡çõ❡s ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✱ ❡♥tã♦ p ≤ q s❡✱ ❡ s♦♠❡♥t❡ s❡✱ pq = qp = p✳

      ▲❡♠❛ ✹✳✸✳✶✶✳ ❙❡❥❛ p ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛ ❡♠ M(A) ❡ s✉♣♦♥❤❛ q✉❡ A ♣♦ss✉❛ ✉♠ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ✐s♦♠❡tr✐❛ ∗ ∗ v = 1 = p ⊗ 1 ♣❛r❝✐❛❧ v ∈ M(A ⊗ K) t❛❧ q✉❡ v ❡ vv ✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ N ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ g ✉♠❛ ❜✐✲ j j j = N ❥❡çã♦ ❡♥tr❡ N ❡ N × N ❡ ❡s❝r❡✈❡♠♦s N = ∪ ∈N ✱ ❡♠ q✉❡ N −1 g (N × {j}) j

      ✳ ❆ss✐♠✱ ♦s N ✬s sã♦ s✉❜❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ❞❡ N✱ ❞✐s❥✉♥✲ j = 1 ⊗ e ii ∈ M (A ⊗ K) P t♦s ❞♦✐s ❛ ❞♦✐s✳ P❛r❛ j ∈ N✱ s❡❥❛ f ✳ ❆ j i ∈N j ♣r♦✈❛ ❞❡ q✉❡ f ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ♣❛r❛ ❝❛❞❛ j✱ é ❢❡✐t❛ ❝♦♠♦ ♥♦ ▲❡♠❛ k ) k

      ❱❛♠♦s ❝♦♥str✉✐r ✐♥❞✉t✐✈❛♠❡♥t❡ ✉♠❛ s❡q✉ê♥❝✐❛ (v ∈N ❞❡ ✐s♦♠❡✲ v k tr✐❛s ♣❛r❝✐❛✐s ❡♠ M(A⊗K) t❛✐s q✉❡ ❛s ♣r♦❥❡çõ❡s v k sã♦ ♠✉t✉❛♠❡♥t❡ k v ♦rt♦❣♦♥❛✐s✱ ❛s ♣r♦❥❡çõ❡s v k sã♦ ♠✉t✉❛♠❡♥t❡ ♦rt♦❣♦♥❛✐s✱ 2n−1 2n +1 n n

      X ∗ ∗

      X X

      X 1 v v k = f j v v k ≤ f j k ✱ k 1 1 12n−1 2n n n

      X ∗ ∗

      X X

      X 1 v k v ≤ (p ⊗ 1) f j v k v = (p ⊗ 1) f j . k ✱ k 1 1 1 P v k ❯♠❛ ✈❡③ q✉❡ ✐st♦ é ❢❡✐t♦✱ ❞❡✜♥✐♠♦s v = ✱ ❝♦♠ ❝♦♥✈❡r❣ê♥❝✐❛ 1

      ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛✳ P❛r❛ ❝♦♥str✉✐r v 2n−1 ✱ ♣r❡❝✐s❛r❡♠♦s ❞❡ ✉♠ ♣❛ss♦ ✐♥t❡r♠❡❞✐ár✐♦✱ q✉❡

      ❝♦♥s✐st❡ ♥❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ ♦✉tr❛ ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ w s❛t✐s❢❛③❡♥❞♦ ∗ ∗ w w = f n ≤ (p ⊗ 1)f n ❡ ww ✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❛♠♦s q✉❡ p ⊗ 1 é

      ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛ ❡♠ M(A⊗K) ❡ A⊗K ♣♦ss✉✐ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✱ ✉♠❛ ✈❡③ q✉❡ A ❡ K ♣♦ss✉❡♠✳ ▲♦❣♦✱ ♣❡❧♦ ▲❡♠❛ ♣❛r❛ n ∈ N u =

      ❡①✐st❡ ✉♠❛ ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ u ∈ M(A ⊗ K ⊗ K) t❛❧ q✉❡ u 1 ⊗ 1 ⊗ e nn = p ⊗ 1 ⊗ 1 ❡ uu ✳ ❈♦♠♦ ♥❛ Pr♦♣♦s✐çã♦ s❡❥❛ ϕ ♦ ∗✲ kl ) = a ⊗ e k 1 l 1 ⊗ e k 2 l 2

      ✐s♦♠♦r✜s♠♦ ❞❡ A⊗K ❡ A⊗K ⊗K t❛❧ q✉❡ ϕ(a⊗e ✱ 1 , k 2 ) = g(k) 1 , l 2 ) = g(l) = ψ(u) −1 ′ ❡♠ q✉❡ (k ❡ (l ✳ ❙❡❥❛ ψ = ϕ ❡ u ✳ ′∗ ′ ′ ′∗ u = f n u ≤ p ⊗ 1 ❆✜r♠❛♠♦s q✉❡ u ❡ u ✳

      P e ii = 1 ❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ ✱ ❝♦♠ ❝♦♥✈❡r❣ê♥❝✐❛ ♥❛ t♦♣♦❧♦❣✐❛ 1

      ψ ❡str✐t❛✳ ❆ss✐♠✱ s❡♥❞♦ e ❛ ❡①t❡♥sã♦ ❞❡ ψ ❛ M(A ⊗ K ⊗ K)✱ t❡♠♦s q✉❡ ′∗ ′

      X e u u = ψ(1 ⊗ 1 ⊗ e nn ) = e ψ( 1 ⊗ e ii ⊗ e nn ) i

      X X e = ψ(1 ⊗ e ii ⊗ e nn ) = 1 ⊗ e ii = f n , i i ∈N n

      ψ ♣♦✐s e é ❝♦♥tí♥✉♦ ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ ✭Pr♦♣♦s✐çã♦ ❆❧é♠ ❞✐ss♦✱ ′ ′∗ ∗ u u = e ψ(uu ) ≤ e ψ(p ⊗ 1 ⊗ 1) = p ⊗ 1. n → N 1 ⊗ e ∈ P

      ❙❡❥❛ σ : N ✉♠❛ ❜✐❥❡çã♦ ❡ ❝♦❧♦❝❛♠♦s a = iσ (i) ∗ ∗ i ∈N n M (A ⊗ K) a = 1 ⊗ 1 = f n

      ✳ ❊♥tã♦✱ a ❡ aa ✳ ❉❡✜♥✐♠♦s w = au ❡ t❡♠♦s✱ ∗ ∗ ∗ w w = u a au = f n ❡ ∗ ∗ ∗ ∗ ∗ ww = auu a ≤ a(p ⊗ 1)a = (p ⊗ 1)aa = (p ⊗ 1)f n .

      P n P 2n−2 = w( f j − v v k )

      P♦r ✜♠✱ ❝♦❧♦❝❛♠♦s v 2n−1 k ✳ ❈♦♠❡ç❛♠♦s 1 1 ♣♦r ♠♦str❛r q✉❡ v 2n−1 é ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧✳

      P 2n−3 P n −1 P 2n−2 P n ∗ ∗ v v k = f j v v k ≤ f j P♦r ❤✐♣ót❡s❡ ❡ ✱ ❞✐ss♦ ∗ ∗ ∗ 1 k 1 1 k 1 v ≤ f n v = v v f n = s❡❣✉❡ q✉❡ v 2n−2 ✱ ❞♦♥❞❡ v 2n−2 2n−2 2n−2 2n−2 2n−2 f n v v 2n−2 2n−2 ✳ ▲♦❣♦✱ n n 2n−2 2n−2 ! ! ∗ ∗ ∗

      X X

      X X v v = f j − v v k f n f j − v v k 2n−1 2n−1 k k 1 ∗ ∗ 1 1 1 = (f n − v v )f n (f n − v v ) 2n−2 2n−2 2n−2 2n−2 = f n − v v , 2n−2 2n−2 v

      ❞♦♥❞❡ v 2n−1 é ♣r♦❥❡çã♦ ❡✱ ♣♦rt❛♥t♦✱ v 2n−1 é ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧✳ 2n−1 ▼❛✐s ❛✐♥❞❛✱ 2n−1 2n−2

      X ∗ ∗ ∗

      X 1 v v k = v v k + v v k k 2n−1 n −1 n 1 2n−1

      X ∗ ∗

      X = f j + v v 2n−2 + (f n − v v 2n−2 ) = f j 1 2n−2 2n−2 1

      ❡✱ ♥♦✈❛♠❡♥t❡ ♣♦r ❤✐♣ót❡s❡✱ 2n−1 2n−2 −1 n

      X ∗ ∗ ∗ ∗

      X X 1 v k v = v k v + v v = (p ⊗ 1) f j + v v . k k 2n−1 2n−1 ✭†✮ 1 2n−1 2n−1 1 ❆❣♦r❛✱ n n 2n−2 2n−2 ! ! ∗ ∗ ∗ ∗

      X X

      X X v v = w f j − v v k f j − v v k w 2n−1 k k 2n−1 1 2n−2 1 1 1 X ∗ ∗ = w(f n − v v k )w 2n−2 1 k

      X ∗ ∗ ∗ = w(f n − v v k )w ww . 1 k ≤ (p ⊗

      ❉❡st❛ ❢♦r♠❛✱ ❝♦♠♦ w é ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ s❛t✐s❢❛③❡♥❞♦ ww ∗ ∗ ∗ 1)f n (p ⊗ 1)f n = ww v (p ⊗ 1)f n =

      ✱ t❡♠♦s q✉❡ ww ✳ ▲♦❣♦✱ v 2n−1 ∗ ∗ 2n−1 v v v ≤ (p ⊗ 1)f n 2n−1 ❡ ❝♦♥❝❧✉í♠♦s q✉❡ v 2n−1 ✳ ❉❡ s❡❣✉❡ q✉❡ 2n−1 2n−1 P P n 2n−1 1 v k v ≤ (p ⊗ 1) f j k ✳ 1 ∗ ∗ v v k

      ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ v 2n−1 é ♦rt♦❣♦♥❛❧ ❛ v k ❡ t❛♠❜é♠ ∗ ∗ 2n−1 v v k v 2n−1 é ♦rt♦❣♦♥❛❧ ❛ v k ✱ ♣❛r❛ 1 ≤ k ≤ 2n−2✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❝♦♠♦ 2n−1 ∗ ∗ v f n = f n v v = ❥á ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ t❡♠♦s q✉❡ v 2n−2 2n−2 2n−2 2n−2 v v 2n−2 ∗ ∗ ∗ ∗ 2n−2 ✱ ❞♦♥❞❡ v v v v = (f n − v v )v v = 0. 2n−1 2n−2 2n−2 2n−2 2n−1 2n−2 2n−2 2n−2

      P 2n−3 P n −1 v v k = f j ❆❧é♠ ❞✐ss♦✱ ❞♦ ❢❛t♦ q✉❡ k ✱ ♦❜t❡♠♦s q✉❡ 1 1 2n−3 !

      X 1 v v k f n = 0 k n v v k ❡✱ ♣♦rt❛♥t♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r f ♣❡❧❛ ❡sq✉❡r❞❛ ❝♦♥❝❧✉í♠♦s q✉❡ ∗ ∗ ∗ k v v k f n = 0 v v k k 2n−1 k ✱ ♣❛r❛ 1 ≤ k ≤ 2n−3✳ ▲♦❣♦✱ v 2n−1 é ♦rt♦❣♦♥❛❧ ❛ v ✱

      ♣❛r❛ 1 ≤ k ≤ 2n − 2✳ ∗ ∗ v k v P❛r❛ ♠♦str❛r♠♦s q✉❡ v 2n−1 2n−1 é ♦rt♦❣♦♥❛❧ ❛ v k ✱ ♣❛r❛ ❝❛❞❛ 1 ≤ k ≤ 2n − 2 2n−1 v =

      ✱ ♦❜s❡r✈❛♠♦s q✉❡✱ ❝♦♠♦ ❥á ❥✉st✐✜❝❛❞♦✱ v 2n−1 ∗ ∗ P P n 2n−2 −1 v 2n−1 v (p ⊗ 1)f n v k v = (p ⊗ 1) f j 2n−1 ❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ k ✳ 1 1 ∗ ∗ ∗ ∗ P 2n−2 v ( v k v ) = 0 v v k v = 0 ❉♦♥❞❡ v 2n−1 ❡ ❛ss✐♠ v 2n−1 ✱ ♣❛r❛ 2n−1 k 2n−1 k 1 1 ≤ k ≤ 2n − 2

      ✳ P❛r❛ ❝♦♥str✉✐r v 2n ✱ ♣r❡❝✐s❛♠♦s ♥♦✈❛♠❡♥t❡ ❞❡ ✉♠ ♣❛ss♦ ✐♥t❡r♠❡✲ ❞✐ár✐♦ q✉❡ ❝♦♥s✐st❡ ♥❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ w t❛❧ q✉❡ ′∗ ′ ′ ′∗ ′ w w ≤ f n w = (p ⊗ 1)f n : N n →

    • +1 ❡ w ✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❡♠♦s σ N n = (p ⊗ 1) 1 ⊗ e iσ ′ +1 ✉♠❛ ❜✐❥❡çã♦ ❡ ❝♦❧♦❝❛♠♦s w (i) ✳ ❊♥tã♦✱

      P i ∈N n ′ ′∗

      X w w = (p ⊗ 1) 1 ⊗ e ii = (p ⊗ 1)f n i ∈N n′∗ ′

      X ′ ′ w w = (p ⊗ 1) 1 ⊗ e σ i ∈N n (i)σ (i)

      X = (p ⊗ 1) 1 ⊗ e ii = (p ⊗ 1)f n ≤ f n . i ∈N n+1 +1 +1 P n P 2n−1 ∗ ′

      = (p ⊗ 1)( f j − v k v )w ❉❡✜♥✐♠♦s v 2n ✳ ❊♥tã♦✱ ❝♦♠♦ 1 1 k

      P P n 2n−2 −1 1 v k v = f j k ✱ t❡♠♦s q✉❡ 1 n n 2n−1 2n−1 ! ! ∗ ∗ ′ ′∗ ∗

      X X

      X X v v = (p ⊗ 1) f j − v k v w w f j − v k v (p ⊗ 1) 2n k k 2n 1 1 ∗ ∗ 1 1 = (p ⊗ 1)(f n − v v )(p ⊗ 1)f n (f n − v v )(p ⊗ 1) 2n−1 2n−1 2n−1 2n−1 = (p ⊗ 1)(f n − v v )(p ⊗ 1) 2n−1 2n−1 = (p ⊗ 1)f n − v v , 2n−1 2n−1 v = v 2n−1 v (p ⊗ 1)f n = (p ⊗ 2n−1 ∗ ∗

      ♣♦✐s✱ ❝♦♠♦ ❥á ♦❜s❡r✈❛❞♦✱ v 2n−1 2n−1 ∗ ∗ 1)f n v 2n−1 v 2n v 2n 2n−1 ✳ ▲♦❣♦✱ v 2n é ♣r♦❥❡çã♦ ❡✱ ♣♦rt❛♥t♦✱ v é ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧✳ n n

      P 2n P +1 P 2n P ∗ ∗ v v k ≤ f j v k v = (p⊗1) f j ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ k ❡ k ✳ 1 1 1 1 P P n 2n−2 −1 v k v = (p ⊗ 1) f j

      ❯s❛♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ k ✱ t❡♠♦s 2n 2n−2 1 1 n

      X ∗ ∗ ∗ ∗

      X X 1 v k v = v k v +v v +(p⊗1)f n −v v = (p⊗1) f j . k k 2n−1 2n−1 1 2n−1 2n−1 1 2n 2n−1 P❛r❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ t❡♠♦s

      X ∗ ∗ ∗

      X 1 k 2n k 2n + v v k = v v v v k 1 n 2n−1 n 2n−1 ! ! ′∗ ∗ ∗ ′

      X X

      X X = w f j − v k v (p ⊗ 1) f j − v k v w 1 1 k k 1 1 n

      X 1 + f j n ′∗ ∗ ′

      X = w − v v + ((p ⊗ 1)f n 2n−1 )w f j n n +1 2n−1 1 X

      X ≤ f j + f n = f j . 1 +1 1 ∗ ∗ v = v v (p ⊗ ❆q✉✐✱ ✉s❛♠♦s ♥♦✈❛♠❡♥t❡ ♦ ❢❛t♦ q✉❡ v 2n−1 2n−1 ∗ ∗ 2n−1 2n−1

      1)f n = (p⊗1)f n v v n −v v ≤ 2n−1 ❜❡♠ ❝♦♠♦ ♦ ❢❛t♦ ❞❡ (p⊗1)f 2n−1 2n−1 2n−1

      1 ✳ ∗ ∗ ∗ v k v v ❆ ♣r♦✈❛ ❞❡ q✉❡ v 2n é ♦rt♦❣♦♥❛❧ ❛ v ❡ v 2n é ♦rt♦❣♦♥❛❧ ❛ 2n k 2n v v k k ✱ 1 ≤ k ≤ 2n−1✱ é ❢❡✐t❛ ♣♦r ❛r❣✉♠❡♥t♦s s❡♠❡❧❤❛♥t❡s ❛♦s ✉t✐❧✐③❛❞♦s

      ♥♦ ❝❛s♦ í♠♣❛r✳ P v k

      ❋✐♥❛❧♠❡♥t❡✱ r❡st❛ ♣r♦✈❛r♠♦s q✉❡ v = ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ ∗ ∗ k v = 1 = p ⊗ 1 q✉❡ t❡♠♦s v ❡ vv ✳ n = v k P n

      ❙❡❥❛ β ✳ ❈♦♠♦ ♣❛r❛ k 6= l t❡♠✲s❡ 1 ∗ ∗ ∗ ∗ n n k ≤ 1 v v l = v v k v v j v v j = 0, k k k j t❡♠♦s q✉❡ β é ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ ❡ ❛ss✐♠ kβ ✱ ♣❛r❛ ❝❛❞❛ n ∈ N✳ β n (b ⊗ e ij ) (b ⊗ e ij )β n

      ❱❛♠♦s ♣r♦✈❛r q✉❡ ♦s ❧✐♠✐t❡s lim ❡ lim ❡①✐st❡♠✱ n n ♣❛r❛ q✉❛✐sq✉❡r b ∈ A ❡ i, j ∈ N✳ N

      ❙❡❥❛ N ∈ N t❛❧ q✉❡ i ∈ N ❡ s❡❥❛ n ≥ N✳ ❊♥tã♦✱ s✉❜st✐t✉✐♥❞♦ n P 2n P ∗ ∗ ∗ 1 v k v f j v v f n k ♣♦r (p⊗1) ❡ v 2n+1 ♣♦r v 2n+1 +1 ✱ ♦❜t❡♠♦s 1 2n+1 2n+1 2n ∗ ∗ ∗

      X β β (b ⊗ e ij ) = v k v (b ⊗ e ij ) + v v (b ⊗ e ij ) 2n+1 k 2n+1 2n+1 2n+1 1

      = (p ⊗ 1)(b ⊗ e ij ) ❡ n

      X β 2n β (b ⊗ e ij ) = (p ⊗ 1)f j (b ⊗ e ij ) = (p ⊗ 1)(b ⊗ e ij ). 2n 1 N ❚♦♠❛♥❞♦ N t❛❧ q✉❡ j ∈ N ❡ r❡♣❡t✐♥❞♦ ♦s ❛r❣✉♠❡♥t♦s ❛❝✐♠❛✱

      (b ⊗ e ij )β n β = (b ⊗ e ij )(p ⊗ 1) ❝♦♥❝❧✉í♠♦s q✉❡ lim n ✳ ❆❣♦r❛✱ ♣❛r❛ n ≥ N✱ n n +1

      X β β 2n+1 (b ⊗ e ij ) = f j (b ⊗ e ij ) = b ⊗ e ij 2n+1 1

      ❡ 2n−1 ∗ ∗ ∗

      X β β (b ⊗ e ij ) = v v k (b ⊗ e ij ) + v v (b ⊗ e ij ) = b ⊗ e ij , 2n 2n 2n k 2n 1 P 2n−1 P n ∗ ∗ ∗ ∗ v v k = f j v = v v f n = f n v v

      ✉♠❛ ✈❡③ q✉❡ ❡ v 2n 2n +1 +1 2n ✳ 1 k 1 2n 2n 2n β β n (b ⊗ e ij ) = b ⊗ e ij (b ⊗

      ▲♦❣♦✱ lim n ❡ ❛♥❛❧♦❣❛♠❡♥t❡ ♣r♦✈❛✲s❡ q✉❡ lim n n e ij )β β n = b ⊗ e ij n ✳ ❆ss✐♠✱ ❝♦♠♦ 2 ∗ ∗ k(β n − β m )(b ⊗ e ij )k = k(b ⊗ e ji )(β β n − β β m )(b ⊗ e ij )k n m2 ∗ ∗ k(b ⊗ e ij )(β n − β m )k = k(b ⊗ e ij )(β n β − β m β )(b ⊗ e ji )k, n m

      P ∗ ∗ v k v = 1 = p ⊗ 1 s❡❣✉❡ q✉❡ v = ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ t❡♠♦s v ❡ vv ✱ k ❝♦♠♦ ❞❡s❡❥❛❞♦✳ ❉❡✜♥✐çã♦ ✹✳✸✳✶✷✳ ❉✐③❡♠♦s q✉❡ ❞✉❛s C ✲á❧❣❡❜r❛s A ❡ B sã♦ ❡st❛✈❡❧✲ ♠❡♥t❡ ✐s♦♠♦r❢❛s s❡ A ⊗ K ❡ B ⊗ K sã♦ ✐s♦♠♦r❢❛s✳ ❈♦r♦❧ár✐♦ ✹✳✸✳✶✸✳ ❙❡ A é ✉♠❛ C ✲á❧❣❡❜r❛ ❝♦♠ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦ ❡ B é ✉♠ ❝❛♥t♦ ❝❤❡✐♦ ❞❡ A✱ ❡♥tã♦ A ❡ B sã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛s✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ p ∈ M(A) t❛❧ q✉❡ B = pAp✳ ■❞❡♥t✐✜❝❛♠♦s B ⊗ K

      ❝♦♠ (p ⊗ 1)A ⊗ K(p ⊗ 1) ❡ s❡❥❛ v ∈ M(A ⊗ K) ✉♠❛ ✐s♦♠❡tr✐❛ ∗ ∗ v = 1 = p ⊗ 1 ♣❛r❝✐❛❧ ❝♦♠♦ ♥♦ ▲❡♠❛ ♦✉ s❡❥❛✱ v ❡ vv ✳

      ❉❡✜♥✐♠♦s π : A ⊗ K → B ⊗ K y 7→ vyv ,

      ❡♠ q✉❡ y ∈ A ⊗ K✳ ∗ ∗ = vv v(a ⊗ ∗ ∗ ∗ ◆♦t❡♠♦s q✉❡✱ s❡ a ∈ A ❡ x ∈ K✱ ❡♥tã♦ v(a ⊗ x)v x)v vv = (p ⊗ 1)v(a ⊗ x)v (p ⊗ 1)

      ✱ ❞♦♥❞❡ π ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ❥á P a ⊗ x q✉❡ s♦♠❛s ✜♥✐t❛s ❞❛ ❢♦r♠❛ ❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡♥s♦ ❡♠ A ⊗ K v = 1

      ✳ ▼❛✐s ❛✐♥❞❛✱ s❡♥❞♦ v ✱ t❡♠♦s q✉❡ π é ∗✲❤♦♠♦♠♦r✜s♠♦✳ = 0

      P❛r❛ ✈❡r q✉❡ π é ✐♥❥❡t✐✈❛✱ s❡❥❛ y ∈ A ⊗ K t❛❧ q✉❡ vyv ✳ ❊♥tã♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r v ♣❡❧❛ ❞✐r❡✐t❛ ❡ ♣♦r v ♣❡❧❛ ❡sq✉❡r❞❛✱ s❡❣✉❡ q✉❡ y = 0✳ ∗ ∗ bvv ❉❛❞♦ b ∈ B ⊗ K✱ t❡♠♦s q✉❡ b = (p ⊗ 1)b(p⊗) = vv ✳ ❚♦♠❛♥❞♦ y = v bv ∈ A ⊗ K

      ✱ s❡❣✉❡ q✉❡ π(y) = b ❡ π é s♦❜r❡❥❡t♦r❛✳ P♦rt❛♥t♦✱ A ❡ B sã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛s✳ ❚❡♦r❡♠❛ ✹✳✸✳✶✹✳ ❙❡❥❛ B ✉♠❛ C ✲á❧❣❡❜r❛ ❤❡r❡❞✐tár✐❛ ❝❤❡✐❛ ❞❡ A ❡ s✉♣♦♥❤❛ q✉❡ A ❡ B ♣♦ss✉❛♠ ❡❧❡♠❡♥t♦s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦s✳ ❊♥tã♦ B

      é ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛ ❛ A✳ (C)

      ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ C ❛ C ✲s✉❜á❧❣❡❜r❛ ❞❡ A⊗M 2 ❝♦♥s✐st✐♥❞♦ ❞❡ P a ij ⊗e ij ∈ B ∈ BA ∈ AB ∈ A s♦♠❛s i,j t❛❧ q✉❡ a ij : i, j ∈ {1.2}} 11 ✱ a 12 ✱ a 21 ❡ a 22 ✱ ❡♠ 2 (C) q✉❡ {e é ✉♠ s✐st❡♠❛ ❞❡ ♠❛tr✐③❡s ✉♥✐❞❛❞❡ ♣❛r❛ M ✳ 11

      ❖❜s❡r✈❛♠♦s q✉❡ B é ✐s♦♠♦r❢♦ ❛♦ ❝❛♥t♦ ❝❤❡✐♦ B ⊗ e ❞❡ C✱ ❞❛❞♦ 11 ∈ M (C) ♣❡❧❛ ♣r♦❥❡çã♦ 1 ⊗ e ✳ 11

      ❉❡ ❢❛t♦✱ ♣❛r❛ ✈❡r q✉❡ 1 ⊗ e é ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛✱ s✉♣♦♥❤❛ q✉❡

      I 11 s❡❥❛ ✉♠ ✐❞❡❛❧ ❢❡❝❤❛❞♦ ❞❡ C ❝♦♥t❡♥❞♦ B ⊗ e ❡ ✈❛♠♦s ♠♦str❛r q✉❡ I = C 22 )

      ✳ ◆♦t❡♠♦s q✉❡ B ⊆ AB ∩ BA ❡ ❛ss✐♠ J = I ∩ (A ⊗ e é ♥ã♦ ♥✉❧♦✱ ♣♦✐s ❞❛❞♦ b ∈ B✱ λ ) λ b ⊗ e = lim (u λ ⊗ e )(b ⊗ e )(u λ ⊗ e ), 22 λ 21 11 12

      ❡♠ q✉❡ (u ∈Λ é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ B✳ ▲♦❣♦✱ J é ✉♠ ✐❞❡❛❧ ❡♠ A ⊗ e 22 ❝♦♥t❡♥❞♦ B ⊗ e 22 ✳ ■❞❡♥t✐✜❝❛♥❞♦ A ❝♦♠ A ⊗ e 22 ✱ ❝❤❡❣❛♠♦s ❛ ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s B é ✉♠❛ C ✲á❧❣❡❜r❛ ❤❡r❡❞✐tár✐❛ ❝❤❡✐❛ ❞❡ A✳ P♦rt❛♥t♦✱ 1 ⊗ e 11 é ♣r♦❥❡çã♦ ❝❤❡✐❛✳

      ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ s❡ ✉♠ ✐❞❡❛❧ I ❝♦♥té♠ A ⊗ e 22 ✱ ♣r♦✈❛✲s❡ q✉❡ I ❝♦♥té♠ B ⊗ e 11 ✱ ❡ ❛ss✐♠ I = C✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ 1 ⊗ e 22 é t❛♠❜é♠ ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛ ❡ A é ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠ ♦ ❝❛♥t♦ ❝❤❡✐♦ A ⊗ e 22 ❞❡ C✳

      ∈ B ∈ A ⊗ e + f ⊗ e ❙❡ f 1 ❡ f 2 ✱ ❡♥tã♦ f = f 1 11 2 22 é ✉♠ ❡❧❡♠❡♥t♦ ❡s✲ tr✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦ ❞❡ C✱ ♣♦✐s s❡ φ ∈ C é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ♣♦s✐t✐✈♦✱ A B ∗ ∗ t❡♠♦s q✉❡ φ | ❡ φ | sã♦ ❢✉♥❝✐♦♥❛✐s ❧✐♥❡❛r❡s ♣♦s✐t✐✈♦s ❡♠ A ❡ B ✱ B (f ⊗e )+φ | A (f ⊗e ) > 0 r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ ♣♦rt❛♥t♦ φ(f) = φ | 1 11 2 22 ✳

      P❡❧♦ ❈♦r♦❧ár✐♦ t❡♠♦s q✉❡ A ❡ B sã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛s ❛ C ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ A ❡ B sã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛s✳

      ❆ ❚❡♦r❡♠❛ ✹✳✸✳✶✺ ✭❇r♦✇♥✲●r❡❡♥✲❘✐❡✛❡❧✮✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s✳ ❙❡ A ❡ B sã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛s✱ ❡♥tã♦ sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ A ❡ B sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s ❡ ❛♠❜❛s ♣♦ss✉❡♠ ❡❧❡♠❡♥t♦s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦s✱ ❡♥tã♦ A ❡ B sã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦✲ ♠♦r❢❛s✳ ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ A ❡ B s❡❥❛♠ ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛s✳

      ∈ K )A ⊗ K ❙❡❥❛ p = e 11 ✳ ❊♥tã♦✱ A ⊗ K(1 ⊗ e 11 é ❞❡♥s♦ ❡♠ A ⊗ K✱ ij

      ♣♦✐s ❝♦♥té♠ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❛ ❢♦r♠❛ a ⊗ e ✱ i, j ∈ N✳ ▲♦❣♦✱ 1 ⊗ e 11 é ♣r♦❥❡çã♦ ❝❤❡✐❛ ❡ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r A ❝♦♠ ♦ ❝❛♥t♦ ❝❤❡✐♦ A ⊗ e 11 ✳ ❙✐♠✐❧❛r♠❡♥t❡✱ ✐❞❡♥t✐✜❝❛♠♦s B ❝♦♠ ♦ ❝❛♥t♦ ❝❤❡✐♦ B ⊗ e 11 ❞❡ B ⊗ K

      ✳ P♦rt❛♥t♦✱ A é ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡ ❛ A ⊗ K ❜❡♠ ❝♦♠♦ B ❡ B ⊗ K sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✱ ♣❡❧♦ ❊①❡♠♣❧♦ P♦r tr❛♥s✐t✐✈✐❞❛❞❡ ❡ ♣❡❧♦ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ A ⊗ K ❡ B ⊗ K✱ s❡❣✉❡ q✉❡ A ❡ B sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✳

      ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ A ❡ B sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s ❡ ❛♠❜❛s ♣♦ss✉❡♠ ❡❧❡♠❡♥t♦s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦s✳ ❙❡❥❛ X ✉♠ A − B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠✲ ♣r✐♠✐t✐✈✐❞❛❞❡ ❡ s❡❥❛ C ❛ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦ ❞❡ X✳ ❊♥tã♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ A ❡ B sã♦ ❝❛♥t♦s ❝❤❡✐♦s ❞❡ C ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛s ♣r♦❥❡çõ❡s 1 0 p = 0 0

      ❡ 0 0 q = , 0 1 r❡s♣❡❝t✐✈❛♠❡♥t❡✳

      ❆❧é♠ ❞✐ss♦✱ s❡♥❞♦ f 1 ❡ f 2 ❡❧❡♠❡♥t♦s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦s ❞❡ A ❡ B

      ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ❡❧❡♠❡♥t♦ f ❞❡ C ❞❛❞♦ ♣♦r f 1 f = f 2

      é ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✱ ♣♦r ✉♠ ❛r❣✉♠❡♥t♦ ❥á ✉s❛❞♦ ♥♦ ❚❡♦r❡♠❛ ▲♦❣♦✱ ❝♦♥❝❧✉í♠♦s ❞♦ ❈♦r♦❧ár✐♦ q✉❡ A ❡ B sã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦✲ ♠♦r❢❛s ❛ C ❡✱ ♣♦rt❛♥t♦✱ sã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛s✳

      ❆ ❤✐♣ót❡s❡ ❞♦ ❚❡♦r❡♠❛ ♦ ❧❡✐t♦r ♣♦❞❡ ❡♥❝♦♥tr❛r ✉♠ ❡①❡♠♣❧♦ ❞❡ ❞✉❛s C ✲á❧❣❡❜r❛s ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✱ ♠❛s q✉❡ ♥ã♦ sã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛s✱ ♠❡s♠♦ ♣❡r♠✐t✐♥❞♦ ♣r♦❞✉t♦s t❡♥✲ s♦r✐❛✐s ❝♦♠ K(H)✱ ♣❛r❛ H ♥ã♦ s❡♣❛rá✈❡❧✳

      ❈❛♣ít✉❧♦ ✺

    Pr♦❞✉t♦s s♠❛s❤

      ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ◆♦ ❈❛♣ít✉❧♦ ♠♦str❛♠♦s q✉❡ é ♣♦ssí✈❡❧ ❝♦♥str✉✐r ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✳ ❆q✉✐✱ q✉❡r❡♠♦s ♠♦str❛r✱ s♦❜ ❝❡r✲ t❛s ❤✐♣ót❡s❡s✱ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❡st❡s ❞♦✐s ❝♦♥❝❡✐t♦s✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♣r♦✈❛♠♦s q✉❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ q✉❛❧q✉❡r✱ ❞❡ ❢❛t♦✱ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛tr❛✈és ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❜❛s❡ s♦❜r❡ s✉❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡✳ ❚❛✐s ❤✐♣ót❡s❡s ❝♦♥s✐st❡♠ ♥❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ✜❜r❛❞♦✱ e ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡ ❞♦ ❣r✉♣♦ ❡ s❡♣❛r❛❜✐❧✐❞❛❞❡ ❞❛ á❧❣❡❜r❛ B ✳ ❱❡r❡♠♦s ❡①❡♠♣❧♦s q✉❡ ❣❛r❛♥t❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡st❛s ❤✐♣ót❡s❡s✳ ❈♦♠❡ç❛♠♦s ❞❡✜♥✐♥❞♦ C ✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s ❡ ♣❛r❛ ❡st❡ t✐♣♦ ❞❡ C

      ✲á❧❣❡❜r❛✱ ❞❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ s♠❛s❤✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ♠♦str❛♠♦s ♦ t❡♦r❡♠❛ ❞❛ ❞✉❛❧✐❞❛❞❡ ❞❡ ❚❛❦❛✐ ♣❛r❛ ♦ ❝❛s♦ ❞✐s❝r❡t♦ ❡ ♥❛ ú❧t✐♠❛ s❡çã♦✱ ♣♦r ✜♠✱ ❛♣r❡s❡♥t❛♠♦s ♥♦ss♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦✳

      ❆♦ ❧♦♥❣♦ ❞❡st❡ ❝❛♣ít✉❧♦✱ q✉❛♥❞♦ ♥❛❞❛ ❢♦r ❞✐t♦ ❛♦ ❝♦♥trár✐♦✱ G é ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❝♦♠ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ e✳

      ∗

      ✺✳✶ C ✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s g } g

      ❉❡✜♥✐çã♦ ✺✳✶✳✶✳ ❙❡❥❛ B ✉♠❛ C ✲á❧❣❡❜r❛✱ G ✉♠ ❣r✉♣♦ ❡ {B ∈G ✉♠❛ g } g ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❢❡❝❤❛❞♦s ❞❡ B✳ ❉✐③❡♠♦s q✉❡ {B ∈G é ✉♠❛ ❣r❛❞✉❛çã♦ ♣❛r❛ B s❡✱ ♣❛r❛ q✉❛✐sq✉❡r g, h ∈ G t❡♠✲s❡

      = B g −1 ✐✮ B g ✱ g B h ⊆ B gh ✐✐✮ B ✱ g } g g B g

      ✐✐✐✮ ❖s s✉❜❡s♣❛ç♦s {B ∈G sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ❛ s♦♠❛ ❞✐r❡t❛ ⊕ ∈G é ❞❡♥s❛ ❡♠ B✳

      ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ B é ✉♠❛ C ✲á❧❣❡❜r❛ G✲❣r❛❞✉❛❞❛✳ ❖❜s❡r✈❛çã♦ ✺✳✶✳✷✳ ❉❛❞❛ ✉♠❛ C ✲á❧❣❡❜r❛ G✲❣r❛❞✉❛❞❛✱ ❡♥tã♦ ❛ ❝♦❧❡✲ g } g çã♦ ❞❡ s✉❜❡s♣❛ç♦s {B ∈G ❢♦r♠❛ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❤❡r❞❛❞❛s ❞❡ B✳ g } g ❊①❡♠♣❧♦ ✺✳✶✳✸✳ ❙❡❥❛ B = {B ∈G ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧✳ ❊♥tã♦ ❛s ∗ ∗

      C (B) (B)

      ✲á❧❣❡❜r❛s s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❡ s❡❝❝✐♦♥❛❧ r❡❞✉③✐❞❛ C ❡ C r sã♦ G

      ✲❣r❛❞✉❛❞❛s✳ c (B) ❉❡♠♦♥str❛çã♦✿ ❱✐♠♦s q✉❡ C ♣♦ss✉✐ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ✜❡❧ ❡ ∗ ∗ C c (B) = ⊕ g j t (B t ) (B) (B) ∈G ✭Pr♦♣♦s✐çã♦ ❞♦♥❞❡ C ❡ C r sã♦ g } g ❣r❛❞✉❛❞❛s ♣♦r ❝ó♣✐❛s ❞❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s {B ∈G ✳ ∗ ∗

      ◆❡♠ s❡♠♣r❡ ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ♦✉ r❡❞✉③✐❞❛ ❞♦ ✜❜r❛❞♦ ❛ss♦❝✐❛❞♦ à s✉❛ ❣r❛❞✉❛çã♦✱ ❝♦♠♦ ♠♦str❛ ♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✳ ❊①❡♠♣❧♦ ✺✳✶✳✹✳ ❈♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❊①❡♠♣❧♦ s❡ A é ✉♠❛ C ✲á❧❣❡❜r❛ ❡ α é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G ❡♠ A✱ ♦s ♣r♦❞✉t♦s ❝r✉③❛✲ α G α,r G ❞♦s ❝❤❡✐♦ ❡ r❡❞✉③✐❞♦ A ⋊ ❡ A ⋊ sã♦ C ✲á❧❣❡❜r❛s G✲❣r❛❞✉❛❞❛s✱ g δ g } g ❝♦♠ ❛ ❣r❛❞✉❛çã♦ ❞❛❞❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s {D ∈G ✳ ❊①❡♠♣❧♦ ✺✳✶✳✺✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ♥ã♦ ❛♠❡♥❛❜❧❡ ✭❊①❡♠♣❧♦ ❱■■✳✷✳✹✱ ∗ ∗

      (G) ⊕ C (G) ✮ ❡ s❡❥❛ B = C r ✳ ❊♥tã♦ B é G ❣r❛❞✉❛❞❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ g } g g = Cδ g ⊕ Cδ g s✉❜❡s♣❛ç♦s {B ∈G ✱ ❡♠ q✉❡ B ✳ ❊♥tr❡t❛♥t♦✱ B ♥ã♦ é ❛

      C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ♦✉ r❡❞✉③✐❞❛ ❞❡ s❡✉ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦

      B = {B g } g ∈G ✳ g } g = ❉❡♠♦♥str❛çã♦✿ ❖ ❢❛t♦ q✉❡ ❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s {B ∈G {Cδ g ⊕ Cδ g } g ∈G é ✉♠❛ ❣r❛❞✉❛çã♦ ♣❛r❛ B s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❛s ♦♣❡r❛✲ çõ❡s ❡ ♥♦r♠❛ q✉❡ ❢♦r♥❡❝❡♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ C ✲á❧❣❡❜r❛ à s♦♠❛ ❞✐r❡t❛ g } g (G) ❞❡ C ✲á❧❣❡❜r❛s ❡ ❞♦ ❢❛t♦ q✉❡ {Cδ ∈G é ✉♠❛ ❣r❛❞✉❛çã♦ ♣❛r❛ C

      (G) ❡ C r ✳ ∗ ∗ ∗ ∗ ∗

      (B) ∼ (G) ⊕ C (G) (B) ∼ (G) ⊕ ❱❛♠♦s ♠♦str❛r q✉❡ C = C ❡ C r = C r C (G) r ✳ ∗ ∗

      (G) (B) ❉❡ ❢❛t♦✱ ❛s ∗✲r❡♣r❡s❡♥t❛çõ❡s π 1 ❡ π 2 ❞❡ C ❡♠ C ♦❜t✐❞❛s✱ g δ g C r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛tr❛✈és ❞❛s ✐♥❝❧✉sõ❡s ❞❡ ⊕ ∈G ❞❛❞❛s ♣♦r

      X X

      X X λ g δ g 7→ (λ g δ g , 0) λ g δ g 7→ (0, λ g δ g ) g g g g

      ∗ ∗

      ⊕ π (G) ⊕ C (G) ❢♦r♥❡❝❡♠ ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ π = π 1 2 ❞❡ C ❡♠

      C (B) (B) ∗ ∗ ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ φ ❞❡ C ❡♠

      C (G) ⊕ C (G) q✉❡ ❡st❡♥❞❡ ❛ ✐♥❝❧✉sã♦ !

      X X

      X g B g (G) ⊕ C (G) g g g ∗ ∗ −1 (α g δ g , λ g δ g ) 7→ α g δ g , λ g δ g ❞❡ ⊕ ∈G ❡♠ C ✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ φ = π ✳ ∗ ∗ ∗

      (B) ∼ (G) ⊕ C (G) c (B) = P❛r❛ ✈❡r q✉❡ C r = C r r ✱ ♥♦t❡♠♦s q✉❡ C

      C c ({Cδ g } g ) ⊕ C c ({Cδ g } g ) ∈G ∈G ❡ ♦❜s❡r✈❛♥❞♦ ❛ ❝♦♥str✉çã♦ ❞❛ r❡♣r❡✲ s❡♥t❛çã♦ r❡❣✉❧❛r✱ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡✱ s❡ S ❡ T ❞❡♥♦t❛♠✱ r❡s♣❡❝t✐✈❛✲ c (B) c ({Cδ g } g ) ♠❡♥t❡✱ ❛s r❡♣r❡s❡♥t❛çõ❡s r❡❣✉❧❛r❡s ❞❡ C ❡ C ∈G ✱ t❡♠♦s q✉❡ ∗ ∗ ∗ ∗ S = T ⊕ T (B) (G) ⊕ C (G)

      ✳ ❉♦♥❞❡ s❡❣✉❡ q✉❡ C r ❡ C r r sã♦ C ✲á❧❣❡❜r❛s ✐s♦♠♦r❢❛s✳

      ❆ss✐♠✱ B ♥ã♦ ♣♦❞❡ s❡r ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ♦✉ r❡❞✉③✐❞❛ ❞❡ B S = e T ⊕ e T

      ✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❛ ❡①t❡♥sã♦ e ❞❛ r❡♣r❡s❡♥t❛çã♦ r❡❣✉❧❛r ❞❡ C c (B) T s❡r✐❛ ✐♥❥❡t✐✈❛ ❡✱ ♣♦rt❛♥t♦✱ e s❡r✐❛ ✐♥❥❡t✐✈❛✱ ❝♦♥tr❛❞✐③❡♥❞♦ ♦ ❢❛t♦ ❞❡ G ♥ã♦ s❡r ❛♠❡♥❛❜❧❡✳

      ❆ss✐♠ ❝♦♠♦ s✉❛ ❞❡♠♦♥str❛çã♦✱ ♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦ é ♦❜t✐❞♦ ❞❡ ❊①❡♠♣❧♦ ✺✳✶✳✻✳ ❙❡❥❛ B ✉♠❛ C ✲á❧❣❡❜r❛ ❡ s✉♣♦♥❤❛ q✉❡ B ❛❞♠✐t❛ ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ρ ❞❡ ✉♠ ❣r✉♣♦ ❝♦♠♣❛❝t♦ ❛❜❡❧✐❛♥♦ Γ✳ ❙❡❥❛ G ♦ ❣r✉♣♦ ❞✉❛❧ g } g ∈G ❞❡ Γ✳ ❊♥tã♦ B é G✲❣r❛❞✉❛❞❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s {B ✱ ❡♠ q✉❡ ♣❛r❛ ❝❛❞❛ g ∈ G✱

      B g = {b ∈ B : ρ γ (b) = g(γ)b, g } g ∈G ♣❛r❛ t♦❞♦ γ ∈ Γ}. ❉❡♠♦♥str❛çã♦✿ ❆✜r♠❛♠♦s q✉❡ {B é ✉♠❛ ❣r❛❞✉❛çã♦ ♣❛r❛ B✳ γ g

      ❯♠❛ ✈❡③ q✉❡ ρ é ✉♠ ∗✲❛✉t♦♠♦r✜s♠♦✱ ❝❛❞❛ s✉❜❡s♣❛ç♦ B é ❢❡❝❤❛❞♦✳ r s ❆❧é♠ ❞✐ss♦✱ s❡❥❛♠ r, s ∈ G✱ a ∈ B ❡ b ∈ B ✳ P❛r❛ γ ∈ Γ✱ t❡♠♦s

      ρ γ (ab) = ρ γ (a)ρ γ (b) = r(γ)s(γ)ab = rs(γ)ab ❡ ∗ ∗ ∗ −1 ∗

      ρ γ (a ) = ρ γ (a) = r(γ)a = r (γ)a , ❞♦♥❞❡ s❡ ✈❡r✐✜❝❛♠ ♦s ✐t❡♥s ✭✐✮ ❡ ✭✐✐✮ ❞❡ ❘❡st❛ ♠♦str❛r♠♦s q✉❡ ♦s g } g g B g s✉❜❡s♣❛ç♦s {B ∈G sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✈❛❧❡ q✉❡ B = ⊕ ∈G ✳

      P b h = 0 h ∈ B h ❙✉♣♦♥❤❛ q✉❡ ✉♠❛ s♦♠❛ ✜♥✐t❛ ✱ ❡♠ q✉❡ b ♣❛r❛ h

      ❝❛❞❛ h✳ ❆ss✐♠✱ γ ∈ Γ✱

      !

      X X h(γ)b h = ρ γ b h = 0. h h ✭‡✮ ❙❡❥❛ φ ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦ s♦❜r❡ B✳ ❆♣❧✐❝❛♥❞♦ φ ❡♠ ❛♠❜♦s

      ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ q✉❡

      X h φ(b h )h(γ) = 0. ◆♦ ❡♥t❛♥t♦✱ ♣❡❧❛ ✐♥✈❛r✐â♥❝✐❛ ❞❛ ♠❡❞✐❞❛ ❞❡ ❍❛❛r✱ s❡ k, l ∈ G = ˆΓ ❡

      γ ∈ Γ t❡♠♦s q✉❡ Z Z −1 −1 Γ Γ k(η)l (η)µ(dη) = k(γη)l (γη)µ(dη) −1 −1 Z −1 = k(γ)l (γ) k(η)l (η)µ(dη). Γ

      (γ) = 1 ❯♠❛ ✈❡③ q✉❡ k(γ)l ♣❛r❛ t♦❞♦ γ ∈ Γ s❡ ❡ s♦♠❡♥t❡ s❡ k = l✱

      ❝♦♥❝❧✉í♠♦s q✉❡ Z −1 Γ k(η)l (η)µ(dη) = [k = l]

      ❡ ✐st♦ ✐♠♣❧✐❝❛✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ q✉❡ ♦ ❝♦♥❥✉♥t♦ {x ∈ C(Γ) : x ∈ G = ˆΓ} h ) = 0 é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡✳ ❉♦♥❞❡ φ(b ✱ ♣❛r❛ ❝❛❞❛ h ❡✱ ❝♦♠♦ φ

      é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦ t♦♠❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡✱ s❡❣✉❡ q✉❡ b h = 0 g } g ∈G ✱ ♣❛r❛ ❝❛❞❛ h✳ ❈♦♠♦ ✉♠ r❡s✉❧t❛❞♦✱ ♦s s✉❜❡s♣❛ç♦s {B sã♦

      ✐♥❞❡♣❡♥❞❡♥t❡s✳ g ∈G B g P♦r ✜♠✱ ♠♦str❡♠♦s q✉❡ ⊕ é ❞❡♥s♦ ❡♠ B✳ P❛r❛ ✐ss♦✱ s❡❥❛

      P g : B → B ❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❞❛❞❛ ♣♦r

      Z ¯

      P g (b) = g(η)ρ η (b)µ(dη), Γ g ❡♠ q✉❡ b ∈ B ❡ g ∈ G✳ ❆ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ P s❡❣✉❡ ❞♦ s❡❣✉✐♥t❡ ❝á❧❝✉❧♦✿

      Z Z kP g (b)k = g(η)ρ η (b)µ(dη) kρ η (b)kµ(dη) = kbk. Γ Γ g ≤ ◆♦t❡♠♦s q✉❡✱ s❡ b ∈ B ✱ ❡♥tã♦

      Z Z P g (b) = g(η)ρ η (b)µ(dη) = bµ(dη) = b. Γ Γ g (B) = B g

      ▼❛✐s ❛✐♥❞❛✱ P ✱ ♣♦✐s ♣❛r❛ b ∈ B ❡ γ ∈ Γ✱ ♦❜té♠✲s❡ ❞❛ ✐♥✈❛r✐â♥❝✐❛ ❞❛ ♠❡❞✐❞❛ ❞❡ ❍❛❛r µ q✉❡

      Z ρ γ (P g (b)) = ρ γ g(η)ρ η (b)µ(dη) Γ

      Z = g(η)ρ γη (b)µ(dη) Γ

      Z −1 = g(γ γη)ρ γη (b)µ(dη) Γ

      Z = g(γ) g(η)ρ η (b)µ(dη) Γ g (b) ∈ B g = g(γ)P g (b),

      ❞♦♥❞❡ P ✳ g g ▲♦❣♦✱ P é ✉♠❛ ♣r♦❥❡çã♦ ❝♦♥tr❛t✐✈❛ ❝✉❥❛ ✐♠❛❣❡♠ é B ✳ ❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦ φ s♦❜r❡ B q✉❡ s❡ g ∈G B g

      ❛♥✉❧❛ ❡♠ ⊕ ❡ ♣r♦✈❡♠♦s q✉❡ φ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✳ : Γ → C

      ❉❛❞♦ b ∈ B✱ ❛ss♦❝✐❛♠♦s ❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ φ ✱ γ 7→ φ(ρ γ (b)).

      ❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ φ ❛✈❛❧✐❛❞❛ ❡♠ g ∈ G é Z b φ (g) = g(η)φ (η)µ(dη) Γ

      Z = g(η)φ(ρ η (b))µ(dη) g

      Z = φ g(η)ρ η (b)µ(dη) = φ(P g (b)). g B g g (b) ∈ B g Γ ❈♦♠♦ φ s❡ ❛♥✉❧❛ ❡♠ ⊕ ∈G ❡ P ✱ s❡❣✉❡ q✉❡ ❛ tr❛♥s❢♦r✲

      ♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ φ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛ ❡✱ ♣♦rt❛♥t♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ P❧❛♥❝❤❡r❡❧ ✭❚❡♦r❡♠❛ ✶✳✻✳✶✱ s❡❣✉❡ q✉❡ φ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ t♦♠❛♥❞♦ γ = 1 ❝♦♥❝❧✉í♠♦s q✉❡ φ(b) = 0✳ ❖✉ s❡❥❛✱ φ é g } g ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ♣r♦✈❛ ❞❡ q✉❡ {B ∈G ❢♦r♠❛ ✉♠❛ ❣r❛❞✉❛çã♦ ♣❛r❛ B✳

      ✺✳✷ Pr♦❞✉t♦s s♠❛s❤ ❡ ❞✉❛❧✐❞❛❞❡ ❞❡ ❚❛❦❛✐ g B g

      ❙❡❥❛ B = ⊕ ∈G ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ♣♦r ✉♠ ❣r✉♣♦ G✳ ❈♦♥s✐❞❡r❡♠♦s ♦ s✉❜❝♦♥❥✉♥t♦

      ( )

      X 2 S := a gh ⊗ e g,h : a gh ∈ B g −1 h ⊆ B ⊗ K(l (G)), g,h (ξ) = hξ, δ h iδ g (G) k finita 2 ❡♠ q✉❡ e ✱ ξ ∈ l ❡ δ ❞❡♥♦t❛ ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ♥♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ k ∈ G✳ 2 (G)).

      Pr♦♣♦s✐çã♦ ✺✳✷✳✶✳ S é ✉♠❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡ B ⊗ K(l ❉❡♠♦♥str❛çã♦✿ P❛r❛ ✈❡r q✉❡ S é s✉❜á❧❣❡❜r❛✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r♠♦s g,h )(b ⊗ e k,l ) ∈ S −1 −1 . q✉❡ (a ⊗ e ♣❛r❛ a ∈ B g h ❡ b ∈ B k l

      ❉❡ ❢❛t♦✱ s❡ h 6= k ♦ r❡s✉❧t❛❞♦ é ❝❧❛r♦✳ ❙❡ h = k✱ ❡♥tã♦ (a ⊗ e g,h )(b ⊗ e k,l ) = ab ⊗ e g,l ∈ S,

      ✉♠❛ ✈❡③ q✉❡ −1 −1 −1 ab ∈ B g h B h l ⊆ B g l . ∗ ∗ −1 ∈ B = B h g

      ❆❧é♠ ❞✐ss♦✱ S é ❛✉t♦❛❞❥✉♥t♦✱ ♣♦✐s a −1 ❡ (a ⊗ ∗ ∗ g h e g,h ) = a ⊗ e h,g . 2 (G))

      ▲♦❣♦✱ S é ∗✲s✉❜á❧❣❡❜r❛ ❞❡ B ⊗ K(l ✳ g B g ❉❡✜♥✐çã♦ ✺✳✷✳✷✳ ❙❡❥❛ B = ⊕ ∈G ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ♣♦r ✉♠

      (G) ❣r✉♣♦ G✳ ❆ C ✲á❧❣❡❜r❛ ♣r♦❞✉t♦ s♠❛s❤ ❞❡ B✱ ❞❡♥♦t❛❞❛ ♣♦r B#C ✱ 2

      (G)) é ♦ ❢❡❝❤♦ ❞❡ S ❡♠ B ⊗ K(l ✳ ❊①❡♠♣❧♦ ✺✳✷✳✸✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❡ s❡❥❛ B = C ❝♦♠ ❛ G✲ e = C g = {0} ❣r❛❞✉❛çã♦ tr✐✈✐❛❧✱ ♦✉ s❡❥❛✱ B ❡ B ✱ ♣❛r❛ t♦❞♦ g ∈ G \ {e}✳

      (G) ∼ = C (G) ❊♥tã♦ C#C ✳

      (G) ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❛♠♦s q✉❡✱ ♥❡st❡ ❝❛s♦✱ C#C é ♦ ❢❡❝❤♦ ❞❡

      X S = { λ g ⊗ e g,g : λ g ∈ C}. finita P

      λ g ⊗ e g,g ❆❧é♠ ❞✐ss♦✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ b = ❡♠ S ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ g g ❢✉♥çã♦ f : G → C ❞❡ s✉♣♦rt❡ ✜♥✐t♦ ❞❛❞❛ ♣♦r f(g) = λ ✱ ❜❡♠ ❝♦♠♦

      P f (g) ⊗ e g,g q✉❛❧q✉❡r ❢✉♥çã♦ f ❞❡ s✉♣♦rt❡ ✜♥✐t♦ t❡♠ b = ❝♦♠♦ ✉♠ g ❡❧❡♠❡♥t♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡♠ S✳

      P ϕ : S 7→ C c (G) λ g ⊗ e g,g 7→ f

      ➱ ❢á❝✐❧ ✈❡r q✉❡ ❛ ❛♣❧✐❝❛çã♦ e ✱ é ✉♠ g

      ϕ ✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t♦r ❡ s♦❜r❡❥❡t♦r✳ P❛r❛ ✈❡r q✉❡ e s❡ ❡st❡♥❞❡ ❛ ✉♠

      (G) (G) ✐s♦♠♦r✜s♠♦ ❡♥tr❡ C#C ❡ C ✱ ♥♦t❡♠♦s q✉❡ S é ✉♠❛ ✉♥✐ã♦ ❞❡ C

      ✲á❧❣❡❜r❛s✱ ✉♠❛ ✈❡③ q✉❡ ♣❛r❛ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ F ❞❡ G✱ t❡♠✲s❡ q✉❡

      X C (F ) = { λ g ⊗ e g,g : λ g ∈ C} g ∈F ϕ

      é ✉♠❛ C ✲á❧❣❡❜r❛✳ P♦rt❛♥t♦✱ e é t❛♠❜é♠ ✐s♦♠étr✐❝♦✱ ❞♦♥❞❡ s❡ ❡st❡♥❞❡ (G) (G)

      ❛ ✉♠ ∗✲✐s♦♠♦r✜s♠♦ ❡♥tr❡ C#C ❡ C ✳ ❊①❡♠♣❧♦ ✺✳✷✳✹✳ ❙❡❥❛ A C ✲á❧❣❡❜r❛✱ G ❞✐s❝r❡t♦ ❡ α ✉♠❛ ❛çã♦ ✭❣❧♦❜❛❧✮ α G)#C (G) ∼ = A ⊗ K(l (G)) 2

      ❞❡ G ❡♠ A✳ ❊♥tã♦✱ (A ⋊ ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ∗ ∗ 2 (G)#C (G) ∼ = K(l (G)) t♦♠❛♥❞♦ A = C✱ t❡♠✲s❡ q✉❡ C ✳ α G

      ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s A⋊ ❝♦♠ ❛ ❣r❛❞✉❛çã♦ ❞❛❞❛ ♣❡❧❛ ❝♦✲ g } g ∈G g = Aδ g ❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s {B ✱ ❡♠ q✉❡ B ✳ ❙❡❥❛ S ❛ ∗✲s✉❜á❧❣❡❜r❛ α G)#C (G) ❞❡♥s❛ ❞❡ (A ⋊ ❡ ❝♦♥s✐❞❡r❡♠♦s 2

      ϕ : S → A ⊗ K(l (G))

      X X f inita f inita a gh δ −1 ⊗ e gh 7→ α g (a gh ) ⊗ e gh . g h ❆✜r♠❛çã♦ ✶✿ ϕ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ é ✐♥❥❡t✐✈❛✳ 1 , . . . ❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ❞❛ ❖❜s❡r✈❛çã♦ ✻✳✸✳✶ ❞❡ s❡❣✉❡ q✉❡ s❡ v 2 v n ∈ K(l (G)) 1 , . . . , b n ∈ A ⋊ α G sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ b ❡

      P n b j ⊗ v j = 0 1 = · · · = b n = 0 1 ✱ ❡♥tã♦ b ✳ ▲♦❣♦✱ ✉♠❛ ✐❣✉❛❧❞❛❞❡ P −1 a gh δ g h ⊗ e g,h =

      ❡♥tr❡ ❞♦✐s ❡❧❡♠❡♥t♦s ❞❡ S✱ ❞✐❣❛♠♦s✱ x = g,h ∈F 1 P k,l ∈F 2 b kl δ −1 ⊗ e k,l = y k l ✱ ❡♠ q✉❡ F 1 ❡ F 2 sã♦ s✉❜❝♦♥❥✉♥t♦s ✜♥✐t♦s ❞❡ G = F gh δ −1 = b gh δ −1

      ✱ ✐♠♣❧✐❝❛ q✉❡ F 1 2 ❡ a g h g h ✱ ♣❛r❛ ❝❛❞❛ g, h ∈ F 1 ✳ ❆ss✐♠✱ x = y ✐♠♣❧✐❝❛ ϕ(x) = ϕ(y)✱ ❞♦♥❞❡ ϕ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳

      ❆❧é♠ ❞✐ss♦✱ ♣❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❛♣❧✐❝❛❞♦ ❛❝✐♠❛✱ t❡♠♦s q✉❡ s❡ P

      α g (a gh ) ⊗ e g,h = 0 g (a gh ) = 0 ✉♠❛ s♦♠❛ ✜♥✐t❛ ✱ ❡♥tã♦ α ✱ ♦✉ s❡❥❛✱ a gh = 0 g

      ✱ ♣♦✐s α é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ A✱ ♣❛r❛ ❝❛❞❛ g ∈ G✳ ▲♦❣♦✱ ϕ é ✐♥❥❡t✐✈❛✳

      ❆✜r♠❛çã♦ ✷✿ ϕ é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ a, b ∈ A✱ g, h, k, l ∈ G✳ ❚❡♠♦s q✉❡

      ϕ((aδ −1 ⊗ e g,h )(bδ −1 ⊗ e k,l )) = [h = k]ϕ(aα −1 (b)δ −1 ⊗ e g,l ) g h k l g h g l

      = [h = k](α g (a)α h (b) ⊗ e g,l ). P♦r ♦✉tr♦ ❧❛❞♦✱

      ϕ(a)ϕ(b) = (α g (a) ⊗ e g,h )(α h (b) ⊗ e k,l ) = [h = k](α g (a)α h (b) ⊗ e g,l ). ▼❛✐s ❛✐♥❞❛✱ −1 −1 −1 ∗ ∗

      ϕ((aδ g h ⊗ e g,h ) ) = ϕ(α h g (a )δ h g ⊗ e h,g ) ∗ ∗ = α g (a ) ⊗ e h,g = ϕ(a) . ▲♦❣♦✱ ϕ é ∗✲❤♦♠♦♠♦r✜s♠♦✳ ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ ϕ é ✐s♦♠étr✐❝❛ ❡ ❛ss✐♠ s❡ ❡st❡♥❞❡ ❛ ✉♠ 2

      ∗ ϕ α G#C (G) (G)) ✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦ e ❞❡ A ⋊ ❡♠ A ⊗ K(l ✳

      P❛r❛ ✐ss♦✱ ♦❜s❡r✈❛♠♦s q✉❡ S é ✉♠❛ ✉♥✐ã♦ ❞❡ C ✲á❧❣❡❜r❛s✱ ✉♠❛ ✈❡③ q✉❡ ♣❛r❛ ❝❛❞❛ ❝♦♥❥✉♥❢♦ ✜♥✐t♦ F ❞❡ G t❡♠✲s❡ q✉❡    

      X −1 C (F ) := b ∈ S : b = a gh δ g h ⊗ e g,h   g,h ∈F

      é ✉♠❛ C ✲á❧❣❡❜r❛✳ ❈♦♠♦ ❥á ✈✐♠♦s q✉❡ ϕ é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦ ∗ ∗ (F ) s♦❜r❡ S ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s♦❜r❡ ❝❛❞❛ C ✲á❧❣❡❜r❛ C ✱ ❝♦♥❝❧✉í♠♦s q✉❡ ϕ é ✐s♦♠étr✐❝❛✳

      ❉❡st❛ ❢♦r♠❛✱ ❡st❡♥❞❡♠♦s ϕ ♣♦r ❝♦♥t✐♥✉✐❞❛❞❡ ❛ ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ 2 ϕ α G#C (G) (G))

      ✐♥❥❡t✐✈♦ e ❞❡ A ⋊ ❡♠ A ⊗ K(l ❡ ♥♦s r❡st❛ ♣r♦✈❛r q✉❡ ϕ e é s♦❜r❡❥❡t✐✈♦✳

      ❉❛❞♦ a ∈ A ❡ g, h ∈ G✱ ❝♦❧♦❝❛♠♦s b = α −1 (a)δ −1 ⊗ e g,h ∈ A α G#C (G). g g h ⋊ ϕ ϕ(b) = a ⊗ e g,h

      ❆♣❧✐❝❛♥❞♦ e ♦❜t❡♠♦s q✉❡ e ✳ ❯♠❛ ✈❡③ q✉❡ s♦♠❛s ✜♥✐t❛s P 2 a ⊗ e g,h (G)) g,h ∈G ❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡♥s♦ ❡♠ A ⊗ K(l ✱ t❡♠♦s

      ϕ q✉❡ e é s♦❜r❡❥❡t✐✈♦✳ α G#C (G) (G)) 2 P♦rt❛♥t♦✱ s❡❣✉❡ q✉❡ A ⋊ ❡ A ⊗ K(l sã♦ ✐s♦♠♦r❢♦s✳ ❊①❡♠♣❧♦ ✺✳✷✳✺✳ ❙❡❥❛ B ✉♠❛ C ✲á❧❣❡❜r❛ ♣♦ss✉✐♥❞♦ ✉♠❛ ❛çã♦ ❝♦♥tí♥✉❛ ρ

      ❞❡ ✉♠ ❣r✉♣♦ ❝♦♠♣❛❝t♦ ❛❜❡❧✐❛♥♦ Γ✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ♣r♦❞✉t♦ s♠❛s❤ B#C (G)

      ❞❡ B r❡❧❛t✐✈♦ à ❣r❛❞✉❛çã♦ ♣❡❧♦ ❣r✉♣♦ ❞✐s❝r❡t♦ G = bΓ ♦❜t✐❞❛

      ♥♦ ❊①❡♠♣❧♦ ♦✉ s❡❥❛✱ B g = {b ∈ B : ρ γ (b) = g(γ)b, ♣❛r❛ t♦❞♦ γ ∈ Γ}.

      (G) ρ Γ ❊♥tã♦ B#C é ✐s♦♠♦r❢♦ ❛♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ B ⋊ ✳

      (G) ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ S ❛ ∗✲s✉❜á❧❣❡❜r❛ ❞❡♥s❛ ❞❡ B#C ❡ s❡❥❛ ψ : S → C(Γ, B) ⊆ B ⋊ ρ Γ g,h )(γ) = h(γ)a g h −1 ❞❛❞❛ ♣♦r ψ(a ⊗ e ✱ ❡♠ q✉❡ a ∈ B ❡ γ ∈ Γ✳

      ❆ ♣r♦✈❛ ❞❡ q✉❡ ψ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ s❡❣✉❡ ❝♦♠♦ ♥♦ ❊①❡♠♣❧♦ ❱❛♠♦s ♠♦str❛r q✉❡ ψ é ✐♥❥❡t✐✈❛✳

      P a gh ⊗ e g,h ∈ B#C (G) ❙❡ ✉♠❛ s♦♠❛ ✜♥✐t❛ a = g,h é t❛❧ q✉❡

      ψ(a) = 0, ❡♥tã♦

      !

      X X

      X g,h h h(γ)a gh = h(γ) a gh = 0, g ♣❛r❛ t♦❞♦ γ ∈ Γ✳

      ❉❛❞♦ φ ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦ s♦❜r❡ B✱ s❡❣✉❡ q✉❡ !

      X X h g h(γ)φ a gh = 0, γ ∈ Γ. ◆♦ ❡♥t❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ {x ∈ C(Γ) : x ∈ G = ˆΓ} é ❧✐♥❡❛r♠❡♥t❡

      ✐♥❞❡♣❡♥❞❡♥t❡✱ ❡♠ q✉❡ x é ♦ ❝❛rát❡r ❝♦♥tí♥✉♦ s♦❜r❡ Γ ❞❛❞♦ ♣♦r γ 7→ x(γ)✱ P a gh ) = 0

      ♣❛r❛ t♦❞♦ γ ∈ Γ✳ ❉♦♥❞❡ s❡❣✉❡ q✉❡ φ( g ✱ ♣❛r❛ ❝❛❞❛ h✳ ❈♦♠♦ P

      φ a gh = 0 é ❛r❜✐trár✐♦✱ s❡❣✉❡ q✉❡ g ♣❛r❛ ❝❛❞❛ h✳ gh = 0

      ❆ss✐♠✱ ✜①❛❞♦ h✱ t❡♠♦s q✉❡ a ✱ ♣❛r❛ ❝❛❞❛ g✳ ❖✉ s❡❥❛✱ a = P g,h ❡ ψ é ✐♥❥❡t✐✈❛✳ a gh ⊗ e g,h = 0

      ▲♦❣♦✱ s❡ ♣r♦✈❛r♠♦s q✉❡ ψ é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦✱ ❝♦♥❝❧✉✐r❡♠♦s q✉❡ ψ

      é ✐s♦♠étr✐❝❛✱ ♣♦✐s ❥á ✈✐♠♦s q✉❡ S é ✉♠❛ ✉♥✐ã♦ ❞❡ C ✲á❧❣❡❜r❛s✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ ψ é ∗✲❤♦♠♦♠♦r✜s♠♦✳ P❛r❛ ✐ss♦✱ é s✉✜❝✐❡♥t❡ ♠♦s✲ tr❛r♠♦s q✉❡

      ψ ((a ⊗ e g,h )(b ⊗ e r,s )) = ψ(a ⊗ e g,h )ψ(b ⊗ e r,s ) ❡ ∗ ∗ g,h , b ⊗ e r,s ∈ B#C (G) ψ(a ⊗ e g,h ) = ψ ((a ⊗ e g,h ) ) , ♣❛r❛ a ⊗ e ✳ P❛r❛ γ ∈ Γ t❡♠♦s ψ ((a ⊗ e g,h )(b ⊗ e r,s )) (γ) = [h = r]s(γ)ab. ❆❣♦r❛✱ ❝♦♠♦ b ∈ B r −1 s ✱ t❡♠♦s q✉❡ ρ η (b) = η(r −1 s)b

      ✱ ♣❛r❛ ❝❛❞❛ η ∈ Γ

      ⋊ ρ Γ ❡ r❡st❛ ♣r♦✈❛r♠♦s q✉❡ e

      P❡❧♦ t❡♦r❡♠❛ ❞❡ ❙t♦♥❡✲❲❡✐❡rstr❛ss✱ t❡♠♦s q✉❡ span{x ∈ C(Γ) : x ∈ G = ˆ Γ}

      ≤ Z U \K kakµ(dη) < εkak.

      ✱ u(K) = 1 ❡ u(Γ \ U) = 0✳ ❉❛í✱ kaχ E − auk B⋊ ρ Γ ≤ kaχ E − auk 1 = Z S 1 kaχ E (η) − au(η)kµ(dη)

      ❡ ✉♠ ❛❜❡rt♦ U ⊆ Γ t❛✐s q✉❡ K ⊆ E ⊆ U ❡ µ(U \ K) < ε✳ P❡❧♦ ❧❡♠❛ ❞❡ ❯r②s♦❤♥✱ ♦❜t❡♠♦s ✉♠❛ ❢✉♥çã♦ u ∈ C(Γ) t❛❧ q✉❡ 0 ≤ u ≤ 1

      ❙❡❥❛ aχ E ∈ B ⋊ ρ Γ ✉♠❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ♥♦ ❝♦♥❥✉♥t♦ ♠❡♥s✉✲ rá✈❡❧ E ⊆ Γ✳ ❙✉♣♦♥❤❛ q✉❡ a ∈ B h ✱ ♣❛r❛ ❛❧❣✉♠ h ∈ G✳ ❉❛❞♦ ε > 0✱ ✉s❛♥❞♦ ❛ r❡❣✉❧❛r✐❞❛❞❡ ❞❛ ♠❡❞✐❞❛ ❞❡ ❍❛❛r µ ❞❡ Γ ♦❜t❡♠♦s ✉♠ ❝♦♠♣❛❝t♦ K ⊆ Γ

      ψ é s♦❜r❡❥❡t✐✈♦✳

      P♦rt❛♥t♦✱ ψ s❡ ❡st❡♥❞❡ ❛ ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t♦r e ψ : B#C (G) → B

      ❡ ❛ss✐♠ t❡♠♦s (ψ(a ⊗ e g,h ) ∗ ψ(b ⊗ e r,s ))(γ) =

      = h(γ)ρ γ (a) = h(γ)g −1 h(γ)a = g(γ)a = ψ(a ⊗ e h,g )(γ).

      ▼❛✐s ❛✐♥❞❛✱ ψ(a ⊗ e g,h ) (γ) = ρ γ ψ(a ⊗ e g,h )(γ −1 )

      ♥♦s ❞✐③ q✉❡ ψ é ❤♦♠♦✲ ♠♦r✜s♠♦✳

      ❖ ❢❛t♦ q✉❡ R S 1 h(η)r −1 (η)µ(dη) = [h = r]

      Z S 1 h(η)r −1 (η)µ(dη) s(γ)ab.

      Z S 1 h(η)as(η −1 γ)r −1 s(η)bµ(dη) =

      Z S 1 h(η)aρ η s(η −1 γ)b µ(dη) =

      é ❞❡♥s♦ ❡♠ C(Γ)✱ ❞♦♥❞❡ s❡❣✉❡ q✉❡ ❡①✐st❡♠ ❡s❝❛❧❛r❡s c g i ✬s✱ i = 1, 2, . . . , n t❛✐s q✉❡ X u − c g g i < ε. i i

      ▲♦❣♦✱ !

      X aχ E − a c g g i < 2εkak. i i B⋊ Γ ρ P −1

      (c g a ⊗ e g h ,g ) ❈♦❧♦❝❛♥❞♦ b = i i i i ✱ t❡♠♦s q✉❡ k e ψ(b) − aχ E k B⋊ Γ < 2εkak. ρ ❆ss✐♠✱ ♦❜s❡r✈❛♥❞♦ q✉❡ ❛s ❢✉♥çõ❡s ❝❛r❛❝t❡ríst✐❝❛s ❢♦r♠❛♠ ✉♠ ❝♦♥✲ ρ Γ g ∈G B g ψ(B#C (G))

      ❥✉♥t♦ ❞❡♥s♦ ❡♠ B ⋊ ❡ B = ⊕ ✱ t❡♠♦s q✉❡ e é ρ Γ ψ(B#C (G)) = B ρ Γ ❞❡♥s♦ ❡♠ B ⋊ ✱ ♦✉ s❡❥❛✱ e ⋊ ✳

      (G) ρ Γ P♦rt❛♥t♦✱ B#C é ✐s♦♠♦r❢♦ ❛ B ⋊ ✳ ▼✉✐t♦ ❡♠❜♦r❛ ♦ ♣ró①✐♠♦ t❡♦r❡♠❛ s❡❥❛ ✈á❧✐❞♦ ♣❛r❛ ♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧

      ❞❡ ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s ❛❜❡❧✐❛♥♦s ✭✈❡❥❛ ❛♣r❡s❡♥t❛♠♦s ❛q✉✐ ✉♠❛ ❞❡♠♦♥str❛çã♦ ♣❛r❛ ♦ ❝❛s♦ ❞✐s❝r❡t♦ ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦ ❝♦♥t❡①t♦ ❞❡ ♣r♦❞✉t♦s s♠❛s❤✳

      ❆ ✜♠ ❞❡ ♣r♦✈❛r♠♦s ♦ ♣ró①✐♠♦ t❡♦r❡♠❛✱ ❧❡♠❜r❡♠♦s q✉❡✱ s❡ G é x ✉♠ ❣r✉♣♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ ❛ ❛♣❧✐❝❛çã♦ x 7→ ξ é ✉♠ ✐s♦♠♦r✜s♠♦ t♦♣♦❧ó❣✐❝♦ ✭✉♠ ✐s♦♠♦r✜s♠♦ ❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✮ ❞❡ G G x (γ) = γ(x) G

      ❡♠ bb ✱ ❡♠ q✉❡ ξ ✱ ♣❛r❛ t♦❞♦ γ ∈ ˆ ✳ ❊ss❡ r❡s✉❧t❛❞♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ t❡♦r❡♠❛ ❞❛ ❞✉❛❧✐❞❛❞❡ ❞❡ P♦♥tr②❛❣✐♥ ✭❚❡♦r❡♠❛ ✶✳✼✳✷✱

      ❚❡♦r❡♠❛ ✺✳✷✳✻ ✭❉✉❛❧✐❞❛❞❡ ❞❡ ❚❛❦❛✐✮✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛✱ G ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❛❜❡❧✐❛♥♦ ❡ α ❛çã♦ ❞❡ G ❡♠ A✳ ❙❡❥❛ ˆα ❛ ❛çã♦ ❞✉❛❧✳ ❊♥tã♦ 2

      ˆ (A ⋊ α G) ⋊ α ˆ G ∼ = A ⊗ K(l (G)). α G G

      ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ B = A ⋊ ✱ Γ = ˆ ❡✱ ♣❛r❛ ❝❛❞❛ g ∈ G✱ s❡❥❛ B g = {b ∈ B : ˆ α γ (b) = γ(g)b, ♣❛r❛ t♦❞♦ γ ∈ Γ}. P❡❧♦ q✉❡ ✜③❡♠♦s ♥♦ ❊①❡♠♣❧♦ ❡ ♣❡❧❛ ♦❜s❡r✈❛çã♦ ❢❡✐t❛ ❛♥t❡s ❞❡ g } g

      ❡♥✉♥❝✐❛r ❡st❡ r❡s✉❧t❛❞♦✱ t❡♠♦s q✉❡ ❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s {B ∈G é ✉♠❛ ❣r❛❞✉❛çã♦ ♣❛r❛ B✳ g = Aδ g

      ❙✉♣♦♥❤❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ B ✱ ♣❛r❛ t♦❞♦ g ❡♠ G✳ ◆❡st❡ 2 (G) ∼ (G)

      ❝❛s♦✱ ✈✐♠♦s ♥♦ ❊①❡♠♣❧♦ q✉❡ B#C = A ⊗ K(l ❡✱ ♣♦r (G) ∼ α Γ

      ♦✉tr♦ ❧❛❞♦✱ ♦ ❊①❡♠♣❧♦ ♥♦s ❞✐③ q✉❡ B#C = B ⋊ ˆ ✳ ❈♦♥s❡✲ α Γ (G)) 2 q✉❡♥t❡♠❡♥t❡✱ ♦❜t❡♠♦s q✉❡ B ⋊ ˆ ❡ A ⊗ K(l sã♦ ✐s♦♠♦r❢♦s✱ ❝♦♠♦ ❞❡s❡❥❛♠♦s✳ g = Aδ g

      P♦rt❛♥t♦✱ ♦ q✉❡ ♣r❡❝✐s❛♠♦s ❢❛③❡r é ✈❡r✐✜❝❛r q✉❡ B ✱ ♣❛r❛ γ (aδ g ) = γ(g)aδ g ❝❛❞❛ g ❡♠ G✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ ˆα ✱ ♣❛r❛ q✉❛✐sq✉❡r γ ∈ Γ g ⊆ B g ❡ a ∈ A✱ ❛ ✐♥❝❧✉sã♦ Aδ é ✐♠❡❞✐❛t❛✳ g

      ❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡ b ∈ B ❡ s❡❥❛ φ ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦ g Aδ g s♦❜r❡ B✳ ❙❛❜❡♠♦s q✉❡ B = ⊕ ∈G ❡ ❛ss✐♠ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ m ) m g Aδ g m → b s❡q✉ê♥❝✐❛ (x ∈N ❡♠ ⊕ ∈G t❛❧ q✉❡ x ✳ ◆❡st❡ ❝❛s♦✱ ♣❛r❛ γ ∈ Γ

      ✱ t❡♠♦s q✉❡ |φ(ˆ α γ (b)) − φ(ˆ α γ (x m ))| ≤ kφkkˆ α γ (b − x m )k = kφkkb − x m k, γ (x m )) γ (b))

      ❞♦♥❞❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s γ 7→ φ(ˆα ❝♦♥✈❡r❣❡ ❛ γ 7→ φ(ˆα ♥❛ ♥♦r♠❛ ❞♦ s✉♣r❡♠♦ k · k ❞❡ C(Γ)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ t❛❧ ❝♦♥✈❡r❣ê♥❝✐❛ 2

      (Γ) 2 t❛♠❜é♠ s❡ ❞á ❡♠ L ❝♦♠ ❛ ♥♦r♠❛ k · k ✳ ◆♦ ❡♥t❛♥t♦✱ ❡s❝r❡✈❡♥❞♦ P x m = a i δ g i m ♣♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ t❛✐s ❢✉♥çõ❡s sã♦ ♣r❡❝✐s❛♠❡♥t❡ im

      P i m ❡ φ(b)ξ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ φ(a i δ g )ξ g g im im ❉❡st❛ ❢♦r♠❛✱ 2 X 0 = lim φ(b)ξ g − φ(a i δ g )ξ g m i m im im 2 2 X 2

      = φ(b)ξ + g − φ(a j δ g )ξ g φ(a i δ g )ξ g , j = g m m jm jm 2 im im i m 6=j m i 2 ❡♠ q✉❡ g m ♣❛r❛ t♦❞♦ m✳ g − φ(a j δ g )ξ g k → 0 2

      ■ss♦ ✐♠♣❧✐❝❛ q✉❡ kφ(b)ξ m ❡✱ ❝♦♠♦ φ é ❛r❜✐✲ jm jm 2 g trár✐♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ b ♣❡rt❡♥❝❡ ❛♦ ❢❡❝❤♦ ❢r❛❝♦ ❞❡ Aδ ✳ ▼❛s✱ ✉♠❛ ✈❡③ g q✉❡ Aδ é ❝♦♥✈❡①♦ ❡ ❢❡❝❤❛❞♦✱ ❛ss✐♠ t❛♠❜é♠ é ♥❛ t♦♣♦❧♦❣✐❛ ❢r❛❝❛✳ ❖✉ g s❡❥❛✱ b ∈ Aδ ✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛✳

      

    ✺✳✸ ❯♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧

    ❡ ❛çõ❡s ♣❛r❝✐❛✐s

      ◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ♥♦ss♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦✳ ❆♣❧✐❝❛♥❞♦ ♦ q✉❡ ❞❡s❡♥✈♦❧✈❡♠♦s ♥❡st❡ ❡ ♥♦ ❝❛♣ít✉❧♦ ♣r❡❝❡❞❡♥t❡✱ ♠♦str❛♠♦s q✉❡ ✉♠

      ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡stá✈❡❧ s♦❜r❡ ✉♠ ❣r✉♣♦ ❡♥✉♠❡rá✈❡❧ ❝✉❥❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ é s❡♣❛rá✈❡❧ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛tr❛✈és ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ e ❣r✉♣♦ ❜❛s❡ ♥❛ á❧❣❡❜r❛ B ✳

      ❈♦♠❡ç❛♠♦s ❛♣r❡s❡♥t❛♥❞♦ ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ q✉❡ ♥ã♦ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ♥❛ s✉❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡✳

      Pr✐♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❛♠♦s q✉❡ s❡ B é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❞❛❞♦ ♣♦r e g } g = {D g δ g } g ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❡♠ B ✱ ❞✐❣❛♠♦s B = {B ∈G ∈G ✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ g ∈ G −1 −1 −1 −1 −1 −1 B g B g = D g δ g D g δ g = α g (α g (D g )D g )δ e = D g δ e . g B g g B g −1 −1

      ▲♦❣♦✱ ♥❡st❡ ❝❛s♦✱ ♦s ✐❞❡❛✐s B ❡ B sã♦ ✐s♦♠♦r❢♦s✳ n } n ❆❣♦r❛✱ s❡❥❛ B = {B ∈Z ♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❞❛❞♦ ♥♦ ❊①❡♠♣❧♦ n = {0} −1 1

      ❖✉ s❡❥❛✱ B ✱ ♣❛r❛ t♦❞♦ n ∈ Z \ {−1, 0, 1} ❡ B ✱ B ✱ B sã♦ ♦s 3 (C) s✉❜❡s♣❛ç♦s ❞❡ M ❣❡r❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r ♠❛tr✐③❡s ❞❛ ❢♦r♠❛       0 0 0 ∗ 0 0 0 ∗ ∗    ∗ 0 0  , ∗ ∗ 0 0 0  .

       ❡ ∗ 0 0 ∗ ∗ 0 0 0

      ❯♠❛ ✈❡③ q✉❡     0 0 0 ∗ 0 0  

      B B = 0 ∗ ∗  , B = 0 0 0  , −1 1 ❡ B 1 −1 0 ∗ ∗ 0 0 0 s❡❣✉❡✱ ❝♦♠♦ ❥✉st✐✜❝❛♠♦s ❛❝✐♠❛✱ q✉❡ B ♥ã♦ é ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❞❛❞♦ ♣♦r ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❡♠ B ✳

      B B P♦❞❡♠♦s ♣❡r❝❡❜❡r✱ ❡♥tr❡t❛♥t♦✱ q✉❡ B −1 1 ❡ B 1 −1 sã♦ ▼♦r✐t❛ g } g

      ❡q✉✐✈❛❧❡♥t❡s✳ ❉❡ ❢❛t♦✱ ❞❛❞♦ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B = {B ∈G ✱ ♦ s✉❜❡s✲ g g B −1 − B −1 B g ♣❛ç♦ B é ✉♠ B g g ✲❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❝♦♠ ❛s B B hx, yi = xy ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ B ❡ ♣r♦❞✉t♦s ✐♥t❡r♥♦s B B = x y g g−1 ❡ hx, yi g ✳ g−1

      ❉❡st❛ ❢♦r♠❛✱ s✉♣♦♥❞♦ ♣♦r ✉♠ ✐♥st❛♥t❡ q✉❡ ♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ é ❡stá✲ e ✈❡❧✱ ♦✉ s❡❥❛✱ q✉❡ B é ✉♠❛ C ✲á❧❣❡❜r❛ ❡stá✈❡❧✱ é ♣♦ssí✈❡❧ ♣r♦✈❛r q✉❡ ❝❛❞❛ −1 B g ✐❞❡❛❧ B g é t❛♠❜é♠ ❡stá✈❡❧✳ ❈♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ t❡♦r❡♠❛ e ❞❡ ❇r♦✇♥✲●r❡❡♥✲❘✐❡✛❡❧✱ ♥♦ ❝❛s♦ ❡♠ q✉❡ B é t❛♠❜é♠ s❡♣❛rá✈❡❧✱ ❡①✐st❡ −1 B g g B −1 ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ✐❞❡❛✐s B g ❡ B g ✱ ♣❛r❛ ❝❛❞❛ g ∈ G✳

      ❆ss✐♠✱ ♥♦ ❝❛s♦ ❞❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ✭s♦❜r❡ ✉♠ ❣r✉♣♦ ❡♥✉♠❡rá✈❡❧✮ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❛❝✐♠❛✱ ♦✉ s❡❥❛✱ ♣❛r❛ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡stá✈❡❧ e (B) ∗ ∗ t❛❧ q✉❡ B é s❡♣❛rá✈❡❧✱ ✈❛♠♦s ♠♦str❛r q✉❡ C é ✐s♦♠♦r❢♦✱ ❝♦♠♦ C ✲

      á❧❣❡❜r❛ ❣r❛❞✉❛❞❛✱ ❛ ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧✳ s B t = ❉❡✜♥✐çã♦ ✺✳✸✳✶✳ ❯♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ B é ❞✐t♦ s❡r s❛t✉r❛❞♦ s❡ B B st

      ✱ ♣❛r❛ q✉❛✐sq✉❡r s, t ∈ G✳ ❊①❡♠♣❧♦ ✺✳✸✳✷✳ ❙❡❥❛ α ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G ❡♠ A ❡ s❡❥❛ B ♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❛ ❛çã♦ α ❝♦♠♦ ♥❛ Pr♦♣♦s✐çã♦ ◆❡st❡ g = D g δ g ❝❛s♦✱ t❡♠♦s q✉❡ B ✱ ♣❛r❛ ❝❛❞❛ g ∈ G✳ ❊♥tã♦ B é s❛t✉r❛❞♦ g = A s❡✱ ❡ s♦♠❡♥t❡ s❡✱ D ✱ ♣❛r❛ t♦❞♦ g ∈ G✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ B é s❛t✉r❛❞♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ α é ✉♠❛ ❛çã♦ ❣❧♦❜❛❧✳ g −1 ❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❛♠♦s q✉❡✱ ♣❛r❛ a ∈ D ❡ b ∈ D g ✱ t❡♠♦s aδ g ∗ bδ −1 = α g (α −1 (a)b)δ e ∈ D g δ e . g g λ ) λ g P♦r ♦✉tr♦ ❧❛❞♦✱ t♦♠❛♥❞♦ (u ∈Λ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ D ✱ s❡❣✉❡ q✉❡ g B −1 = D g δ g D −1 δ −1 = D g δ e aδ e = lim au λ δ e = lim aδ g ∗ α −1 (u λ )δ −1 . λ λ g g

      ▲♦❣♦✱ B g g g ✳ ❉♦♥❞❡ s❡ B é s❛t✉r❛❞♦✱ g = A g = A ❞❡✈❡♠♦s t❡r D ✱ ♣❛r❛ ❝❛❞❛ g✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ D ✱ ♣❛r❛ ❝❛❞❛ g ∈ G✱ ❡♥tã♦ g B s B t = Aδ s Aδ t = Aδ st , ♣♦✐s ❝❛❞❛ α é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ A✳ g ∈G B g Pr♦♣♦s✐çã♦ ✺✳✸✳✸✳ ❙❡❥❛ B = ⊕ ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ❡ s❡❥❛

      I ♦ ❢❡❝❤♦ ❞❡

      X −1 −1 −1 S := { b g h ⊗ e g,h : b g h ∈ B g B h } ∗ ∗ finita

      (G) ♥❛ ♥♦r♠❛ ❞❡ B#C ✳ ❊♥tã♦ I é ✉♠ ✐❞❡❛❧ ❞❛ C ✲á❧❣❡❜r❛ ♣r♦❞✉t♦

      (G) e s♠❛s❤ B#C ✳ ▼❛✐s ❛✐♥❞❛✱ I é ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡ ❛ B ✳ ❉❡♠♦♥str❛çã♦✿ ➱ s✉✜❝✐❡♥t❡ ♠♦str❛r♠♦s q✉❡ −1 −1

      (b g b h ⊗ e g,h )(a kl ⊗ e k,l ) ∈ I st ⊗ e s,t )(b g b h ⊗ e g,h ) ∈ I, st ∈ B s t g ∈ B g h ∈ B h −1 −1 −1 ❡ (a ❡♠ q✉❡ a ✱ b ❡ b ✳

      ❉❡ ❢❛t♦✱ (b −1 b h ⊗ e g,h )(a st ⊗ e s,t ) = [h = s]b −1 b h a st ⊗ e g,t . g g

      ❙❡ h 6= s ♦ r❡s✉❧t❛❞♦ é ❝❧❛r♦✳ ❙✉♣♦♥❤❛ q✉❡ h = s✱ ❡♥tã♦ b g −1 b h a st ∈ B g −1 B s B s −1 t ⊆ B g −1 B t .

      ❙✐♠✐❧❛r♠❡♥t❡✱ −1 −1 (a st ⊗ e s,t )(b g b h ⊗ e g,h ) = [t = g]a st b g b h ⊗ e s,h , st b g b h ∈ B s B h −1 −1

      ❡ ♥♦ ❝❛s♦ ❡♠ q✉❡ g = t✱ t❡♠✲s❡ a ✳ (G)

      ▲♦❣♦✱ I é ✉♠ ✐❞❡❛❧ ❡♠ B#C ✳ M B e ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ I ∼ ✳ e e ⊗ e e,e ⊆ I e,e ∈ M (I) ■❞❡♥t✐✜❝❛♠♦s B ❝♦♠ B ❡ s❡❥❛ p = 1 ⊗ e ✳ e ⊗ e e,e = pIp

      ❆✜r♠❛♠♦s q✉❡ B ❡ p é ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛✳ e ⊗ e e,e = pIp ❉❡ ❢❛t♦✱ é ❢á❝✐❧ ✈❡r q✉❡ B ✳ P❛r❛ ✈❡r q✉❡ p é ♣r♦❥❡çã♦ −1 b h ⊗ e g,h ∈ I

      ❝❤❡✐❛✱ ♥♦t❡♠♦s q✉❡ ❞❛❞♦ a = b g ✱ t❡♠✲s❡ a = (b −1 ⊗ e g,e )p(b h ⊗ e e,h ), g ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ IpI é ❞❡♥s♦ ❡♠ I✳ ▲♦❣♦✱ p é ♣r♦❥❡çã♦ ❝❤❡✐❛✳ e ❆ss✐♠✱ B ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ❝❛♥t♦ ❝❤❡✐♦ ❞❡ I ❡✱ ♣♦rt❛♥t♦✱ sã♦ C

      ✲á❧❣❡❜r❛s ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✳ g B g ❖❜s❡r✈❛çã♦ ✺✳✸✳✹✳ ❙❡ B = ⊕ ∈G é ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ❡ ♦ g } g ✜❜r❛❞♦ ❞❡ ❋❡❧❧ {B ∈G é s❛t✉r❛❞♦✱ s❡❣✉❡ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r q✉❡ B e (G)

      ❡ B#C sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ ❛❣♦r❛ é ❝♦♥str✉✐r ✉♠❛ ❛çã♦ ❞❡ G ♥❛ C ✲á❧❣❡❜r❛ ♣r♦✲

      (G) ❞✉t♦ s♠❛s❤ B#C ✱ ❡ ❛ss✐♠ ♦❜t❡r ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❡♠ I ♣♦r r❡s✲ tr✐çã♦✳ g ∈ B(l (G)) 2

      ❙❡❥❛ λ ❞❛❞♦ ♣♦r −1 2 ∗ λ g (ξ)(h) = ξ(g

      h), −1 (G) g = λ g

      ❡♠ q✉❡ ξ ∈ l ✳ ❊♥tã♦ ❝❛❞❛ λ é ✉♥✐tár✐♦ ❡ λ g ✳ ▼❛✐s ❛✐♥❞❛✱ 2 λ gh = λ g λ h

      (G)) ✱ ♣❛r❛ q✉❛✐sq✉❡r g, h ∈ G✳ ❆ ❛♣❧✐❝❛çã♦ λ : G → U(l ✱ g 7→ λ g

      é ❝❤❛♠❛❞❛ r❡♣r❡s❡♥t❛çã♦ r❡❣✉❧❛r à ❡sq✉❡r❞❛✳ ◆❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦✱ ✈❛♠♦s ♠♦str❛r q✉❡✱ ♣❛r❛ ❝❛❞❛ g ∈ G✱ ♦ g (G)

      ❡❧❡♠❡♥t♦ 1 ⊗ λ ❞á ♦r✐❣❡♠ ❛ ✉♠ ∗✲❛✉t♦♠♦r✜s♠♦ ❞❡ B#C ✳ g : B#C (G) → B#C (G) ∗ ∗ Pr♦♣♦s✐çã♦ ✺✳✸✳✺✳ ❙❡❥❛ θ ❞❛❞❛ ♣♦r a 7→ (1 ⊗ λ g )a(1 ⊗ λ −1 ). g

      g

      (G) ❊♥tã♦✱ θ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ ❞❡✜♥❡ ✉♠ ∗✲❛✉t♦♠♦r✜s♠♦ ❞❡ B#C ✳ g (G) ▼❛✐s ❛✐♥❞❛✱ ❛ ❛♣❧✐❝❛çã♦ g 7→ θ é ✉♠❛ ❛çã♦ ❞❡ G ❡♠ B#C ✳ g )a(1 ⊗ ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ t❡♠♦s ❞❡ ♠♦str❛r q✉❡ (1 ⊗ λ ∗ ∗ λ g −1 ) ∈ B#C (G) (G)

      ✱ ♣❛r❛ t♦❞♦ a ∈ B#C ✳ ❖❜s❡r✈❛♠♦s q✉❡ ♣❛r❛ h ∈ G ✱ −1 −1 −1 e s,t (λ g (δ h )) = e s,t (δ g h ) = [g h = t]δ s = [h = gt]δ s , s,t λ g = e s,gt −1

      ❞♦♥❞❡ s❡❣✉❡ q✉❡ e ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ λ g (e s,t (δ h )) = [h = t]λ g (δ s ) = δ gs g e s,t = e gs,t

      ❡ ❝♦♥❝❧✉í♠♦s q✉❡ λ ✳ s,t ∈ B#C (G) ❆ss✐♠✱ s❡ a = b ⊗ e ✱ s❡❣✉❡ q✉❡

      (1⊗ λ g )a(1⊗ λ g −1 ) = (1⊗ λ g )b⊗ e s,t (1⊗ λ g −1 ) = b⊗ e gs,gt ∈ B#C (G), s −1 t = B −1 ♣♦✐s b ∈ B (gs) (gt) ✳ ❈♦♠♦ ❡❧❡♠❡♥t♦s ❞❡st❛ ❢♦r♠❛ ❣❡r❛♠ B#C (G) g

      ✱ s❡❣✉❡ q✉❡ θ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ g (G)) g 2 ❯♠❛ ✈❡③ q✉❡ ❝❛❞❛ λ é ✉♠ ✉♥✐tár✐♦ ❡♠ B(l ✱ t❡♠✲s❡ q✉❡ θ é

      (G) ✉♠ ∗✲❛✉t♦♠♦r✜s♠♦ ❞❡ B#C ✳ P♦rt❛♥t♦✱ r❡st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ ❛ g (G) ❛♣❧✐❝❛çã♦ g 7→ θ é ✉♠❛ ❛çã♦ ❞❡ G ❡♠ B#C ✳

      (G) ❉❡ ❢❛t♦✱ s❡❥❛♠ a ∈ B#C ❡ g, h ❡♠ G✳ ❊♥tã♦

      θ g (θ h (a)) = (1 ⊗ λ g )(1 ⊗ λ h )a(1 ⊗ λ h −1 )(1 ⊗ λ g −1 ) −1 −1 = (1 ⊗ λ gh )a(1 ⊗ λ h g ) = θ gh (a). ❖ s❡❣✉✐♥t❡ é ♥♦ss♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦✿ g } g

      ❚❡♦r❡♠❛ ✺✳✸✳✻✳ ❙❡❥❛ B = {B ∈G ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❡stá✈❡❧ s♦❜r❡ ✉♠ ❣r✉♣♦ ❡♥✉♠❡rá✈❡❧ G ❝✉❥❛ á❧❣❡❜r❛ r❡❧❛t✐✈❛ à ✜❜r❛ ✉♥✐❞❛❞❡ é s❡♣❛rá✈❡❧✳ e ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G ♥❛ á❧❣❡❜r❛ B t❛❧ q✉❡ ♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦ é ✐s♦♠♦r❢♦ ❛ B✳ ∗ ∗

      (B) ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ C ❛ C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❞♦ ✜❜r❛❞♦ ∗ ∗ B (B)

      ✳ ▲❡♠❜r❡♠♦s q✉❡ C é ✉♠❛ C ✲á❧❣❡❜r❛ G✲❣r❛❞✉❛❞❛✱ ❝♦♠ ❛ ❣r❛❞✉✲ g } g ❛çã♦ ❞❛❞❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s {B ∈G ✳ ❙❡❥❛ I ♦ ✐❞❡❛❧ ❞❛ C ✲

      (G) á❧❣❡❜r❛ ♣r♦❞✉t♦ s♠❛s❤ B#C ❞❡✜♥✐❞♦ ❝♦♠♦ ♥❛ Pr♦♣♦s✐çã♦ g } g , {α g } g ) ❡ s❡❥❛ α = ({D ∈G ∈G ❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G ❡♠ I✱ ❝♦♠♦ ♥❛ ∗ ∗

      (B) ❖❜s❡r✈❛çã♦ ❱❛♠♦s ♠♦str❛r q✉❡ C é ✐s♦♠♦r❢❛✱ ❝♦♠♦ C ✲ α G á❧❣❡❜r❛ ❣r❛❞✉❛❞❛✱ ❛♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ (I ⊗ K) ⋊ ⊗1 ✱ ❡♠ q✉❡

      α ⊗ 1 = ({D g ⊗ K} g , {α g ⊗ 1} g ) ∈G ∈G é ❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G ❡♠ I ⊗ K ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡ α✳ α G

      ❖❜s❡r✈❛♠♦s q✉❡✱ ❝♦♠♦ ❛ ✐♥❝❧✉sã♦ ι ❞❡ I ❡♠ I ⋊ é ♥ã♦ ❞❡❣❡✲ α G ♥❡r❛❞❛ ❡ ✐♥❥❡t✐✈❛ ✭I ❝♦♥té♠ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ I ⋊ ✮✱ ♣♦❞❡♠♦s ✈❡r s✉❛ á❧❣❡❜r❛ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s M(I) ❝♦♠♦ ✉♠❛ s✉❜á❧✲ α

      G) α

      G) ❣❡❜r❛ ❞❡ M(I ⋊ ✱ ❝♦♥t❡♥❞♦ ❛ ✉♥✐❞❛❞❡ ❞❡ M(I ⋊ ✱ ❛tr❛✈és ❞❛ ❡①t❡♥sã♦ eι ✭Pr♦♣♦s✐çã♦ ❆❧é♠ ❞✐ss♦✱ é ✐♠♣♦rt❛♥t❡ ♣❡r❝❡❜❡r q✉❡✱ h ∈ D h δ h s❡ v ∈ M(I) ❡ aδ ✱

      ˜ι(v)aδ h = lim ι(vu λ )aδ h = lim vu λ δ e aδ h = lim vu λ aδ h = vaδ h , λ λ λ−1 aδ h ˜ι(v) = aδ h lim ι(u λ v) = lim aδ h u λ vδ e = lim α h (α h (a)u λ v)δ h λ λ λ −1 = α h (α h (a)v)δ h . e

      ❊s❝r❡✈❡♠♦s vδ ♣❛r❛ ˜ι(v) ❡ ❛ss✐♠✱ −1 vδ e aδ h = vaδ h h vδ e = α h (α h (a)v)δ h , ❡ aδ e D h δ h ⊆ D h δ h

      ♣❛r❛ t♦❞♦ v ∈ M(I)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡ vδ ❡ D h δ h vδ e ⊆ D h δ h

      ✳ ❊st❡ ❢❛t♦ s❡rá ✐♠♣♦rt❛♥t❡ ❛♦ ❧♦♥❣♦ ❞❛ ❞❡♠♦♥str❛çã♦✱ ✉♠❛ ✈❡③ q✉❡ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ✐s♦♠♦r✜s♠♦s ❡♥tr❡ C ✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s✳ e,e ∈ M (I) e ∈ M (I ⋊ α

      G) ❈♦♥s✐❞❡r❡♠♦s p = 1 ⊗ e ❡ s❡❥❛ pδ ✳ ▼♦s✲ e (B) tr❡♠♦s q✉❡ pδ é ♣r♦❥❡çã♦ ❝❤❡✐❛ ❡ C é ✐s♦♠♦r❢❛ ❛♦ ❝❛♥t♦ ❝❤❡✐♦ pδ e (I ⋊ α G)pδ e

      ✳ α G)pδ e (I ⋊ α

      G) ⊇ (Iδ e )pδ e (Iδ e ) = (IpI)δ e ❉❡ ❢❛t♦✱ (I ⋊ ✳ ❏á ✈✐♠♦s q✉❡ p é ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛ ✈✐st❛ ❝♦♠♦ ✉♠ ♠✉❧t✐♣❧✐❝❛❞♦r ❞❡ I ❡ ❛ss✐♠ s❡❣✉❡ q✉❡

      (I α G)pδ e (I α G) ⊇ Iδ e . α G)pδ e (I α ⋊ ⋊

      G) ◆❡st❡ ❝❛s♦✱ ♦ ✐❞❡❛❧ (I ⋊ ⋊ ❝♦♥té♠ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐✲ α G e ♠❛❞❛ ♣❛r❛ I ⋊ ✱ ♦ q✉❡ ♥♦s ❞✐③ q✉❡ pδ é t❛♠❜é♠ ✉♠❛ ♣r♦❥❡çã♦ ❝❤❡✐❛✳ e (I α G)pδ e = B

      ❆✜r♠❛♠♦s q✉❡ pδ ⋊ ✱ ❡♠ q✉❡ B := span{b h ⊗ e e,h δ h : h ∈ G, b h ∈ B h }. r ∈ D r δ r e (I α G)pδ e ❙❡❥❛ y = aδ t❛❧ q✉❡ y ∈ pδ ⋊ ✳ P♦r ✉♠ ❧❛❞♦✱ ✐ss♦ s✐❣♥✐✜❝❛ q✉❡ aδ r = (1 ⊗ e e,e δ e )aδ r = (1 ⊗ e e,e )aδ r . e,e )a h ⊗e e,h :

      ❖✉ s❡❥❛✱ a = (1⊗e ❡ ♥❡st❡ ❝❛s♦✱ ♦❜s❡r✈❛♠♦s q✉❡ a ∈ span{b h ∈ G, b h ∈ B h } ✳

      P♦r ♦✉tr♦ ❧❛❞♦✱ −1 aδ r = aδ r (pδ e ) = α r (α r (a)(1 ⊗ e e,e ))δ r , r (a)(1⊗e e,e ) = α r (a) r (a) ∈ span{b h ⊗ −1 −1 −1 ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ α ✱ ❞♦♥❞❡ α e h,e : h ∈ G, b h ∈ B −1 }. −1 (a) = θ −1 (a) −1 (b h ⊗ h ▲❡♠❜r❛♥❞♦ q✉❡ α r r ❡ θ r e e,h ) = b h ⊗e −1 −1 r ⊗e e,r r ∈ B r r ,r h ✱ ❝♦♥❝❧✉í♠♦s q✉❡ a = b ✱ ♣❛r❛ ❛❧❣✉♠ b ✳ α G a g δ g ∈ ⊕ g D g δ g P

      ❆❣♦r❛✱ s❡❥❛ a ∈ I ⋊ ❡ ε > 0✳ ❙❡❥❛ x = g ∈G t❛❧ q✉❡ ka − xk < ε. ❉❡st❛ ❢♦r♠❛✱ kpδ e apδ e − pδ e xpδ e k < ε. ◆♦ ❡♥t❛♥t♦✱

      !

      X X pδ e xpδ e = pδ e a g δ g pδ e = pδ e (a g δ g )pδ e . e g g e e α e g g g g (a δ )pδ ∈ pδ (I ⋊ G)pδ δ

      P❛r❛ ❝❛❞❛ g✱ pδ é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ D e D h δ h ⊆ D h δ h h δ h vδ e ⊆ D h δ h ✭❛q✉✐ ❡♥tr❛ ♦ ❢❛t♦ q✉❡ vδ ❡ D ✱ ♣❛r❛ t♦❞♦ v ∈ M (I) g ∈ B g

      ✮✳ P❡❧♦ q✉❡ ❥á ❢♦✐ ❢❡✐t♦✱ s❡❣✉❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ g✱ ❡①✐st❡ b e (a g δ g )pδ e = b g ⊗ e e,g δ g t❛❧ q✉❡ pδ ✳ ▲♦❣♦✱ pδ e xpδ e ∈ span{b h ⊗ e e,h δ h : b h ∈ B h } ⊆ B ❡ kpδ e apδ e − pδ e xpδ e k < ε. e apδ e ∈ B e (I ⋊ α

      ❈♦♠♦ ε é ❛r❜✐trár✐♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ pδ ✳ ❉♦♥❞❡ pδ G)pδ e ⊆ B

      ✳ P❛r❛ ❛ ✐♥❝❧✉sã♦ ✐♥✈❡rs❛✱ é s✉✜❝✐❡♥t❡ ♦❜s❡r✈❛r♠♦s q✉❡ pδ e (b h ⊗ e eh δ h ) = (1 ⊗ e ee )(b h ⊗ e eh )δ h = b h ⊗ e eh δ h

      ❡ b h ⊗ e e,h δ h (1 ⊗ e e,e δ e ) = α h (α h −1 (b h ⊗ e e,h )1 ⊗ e e,e )δ h −1 h ∈ B h = α h (b h ⊗ e h ,e )δ h = b h ⊗ e e,h δ h , ♣❛r❛ q✉❛✐sq✉❡r h ∈ G ❡ b ✳ e (I α

      ❯s❛♥❞♦ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣r♦✈❛❞❛ ❛❝✐♠❛✱ ✈❛♠♦s ♠♦str❛r q✉❡ pδ ⋊ ∗ ∗ G)pδ e (B) (B)

      é ✐s♦♠♦r❢♦ ❛ C ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ t❡r❡♠♦s q✉❡ C ❡ I α G ⋊ sã♦ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✳ ❆ ✜♠ ❞❡ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦✱ ❛ ♣❛rt✐r e (I α G)pδ e

      ❞❡ ❛❣♦r❛ ❡s❝r❡✈❡♠♦s B ❡♠ ✈❡③ ❞❡ pδ ⋊ ✳ ❈♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦

      ϕ : ⊕ g B g → B ∈G

      X X g g ∈G ∈G b g 7→ b g ⊗ e e,g δ g . g ∈G B g ❆✜r♠❛♠♦s q✉❡ ϕ é ✉♠❛ ∗✲r❡♣r❡s❡♥t❛çã♦ ❞❡ ⊕ ✳

      ❚❡♠♦s q✉❡ ϕ(b g )ϕ(b h ) = (b g ⊗ e e,g δ g )(b h ⊗ e e,h δ h ) −1

      = α g ((b g ⊗ e g e )(b h ⊗ e e,h ))δ gh = b g b h ⊗ e e,gh δ gh = ϕ(b g b h ),

      ❡ ∗ ∗ ∗ ∗ −1 −1 ϕ(b h ) = (b h ⊗ e e,h δ h ) = b ⊗ e e,h δ h = ϕ(b ). h h (B), ϕ

      P♦rt❛♥t♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❡ C s❡ ❡st❡♥❞❡ ❛ ✉♠ ∗ ′ ∗ ϕ : C (B) → B

      ✲❤♦♠♦♠♦r✜s♠♦ e ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♥s✐❞❡r❡♠♦s ❛ ❛♣❧✐❝❛çã♦ 2

      ψ : ⊕ g D g δ g → C (B) ⊗ K(l (G)) ∈G

      X a g δ g 7→ a g (1 ⊗ λ g ), ❡ ♠♦str❡♠♦s q✉❡ ψ é ✉♠❛ ∗✲❤♦♠♦♠♦r✜s♠♦✳

      ❉❡ ❢❛t♦✱ ✐ss♦ s❡❣✉❡ ❞♦s s❡❣✉✐♥t❡s ❝á❧❝✉❧♦s✿ ψ(aδ r )ψ(bδ s ) = a(1 ⊗ λ r )b(1 ⊗ λ s ) −1

      = a(1 ⊗ λ r )b(1 ⊗ λ r )(1 ⊗ λ r )(1 ⊗ λ s ) = aθ r (b)(1 ⊗ λ rs ) −1 = θ r (θ r (a)b)(1 ⊗ λ rs ) −1 = α r (α r (a)b)(1 ⊗ λ rs ) = ψ ((aδ r )(bδ s ))

      ❡ ∗ ∗ ∗ ψ(aδ r ) = [a(1 ⊗ λ r )] = (1 ⊗ λ −1 )a r

      = (1 ⊗ λ −1 )a (1 ⊗ λ r )(1 ⊗ λ −1 ) r r ∗ ∗ = α −1 (a )(1 ⊗ λ −1 ) = ψ ((aδ r ) ) , r r r s

      ❡♠ q✉❡ a ∈ D ❡ b ∈ D ✳ α ❆❣♦r❛✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ I ⋊ 2 G ψ : I α G → C (B)⊗K(l (G))

      ✱ ψ s❡ ❡st❡♥❞❡ ❛ ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ e ⋊ ✳ h ∈ B h P♦r ✜♠✱ ♦❜s❡r✈❛♠♦s q✉❡ ♣❛r❛ b ✱ e

      ψ(b h ⊗ e e,h δ h ) = b h ⊗ e e,e ∈ C (B) ⊗ e e,e , ∗ ∗ −1 (B) (B) ⊗ e e,e ϕ

      ❡ ✐❞❡♥t✐✜❝❛♥❞♦ C ❝♦♠ C ✱ t❡♠♦s q✉❡ e ψ = e ✳ ❖✉ s❡❥❛✱ ∗ ′ ∗ (B)

      ✜❝❛ ♣r♦✈❛❞♦ q✉❡ C ❡ B sã♦ C ✲á❧❣❡❜r❛s ✐s♦♠♦r❢❛s✳ e ❈♦♠♦ G é ❡♥✉♠❡rá✈❡❧ ❡ B é s❡♣❛rá✈❡❧✱ s❡❣✉❡ q✉❡ I é ♣♦ss✉✐ ❡❧❡✲

       ee ∈ M (I)

      ♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❈♦♠♦ p = 1 ⊗ e é ✉♠❛ ♣r♦✲ ❥❡çã♦ ❝❤❡✐❛✱ ♦ ▲❡♠❛ ♥♦s ❞✐③ q✉❡ ❡①✐st❡ ✉♠❛ ✐s♦♠❡tr✐❛ ♣❛r❝✐❛❧ ∗ ∗ v ∈ M (I ⊗ K) v = 1 ⊗ 1 = p ⊗ 1 t❛❧ q✉❡ v ❡ vv ✳ α

      G) ⊗ K) ❙❡❥❛ ι ⊗ 1 ❛ ✐♥❝❧✉sã♦ ❞❡ I ⊗ K ❡♠ M((I ⋊ ✳ ❖✉ s❡❥❛✱ ❡♠ e ⊗ k ι ⊗ 1

      ✉♠ t❡♥s♦r ❡❧❡♠❡♥t❛r a ⊗ k t❡♠✲s❡ (ι ⊗ 1)(a ⊗ k) = aδ ✳ ❙❡❥❛ ] ❛ ❡①t❡♥sã♦ ❞❡ ι⊗1 à á❧❣❡❜r❛ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s M(I ⊗K)✳ ❈♦♥s✐❞❡r❛♥❞♦ ∗ ∗ u = (] ι ⊗ 1)(v) ∈ M ((I ⋊ α

      G) ⊗ K) u = 1 = pδ e ⊗ 1 ✱ t❡♠♦s q✉❡ u ❡ uu ✳

      ❈♦♠♦ ♣♦❞❡♠♦s ✈❡r ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❈♦r♦❧ár✐♦ ❛ ✐s♦♠❡tr✐❛ ⊗ K α

      G) ⊗ K ♣❛r❝✐❛❧ u ❞á ♦r✐❣❡♠ ❛ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ B ❡ (I ⋊ ∗ ′ yu ⊗ K ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ y 7→ u ✱ ♣❛r❛ y ∈ B ✳ ❚❛❧ ✐s♦♠♦r✜s♠♦ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ C ✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s✱ ♣♦✐s u é ✐♠❛❣❡♠ ❞❡ ✉♠

      ι ⊗ 1 ♠✉❧t✐♣❧✐❝❛❞♦r ❞❡ I ⊗ K ♣♦r ] ❡✱ ♣♦rt❛♥t♦✱ ♣r❡s❡r✈❛ ❛ ❣r❛❞✉❛çã♦ ✭❛♥á❧♦❣♦ ❛♦ ❢❡✐t♦ ♥♦ ✐♥í❝✐♦ ❞❡st❛ ❞❡♠♦♥str❛çã♦✮✳ α

      G) ⊗ K ∼ α G P❡❧❛ Pr♦♣♦s✐çã♦ s❡❣✉❡ q✉❡ (I ⋊ = I ⊗ K ⋊ ⊗1 ∗ ∗ ∗

      (B) ❡ ♣❡❧❛ Pr♦♣♦s✐çã♦ C é ✉♠❛ C ✲á❧❣❡❜r❛ ❡stá✈❡❧ ❝♦♠♦ C ✲ ∗ ′

      (B) á❧❣❡❜r❛ ❣r❛❞✉❛❞❛✳ ❈♦♠♦ ♦ ✐s♦♠♦r✜s♠♦ ❝♦♥str✉í❞♦ ❡♥tr❡ C ❡ B

      (B) α G t❛♠❜é♠ ♣r❡s❡r✈❛ ❛ ❣r❛❞✉❛çã♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ C ❡ (I ⊗K)⋊ ⊗1 sã♦ C ✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s ✐s♦♠♦r❢❛s✳ e ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G s♦❜r❡ ❛ C ✲á❧❣❡❜r❛ B ❝♦rr❡s✲

      ♣♦♥❞❡♥t❡ ❛ α ⊗ 1✱ ♦❜t❡♠♦s ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ✐s♦♠♦r❢♦ ❛ B ✭❊①❡♠♣❧♦ ❖❜s❡r✈❛çã♦ ✺✳✸✳✼✳ ➱ ♣♦ssí✈❡❧ ❡♥❢r❛q✉❡❝❡r ❛ ❤✐♣ót❡s❡ ❞♦ ❚❡♦r❡♠❛ g −1 B g ❡①✐❣✐♥❞♦ q✉❡ ❝❛❞❛ ✐❞❡❛❧ B ♣♦ss✉❛ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✱ e ❡♠ ✈❡③ ❞❛ s❡♣❛r❛❜✐❧✐❞❛❞❡ ❞❡ B ✳

      ❆ss✉♠✐♥❞♦ ✉♠ ❝♦♥tr❛✲❡①❡♠♣❧♦ ♣❛r❛ ♦ t❡♦r❡♠❛ ❞❡ ❇r♦✇♥✲●r❡❡♥✲ ❘✐❡✛❡❧ ✭❚❡♦r❡♠❛ ✱ ✜❝❛ ❢á❝✐❧ ❡♥❝♦♥tr❛r ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ ✜❜r❛❞♦ g

      ✶ , g , . . . , g , . . .

      ❙❡❥❛ {g n } ✉♠❛ ❡♥✉♠❡r❛çã♦ ♣❛r❛ G ❡ s❡❥❛ (v n ) n∈N ✉♠❛ ✉♥✐❞❛❞❡ 1 2 g B g n = v ⊗ P n i e ❛♣r♦①✐♠❛❞❛ ❡♥✉♠❡rá✈❡❧ ♣❛r❛ B −1 ✳ P❛r❛ n ∈ N✱ ❝♦❧♦❝❛♠♦s u n g g ✳ ❊♥tã♦ (u n ) n∈N é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ I✳ i i g i =1

      ❞❡ ❋❡❧❧ ❡stá✈❡❧ s♦❜r❡ ✉♠ ❣r✉♣♦ ❡♥✉♠❡rá✈❡❧ q✉❡ ♥ã♦ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✳

      ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t❡s✱ ♠❛s ♥ã♦ ❡st❛✈❡❧♠❡♥t❡ ✐s♦♠♦r❢❛s✳ ❈❧❛r♦✱ ✉♠❛ ❞❡❧❛s ♥ã♦ ❝✉♠♣r❡ ❛ ❤✐♣ót❡s❡ ❞❡ ♣♦ss✉✐r ✉♠ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✳ ❙❡❥❛ X ✉♠ A⊗K −B ⊗K✲ ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❡ s❡❥❛ C ❛ á❧❣❡❜r❛ ❞❡ ❧✐❣❛çã♦ ❞❡ X✳ ❙❡❥❛ {B n } n ∈Z ♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❝♦♠♦ ♥♦ ❊①❡♠♣❧♦ ❖✉ s❡❥❛✱ B ✱ n = {0} ♣❛r❛ ❝❛❞❛ n ∈ Z \ {−1, 0, 1}✱ ❡ B −1 ✱ B ✱ B 1 sã♦ ♦s s✉❜❡s♣❛ç♦s ❞❡ C ❞❡✜♥✐❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r

      A ⊗ K

      X , . ❡ e

      X B ⊗ K n } n ∈Z ❈♦♠♦ ❥á ❛r❣✉♠❡♥t❛♠♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ s❡ B = {B é ✉♠ e

      ✜❜r❛❞♦ ❞❡ ❋❡❧❧ r❡❧❛t✐✈♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ Z ❡♠ B ✱ ❡♥tã♦ ❞❡✈❡ B B

      ❡①✐st✐r ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ✐❞❡❛✐s B −1 1 ❡ B 1 −1 ✳ ◆♦ ❡♥t❛♥t♦✱

      0 X B B = = −1 1 e

      X

      0 B ⊗ K ❡

      0 X A ⊗ K B B = = , 1 −1 e

      X q✉❡ ♥ã♦ sã♦ C ✲á❧❣❡❜r❛s ✐s♦♠♦r❢❛s✳ ❱❛♠♦s ❛♣r❡s❡♥t❛r ❛❣♦r❛ ❞♦✐s ❝♦r♦❧ár✐♦s ❞♦ ❚❡♦r❡♠❛ ◆♦ ♣r✐✲

      ♠❡✐r♦✱ ❛♣❧✐❝❛♠♦s ♦ t❡♦r❡♠❛ ♣❛r❛ ❛ ❡st❛❜✐❧✐③❛çã♦ ❞❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ s❛t✐s❢❛③❡♥❞♦ ❛s ❤✐♣ót❡s❡s ❞❡ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡ ❞♦ ❣r✉♣♦ ❡ s❡♣❛r❛❜✐❧✐❞❛❞❡ ❞❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡✳ g } g ❈♦r♦❧ár✐♦ ✺✳✸✳✽✳ ❙❡❥❛ B = {B ∈G ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ s♦❜r❡ ✉♠ ❣r✉♣♦ ❡♥✉♠❡rá✈❡❧ G ❝✉❥❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ é s❡♣❛rá✈❡❧ ❡ s❡❥❛ B⊗K s✉❛ e ⊗ K ❡st❛❜✐❧✐③❛çã♦✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G ♥❛ á❧❣❡❜r❛ B t❛❧ q✉❡ ♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦ é ✐s♦♠♦r❢♦ ❛ B ⊗ K✳ ❉❡♠♦♥str❛çã♦✿ ❇❛st❛ ♦❜s❡r✈❛r♠♦s q✉❡ B ⊗ K s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s e ⊗ K ❞♦ ❚❡♦r❡♠❛ ✉♠❛ ✈❡③ q✉❡ B ⊗ K é ❡stá✈❡❧ ❡ B é s❡♣❛rá✈❡❧✱ e ✉♠❛ ✈❡③ q✉❡ B ❡ K sã♦ s❡♣❛rá✈❡✐s✳

      ❖ ♣ró①✐♠♦ ❝♦r♦❧ár✐♦ é ✉♠ r❡s✉❧t❛❞♦ ❞❡ q✉❡ ♥❡st❡ tr❛❜❛❧❤♦ ♦❜t✐✲ ✈❡♠♦s ❝♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ g } g ❈♦r♦❧ár✐♦ ✺✳✸✳✾✳ ❙❡❥❛ B = {B ∈G ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ s❛t✉r❛❞♦ ❡ ❡stá✈❡❧ s♦❜r❡ ✉♠ ❣r✉♣♦ ❡♥✉♠❡rá✈❡❧ G ❝✉❥❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ é e

      s❡♣❛rá✈❡❧✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❛çã♦ ❣❧♦❜❛❧ ❞❡ G ♥❛ á❧❣❡❜r❛ B t❛❧ q✉❡ ♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦ é ✐s♦♠♦r❢♦ ❛ B✳ ❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❚❡♦r❡♠❛ ❡①✐st❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ G e ❡♠ B t❛❧ q✉❡ ♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦ é ✐s♦♠♦r❢♦ ❛ B✳ ❙❡ α = ({D g } g , {α g } g ) ∈G ∈G é t❛❧ ❛çã♦ ♣❛r❝✐❛❧✱ s❡❣✉❡ ❞♦ ❊①❡♠♣❧♦ q✉❡ α é ✉♠❛ ❛çã♦ ❣❧♦❜❛❧✳

      ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s

      ❈♦♠❜✐♥❛♥❞♦ ♦ q✉❡ ❛♣r❡s❡♥t❛♠♦s ♥❡st❡ tr❛❜❛❧❤♦ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❡♠ ♣♦❞❡♠♦s t✐r❛r ✐♠♣♦rt❛♥t❡s ❝♦♥❝❧✉sõ❡s ❡♥✈♦❧✈❡♥❞♦ C ✲ á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s✳

      ❉❡ ❛❝♦r❞♦ ❝♦♠ ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ B é ❞✐t❛ s❡r t♦♣♦❧♦✲ ❣✐❝❛♠❡♥t❡ ❣r❛❞✉❛❞❛ s❡ ❡①✐st❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❧✐♠✐t❛❞❛ ❞❡ B e e ❡♠ B q✉❡ é ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ ❡♠ B ❡ s❡ ❛♥✉❧❛ ❡♠ ❝❛❞❛ s✉❜❡s✲ t ♣❛ç♦ B ✱ ♣❛r❛ t 6= e✳ ❆✐♥❞❛ ❡♠ ✉♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ é q✉❡✱ ♣❛r❛ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛♠❡♥❛❜❧❡ B✱ q✉❛❧q✉❡r C ✲á❧❣❡❜r❛ t♦♣♦✲ ❧♦❣✐❝❛♠❡♥t❡ ❣r❛❞✉❛❞❛ ❝✉❥♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦ ❝♦✐♥❝✐❞❡ ❝♦♠ B✱ é ✐s♦♠♦r❢❛ à C ✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ r❡❞✉③✐❞❛ ❞❡ B✳

      ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛tr❛✈és ❞♦ ❚❡♦r❡♠❛ ❡ ❞♦s r❡s✉❧t❛❞♦s ❝✐t❛❞♦s ❛❝✐♠❛✱ t❡♠♦s ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ✉♠❛ C ✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ é ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❞❛ s✉❛ á❧❣❡❜r❛ ❞❛ ✜❜r❛ ✉♥✐❞❛❞❡ ♣❡❧♦ ❣r✉♣♦ ❜❛s❡✳ ❊♠ s✉♠❛✱ s❡ ♦ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ C ✲ á❧❣❡❜r❛ t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ❣r❛❞✉❛❞❛ é ❛♠❡♥❛❜❧❡ ❡ s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ❡♥tã♦ t❛❧ C ✲á❧❣❡❜r❛ é✱ ❞❡ ❢❛t♦✱ ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❝♦♠♦ ❞❡✜♥✐♠♦s ♥❛ ❙❡çã♦

      ❆♣ê♥❞✐❝❡ ❆ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ❛✉①✐❧✐❛r❡s

      ◆❡st❡ ❛♣ê♥❞✐❝❡✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❝♦♥str✉çã♦ ❞❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✲ ✈❡♥t❡ ❞❡ ✉♠❛ ∗✲á❧❣❡❜r❛ ❛❞♠✐ssí✈❡❧✱ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡q✉✐✈❛❧❡♥t❡s ❞❡ ❡❧❡♠❡♥t♦s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦s ❡♠ ✉♠❛ C ✲á❧❣❡❜r❛✱ ❛❧é♠ ❞❡ ❛❧❣✉♥s ❢❛t♦s s♦❜r❡ ❛ á❧❣❡❜r❛ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛✳

      ∗

      ❆✳✶ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡

      ◆❡st❛ s❡çã♦✱ ❞❡✜♥✐♠♦s ❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ ✉♠❛ ∗✲á❧❣❡❜r❛ ❡ ♠♦str❛♠♦s q✉❡ s❡♠♣r❡ ❡①✐st❡ ❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ ✉♠❛ ∗✲á❧❣❡❜r❛ ❛❞♠✐ssí✈❡❧✳ ❈♦♠♦ r❡❢❡rê♥❝✐❛s✱ ❝✐t❛♠♦s

      ❆♦ ❧♦♥❣♦ ❞❡st❛ s❡çã♦✱ B é ✉♠❛ ∗✲á❧❣❡❜r❛✳ ∗ ∗ ❉❡✜♥✐çã♦ ❆✳✶✳✶✳ ❯♠❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ B é ✉♠❛ C ✲á❧❣❡❜r❛ A

      ❡q✉✐♣❛❞❛ ❝♦♠ ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ι : B → A t❛❧ q✉❡ ♣❛r❛ t♦❞❛ C ✲ á❧❣❡❜r❛ C ❡ ♣❛r❛ t♦❞♦ ∗✲❤♦♠♦♠♦r✜s♠♦ ϕ : B → C✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ∗ ϕ : A → C

      ✲❤♦♠♦♠♦r✜s♠♦ e t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛ ι // A

      B ϕ ϕ

      e

      C ϕ ◦ ι = ϕ

      ❝♦♠✉t❛✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ e ✳

      ❉❡✜♥✐çã♦ ❆✳✶✳✷✳ ❉✐③❡♠♦s q✉❡ B é ❛❞♠✐ssí✈❡❧ s❡ ♣❛r❛ ❝❛❞❛ b ∈ B✱ ❡①✐st❡ M > 0 t❛❧ q✉❡ s❡ C é ✉♠❛ C ✲á❧❣❡❜r❛ ❡ ϕ : B → C é ✉♠ ∗

      ✲❤♦♠♦♠♦r✜s♠♦✱ ❡♥tã♦ kϕ(b)k ≤ M✳ kπ(b)k ❙❡ B é ✉♠❛ ∗✲á❧❣❡❜r❛ ❛❞♠✐ssí✈❡❧✱ ✈❛♠♦s ❞❡♥♦t❛r sup π ♦ s✉✲

      ♣r❡♠♦ ❞♦ ❝♦♥❥✉♥t♦ {kπ(b)k : π : B → C, C é ✉♠❛ C ✲á❧❣❡❜r❛✱ π é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦}. ❚❡♦r❡♠❛ ❆✳✶✳✸✳ ❙❡❥❛ B ✉♠❛ ∗✲á❧❣❡❜r❛ ❛❞♠✐ssí✈❡❧✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ C

      ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ♣❛r❛ B✱ q✉❡ é ú♥✐❝❛ ❛ ♠❡♥♦s ❞❡ ∗✲✐s♦♠♦r✜s♠♦s✳ ❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❛♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ k| · k| : B → R ❞❛❞❛ + kπ(b)k ♣♦r b 7→ sup π ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ ❞❡✜♥❡ ✉♠❛ C ✲s❡♠✐♥♦r♠❛ ❡♠ B✳ ❙❡❥❛ N = {b ∈ B : k|bk| = 0}✳ ❊♥tã♦✱ I é ✉♠ ∗✲✐❞❡❛❧ ❞❡ B ❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ q✉♦❝✐❡♥t❡ B/N✱ q✉❡ é ✉♠❛ ∗✲á❧❣❡❜r❛ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈♦❧✉çã♦ ❞❡✜♥✐❞❛s ♣♦r ′ ′

      (b + N )(b + N ) := bb + N, ❡ ∗ ∗ (b + N ) := b + N, ♣❛r❛ b✱ b ✳ +

      ∈ B ❉❡✜♥✐♠♦s ❡♠ B/N ❛ ❛♣❧✐❝❛çã♦ k · k : B/N → R ♣♦r kb + N k = sup kπ(b)k. π

    • N ◆♦t❡♠♦s q✉❡ k · k ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s s❡ b + N = b ✱ s❡❣✉❡ ′ ′

      ∈ N ) q✉❡ b − b ✳ ▲♦❣♦✱ π(b) = π(b ✱ ♣❛r❛ t♦❞♦ ∗✲❤♦♠♦♠♦r✜s♠♦ π ❞❡ B ∗ ′

    • N k ❡♠ ✉♠❛ C ✲á❧❣❡❜r❛ C✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ kb + Nk = kb ✳

      ➱ ❢á❝✐❧ ✈❡r q✉❡ k·k ❞❡✜♥❡ ✉♠❛ C ✲♥♦r♠❛ ❡♠ B/N✱ ❥á q✉❡ kb+Nk = 0 kπ(b)k = 0 ✐♠♣❧✐❝❛ q✉❡ sup π ❡✱ ♣♦rt❛♥t♦✱ b ∈ N✳

      ❙❡❥❛ A ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ (B/N, k · k)✳ ❙❡❣✉❡ q✉❡ A é ✉♠❛ C ✲ á❧❣❡❜r❛✳ ❙❡♥❞♦ ι : B → A ❛ ❛♣❧✐❝❛çã♦ q✉♦❝✐❡♥t❡✱ ♦✉ s❡❥❛✱ ι(b) = b + N✱ ♣❛r❛ t♦❞♦ b ∈ B✱ ✈❛♠♦s ♠♦str❛r q✉❡ (A, ι) é ✉♠❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ♣❛r❛ B✳

      ❉❡ ❢❛t♦✱ s❡❥❛ C ✉♠❛ C ✲á❧❣❡❜r❛ ❡ ϕ : B → C ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦✳ ❉❡✜♥✐♠♦s

      ϕ : B/N → C e b + N 7→ ϕ(b).

      ϕ + N ❙❡❣✉❡ q✉❡ e ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s s❡ b + N = b ✱ t❡♠♦s q✉❡

      ′ ′

      ϕ(b − b ) = 0 kπ(b − b )k = 0 ✱ ❥á q✉❡ sup π ✳ ❈♦♠♦ ϕ é ❧✐♥❡❛r✱ ♦❜t❡♠♦s ϕ(b) = ϕ(b ) ϕ

      ✳ ❆❧é♠ ❞✐ss♦✱ e é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦✳ ❆❣♦r❛✱ ✉♠❛ ✈❡③ q✉❡

      ϕ(b + N )k = kϕ(b)k ≤ sup kπ(b)k = kb + N k, k e π ϕ

      ❝♦♥❝❧✉í♠♦s q✉❡ e s❡ ❡st❡♥❞❡ ❛ ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ❞❡ A ❡♠ C✱ q✉❡ ϕ

      ❝♦♥t✐♥✉❛r❡♠♦s ❞❡♥♦t❛♥❞♦ ♣♦r e ✳ ▼❛✐s ❛✐♥❞❛✱ ♣❛r❛ t♦❞♦ b ∈ B✱ ϕ(b + N ) = ϕ(b)

      ϕ(ι(b)) = e e ϕ ◦ ι = ϕ ϕ

      ❡ ❛ss✐♠ e ✳ ❆ ✉♥✐❝✐❞❛❞❡ ❞❡ e s❡❣✉❡ ❞♦ ❢❛t♦ ❞❡ ι(B) s❡r ❞❡♥s♦ ❡♠ A✳

      ▲♦❣♦✱ r❡st❛ s♦♠❡♥t❡ ♠♦str❛r♠♦s ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ A✱ ❛ ♠❡♥♦s ❞❡ ∗✲ ✐s♦♠♦r✜s♠♦s✳

      ❙❡❥❛ (C, ) ✉♠❛ ♦✉tr❛ C ✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ♣❛r❛ B✳ Pr✐♠❡✐r❛✲ C ♠❡♥t❡✱ ♥♦t❡♠♦s q✉❡ ♦ ∗✲❛✉t♦♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ Id ❢❛③ ♦ ❞✐❛❣r❛♠❛

      B // C Id C C

      ❝♦♠✉t❛r✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♦ ❞✐❛❣r❛♠❛ ι B // A ι Id A

      A ❝♦♠✉t❛✳

      P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ✉♥✐✈❡rs❛✐s ❞❡ A ❡ C✱ r❡s✲ ♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦❞❡♠♦s ♦❜t❡r ú♥✐❝♦s ∗✲❤♦♠♦♠♦r✜s♠♦s ϕ : A → C ❡ ϕ : C → A t❛✐s q✉❡ ♦s ❞✐❛❣r❛♠❛s ι

      // A B ϕ

      C

      ❡ B // C ι ϕ

      A ′ ′ ◦  =  ◦ ϕ ◦ ι = ι

      ❝♦♠✉t❛♠✳ ❉❛í✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ϕ ◦ ϕ ❡ ϕ ✳ ′ ′ = Id C ◦ ϕ = Id A

      P♦rt❛♥t♦✱ ♣♦r ✉♥✐❝✐❞❛❞❡✱ ❞❡✈❡♠♦s t❡r ϕ ◦ ϕ ❡ ϕ ✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛✳

      ❆✳✷ ❊❧❡♠❡♥t♦s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦s

      ◆❡st❛ s❡çã♦✱ ❞❡✜♥✐♠♦s ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦ ❡✱ ❡♥tr❡ ♦✉✲ tr❛s ❝♦✐s❛s✱ ♠♦str❛♠♦s q✉❡ ✉♠❛ C ✲á❧❣❡❜r❛ ♣♦ss✉✐ ✉♠ ❡❧❡♠❡♥t♦ ❡s✲ tr✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣♦ss✉✐ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ s❡q✉❡♥❝✐❛❧✳ ❈♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ t♦❞❛ C ✲á❧❣❡❜r❛ s❡♣❛rá✈❡❧ ♣♦ss✉✐ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✳

      ❆ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ ✉s❛❞❛ ♥❡st❛ s❡çã♦ é ❉❡✜♥✐çã♦ ❆✳✷✳✶✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❡❧❡♠❡♥t♦ ♣♦s✐t✐✈♦ e ❞❡ ✉♠❛ C ✲ á❧❣❡❜r❛ A é ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✱ s❡ eAe = A✳ Pr♦♣♦s✐çã♦ ❆✳✷✳✷✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛ ❡ e ∈ A ✉♠ ❡❧❡♠❡♥t♦ ♣♦s✐t✐✈♦✳ ❙ã♦ ❡q✉✐✈❛❧❡♥t❡s✿ ✭✐✮ Ae é ❞❡♥s♦ ❡♠ A❀ ✭✐✐✮ φ(e) > 0 ♣❛r❛ t♦❞♦ ❡st❛❞♦ φ ❞❡ A❀ ✭✐✐✐✮ e é ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✳ ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ✈❛❧❤❛ ✭✐✮ ❡ s❡❥❛ φ ✉♠ ❡st❛❞♦ ❞❡ A✳ ❯♠❛ c ✈❡③ q✉❡ e é ♣♦s✐t✐✈♦✱ ❡s❝r❡✈❡♠♦s e = c ✱ ❝♦♠ c ∈ a✳ P❡❧♦ ❚❡♦r❡♠❛ ✸✳✸✳✼

      c) = 0 ❞❡ ❞✐③❡r q✉❡ φ(e) = φ(c ✱ ✐♠♣❧✐❝❛ q✉❡ φ(ac) = 0✱ ♣❛r❛ t♦❞♦ a ∈ A

      ✳ ❯♠❛ ✈❡③ q✉❡ Ac é ❞❡♥s♦ ❡♠ A✱ t❡♠♦s φ = 0✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ▲♦❣♦✱ φ(e) > 0 ❡ s❡❣✉❡ ✭✐✮⇒✭✐✐✮✳

      P❛r❛ ✭✐✐✮⇒✭✐✐✐✮✱ ✈❛♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ✺✳✸✳✶ ❞❡ q✉❡ ❛✜r♠❛ q✉❡ ∗ ∗ s❡ B 1 ❡ B 2 sã♦ C ✲s✉❜á❧❣❡❜r❛s ❤❡r❡❞✐tár✐❛s ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ A t❛✐s ⊆ B q✉❡ B 1 2 ✱ ❡♥tã♦ s❡ t♦❞♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ♣♦s✐t✐✈♦ τ ❞❡ A s❡ ❛♥✉❧❛

      = B ❡♠ B 1 t❛♠❜é♠ s❡ ❛♥✉❧❛ ❡♠ B 2 ✱ s❡❣✉❡ q✉❡ B 1 2 ✳

      ❈♦♠ ❡st❡ r❡s✉❧t❛❞♦ ❡♠ ♠ã♦s✱ s❡ B = eAe✱ ❞❡✈❡♠♦s t❡r B = A✳ ❈♦♠ ❡❢❡✐t♦✱ ❝❛s♦ ❝♦♥trár✐♦ ❡①✐st✐r✐❛ ✉♠ ❡st❛❞♦ φ ❞❡ A t❛❧ q✉❡ φ(B) = 0✳ ◆❡st❡

      ❝❛s♦✱ φ(ee) = 0 ❡ ♥♦✈❛♠❡♥t❡ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✸✳✸✳✼ ❞❡ t❡rí❛♠♦s φ(e) = 0

      ✱ ❝♦♥tr❛❞✐③❡♥❞♦ ✭✐✐✮✳ P❛r❛ ✭✐✐✐✮⇒✭✐✮ ❜❛st❛ ♦❜s❡r✈❛r q✉❡ eAe ⊆ Ae✳

      Pr♦♣♦s✐çã♦ ❆✳✷✳✸✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛✳ ❙ã♦ ❡q✉✐✈❛❧❡♥t❡s✿ ✭✐✮ A ♣♦ss✉✐ ✉♠ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦❀ ✭✐✐✮ A ♣♦ss✉✐ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ s❡q✉❡♥❝✐❛❧✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♠❡ç❛♠♦s ♣♦r ♠♦str❛r ❛ ✐♠♣❧✐❝❛çã♦ ✭✐✮⇒✭✐✐✮✳ ❙❡❥❛ e

      ✉♠ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦ ❞❡ A✳ P❛r❛ ❝❛❞❛ n ∈ N✱ ❝♦❧♦❝❛♠♦s −1

      1 u n = e e + . n ) n ∈N n ▼♦str❡♠♦s q✉❡ (u é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ❡♥✉♠❡rá✈❡❧ ♣❛r❛ A

      ✳ n : σ(e) → R P❛r❛ ❝❛❞❛ n ∈ N✱ s❡❥❛ g ❛ ❢✉♥çã♦ 2 t t 7→ . 1 n ) n t + n

      ❊♥tã♦✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s (g ∈N é ♣♦♥t✉❛❧♠❡♥t❡ ❝r❡s❝❡♥t❡ ❡ ❝♦♥✲ ✈❡r❣❡ ♣♦♥t✉❛❧♠❡♥t❡ à ❢✉♥çã♦ ✐♥❝❧✉sã♦ z : σ(e) → R✳ ❈♦♠♦ σ(e) é n → z ❝♦♠♣❛❝t♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❉✐♥✐✱ s❡❣✉❡ q✉❡ g ✉♥✐❢♦r♠❡♠❡♥t❡✳

      (e, 1) ◆♦ ❡♥t❛♥t♦✱ ♦ ∗✲✐s♦♠♦r✜s♠♦ ϕ ❡♥tr❡ C(σ(e)) ❡ C é t❛❧ q✉❡

      ϕ(z) = e n ) → e n ) = eu n ✳ ❙❡❣✉❡ q✉❡ ϕ(g ✳ ❈♦♠♦ ϕ(g ✱ ❝♦♥❝❧✉í♠♦s q✉❡ eu n → e n → a

      ✳ ▼❛s✱ Ae é ❞❡♥s♦ ❡♠ A✱ ❡ ❛ss✐♠ ♦❜t❡♠♦s q✉❡ au ✱ ♣❛r❛ t♦❞♦ a ∈ A✳ n ) n ∈N P❛r❛ ✭✐✐✮⇒✭✐✮✱ s❡❥❛ (u ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ s❡q✉❡♥❝✐❛❧

      ♣❛r❛ A✳ ❙❡❥❛

      X u n e = n n =1

      2 ❡ ♣r♦✈❡♠♦s q✉❡ e é ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s ♠♦str❛r q✉❡ φ(e) > 0✱ ♣❛r❛ t♦❞♦ ❡st❛❞♦ φ ❞❡ A✱ ❡ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ❝♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ Pr♦♣♦s✐çã♦ n φ(u n ) = 1

      ❙✉♣♦♥❤❛ q✉❡ φ s❡❥❛ ✉♠ ❡st❛❞♦ ❞❡ A✳ ❈♦♠♦ kφk = lim ✱ N ) > 0 n ) ≥ 0 ❞❡✈❡ ❡①✐st✐r N ∈ N t❛❧ q✉❡ φ(u ✳ ▲♦❣♦✱ ❝♦♠♦ φ(u ♣❛r❛ t♦❞♦ n ∈ N✱

      X φ(u n ) φ(u N ) φ(e) = ≥ > 0. n N n =1

      2

      2

      ∗

      ❯♠❛ C ✲á❧❣❡❜r❛ q✉❡ ♣♦ss✉✐ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ s❡q✉❡♥❝✐❛❧ é ❞✐t❛ s❡r σ✲✉♥✐t❛❧✳

      ❖ s❡❣✉✐♥t❡ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ❡ s✉❛ ❞❡✲ ♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❖❜s❡r✈❛çã♦ ✸✳✶✳✶✳ ❈♦r♦❧ár✐♦ ❆✳✷✳✹✳ ❙❡❥❛ A ✉♠❛ C ✲á❧❣❡❜r❛ s❡♣❛rá✈❡❧✳ ❊♥tã♦ A ♣♦ss✉✐ ❡❧❡♠❡♥t♦ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦✳

      ❆✳✸ ➪❧❣❡❜r❛ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s

      ◆❡st❛ s❡çã♦✱ ❞❡✜♥✐♠♦s ❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ ❞❛ á❧❣❡❜r❛ ❞❡ ♠✉❧t✐♣❧✐✲ ❝❛❞♦r❡s✱ ♠♦str❛♠♦s q✉❡ ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ♥ã♦❞❡❣❡♥❡r❛❞♦ ❞❡ ✉♠❛ C

      ✲á❧❣❡❜r❛ s❡ ❡st❡♥❞❡ ✉♥✐❝❛♠❡♥t❡ ❛ ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ✉♥✐t❛❧ ❞❛ s✉❛ á❧❣❡❜r❛ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❡ t❛❧ ∗✲❤♦♠♦♠♦r✜s♠♦ é t❛♠❜é♠ ❡str✐t❛♠❡♥t❡ ❝♦♥tí♥✉♦✳ ❆❧é♠ ❞✐ss♦✱ ❞❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❡s♣❛❝✐❛❧ ❞❡ C ✲ á❧❣❡❜r❛s ❡✱ ♥❡st❡ ❝♦♥t❡①t♦✱ ♠♦str❛♠♦s ✉♠ r❡s✉❧t❛❞♦ q✉❡ r❡❧❛❝✐♦♥❛ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❛s á❧❣❡❜r❛s ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❝♦♠ ❛ á❧❣❡❜r❛ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✳

      ❆✳✸✳✶ ❚♦♣♦❧♦❣✐❛ ❡str✐t❛

      ❈♦♥s✐❞❡r❡ ❛ t♦♣♦❧♦❣✐❛ s♦❜r❡ M(A) ❣❡r❛❞❛ ♣❡❧❛s s❡♠✐♥♦r♠❛s kbk a = kbak + kabk, ❡♠ q✉❡ b ∈ M(A) ❡ a ∈ A✳ ❚❛❧ t♦♣♦❧♦❣✐❛ é ❝❤❛♠❛❞❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛ λ ) λ ❞❡ M(A)✳ ❆ss✐♠✱ ✉♠ ♥❡t (µ ∈Λ ❡♠ M(A) ❝♦♥✈❡r❣❡ ❛ µ ♥❛ t♦♣♦❧♦❣✐❛ λ a → µa λ → aµ ❡str✐t❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ µ ❡ aµ ✱ ♣❛r❛ t♦❞♦ a ∈ A✳ ❖❜s❡r✈❛çã♦ ❆✳✸✳✶✳ A é ❞❡♥s♦ ❡♠ M(A) ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛✳ λ µ ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ♣❛r❛ µ ∈ M(A)✱ t❡♠♦s q✉❡ u ❝♦♥✈❡r❣❡ λ ) λ ❛ µ ♥❛ t♦♣♦❧♦❣✐❛ ❡str✐t❛✱ ❡♠ q✉❡ (u ∈Λ é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ A✳ Pr♦♣♦s✐çã♦ ❆✳✸✳✷✳ ❙❡❥❛ π : A → M(B) ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ♥ã♦❞❡✲ ❣❡♥❡r❛❞♦ ✭ π(A)B = B✮✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❡①t❡♥sã♦ ❞❡ π ❛ ✉♠ ∗

      ✲❤♦♠♦♠♦r✜s♠♦ ✉♥✐t❛❧ eπ : M(A) → M(B)✳ λ )b ❉❡♠♦♥str❛çã♦✿ ❯♠❛ ✈❡③ q✉❡ π é ♥ã♦✲❞❡❣❡♥❡r❛❞❛✱ s❡❣✉❡ q✉❡ π(u ❝♦♥✈❡r❣❡ ❛ b✱ ♣❛r❛ q✉❛❧q✉❡r b ∈ B✳ ❆ss✐♠✱ ♦❜t❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛

      {π(a n )b} n n π(a n )b = b ∈N t❛❧ q✉❡ lim ✳ ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ♦❜t❡r ′ ′ )} n n bπ(a ) = b

      ✉♠❛ s❡q✉ê♥❝✐❛ {bπ(a n ∈N t❛❧ q✉❡ lim n ✳ ❈♦♠ ✐ss♦✱ ♣❛r❛ µ ∈ M (A) L π R (µ))

      ✱ ❞❡✜♥✐♠♦s eπ(µ) = (eπ (µ), e ✱ ❡♠ q✉❡ π L (µ)(b) = lim π(µa n )b, R (µ)(b) = lim bπ(a µ). e ❡ eπ n n n

      Pr♦✈❡♠♦s q✉❡ eπ(µ) ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ M(B)✳ 2 µ ≤ kµk

      Pr✐♠❡✐r❛♠❡♥t❡✱ s❛❜❡♠♦s q✉❡ µ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ∗ ∗ ∗ 2 ∗ ∗ b π(a µ µa)b ≤ kµk b π(a a)b. ▲♦❣♦✱ kπ(µa)bk ≤ kµkkπ(a)bk.

      ❆♥❛❧♦❣❛♠❡♥t❡✱ ∗ ∗ ∗ kbπ(aµ)k = kπ(µ a )b k ≤ kµkkbπ(a)k. ❊st❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡rá ✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ❢✉♥❞❛♠❡♥t❛❧ ❛♦ ❧♦♥❣♦ ❞❛ ❞❡✲ ♠♦♥str❛çã♦✳

      ❉❛✐✱ s❡❣✉❡ q✉❡ kπ(µa n )b − π(µa m )bk ≤ kµkkπ(a n )b − π(a m )bk n π(µa n )b ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ❧✐♠✐t❡ lim ❡①✐st❡✳ ❈♦♠ ♦ ♠❡s♠♦ ❛r✲ ❣✉♠❡♥t♦ ♣r♦✈❛✲s❡ q✉❡ t❛❧ ❧✐♠✐t❡ ✐♥❞❡♣❡♥❞❡ ❞❛ s❡q✉ê♥❝✐❛ q✉❡ t♦♠❛r♠♦s ❞❡st❛ ❢♦r♠❛ ❝♦♥✈❡r❣✐♥❞♦ ❛ b✳ ❯s❛♥❞♦ q✉❡ Bπ(A) é ❞❡♥s♦ ❡♠ B ❡ ❛r❣✉✲ n bπ(a n µ) ♠❡♥t❛♥❞♦ s✐♠✐❧❛r♠❡♥t❡✱ ♦❜t❡♠♦s q✉❡ ♦ ❧✐♠✐t❡ lim ❡①✐st❡✱ ❡♠ n ) → b q✉❡ bπ(a ✳ n )b → b )c → c

      ❖❜s❡r✈❛♠♦s t❛♠❜é♠ q✉❡ s❡ b, c ∈ B✱ π(a ✱ π(a n ❡ π(c n )(b+c) → b+c n ) n ∈N ) n ∈N n ) n ∈N

      ✱ ❡♠ q✉❡ (a ✱ (a n ❡ (c sã♦ s❡q✉ê♥❝✐❛s ❡♠ A✱ ❡♥tã♦✱ ♣❛r❛ λ ∈ Λ ✜①❛❞♦✱ kπ(µu λ )(π(a n )b + π(a )c − π(c n )(b + c))k ≤ (△), n )b + π(a )c − π(c n )(b + c)k n

      ❡♠ q✉❡ (△) = kµkkπ(a n ✱ ❡ ✐st♦ s❡❣✉❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ q✉❡ ❛♣r❡s❡♥t❛♠♦s ♥♦ ✐♥í❝✐♦ ❞❛ ❞❡♠♦♥str❛çã♦✳

      P❛ss❛♥❞♦ ❛♦ ❧✐♠✐t❡ s♦❜r❡ λ ❡ ❞❡♣♦✐s s♦❜r❡ n✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❛♣❧✐✲ L (µ) n π(µa n )b ❝❛çã♦ eπ ❞❛❞❛ ♣♦r b 7→ lim é ❧✐♥❡❛r✳ ❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ π(µ) R n bπ(a n µ) n ) → b e ❞❡✜♥✐❞❛ ♣♦r b 7→ lim ✱ ❡♠ q✉❡ bπ(a ✳

      ∈ B ▼❛✐s ❛✐♥❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r b, b ✱ ′ ′ ′

      π(µ) L (bb ) = lim π(µa n )bb π(µ) L (b)b , e = e n

      n )bb → bb n )b → b ′ ′

      ❥á q✉❡ π(a ✱ s❡♠♣r❡ q✉❡ π(a ✳ ❚❛♠❜é♠ t❡♠♦s q✉❡ ′ ′ π(µ) R (bb π(µ) R (b ). e ) = be n ) → b )b → b ′ ′ ′

      P♦r ✜♠✱ s✉♣♦♥❤❛ q✉❡ bπ(a ❡ π(a n ✱ ❊♥tã♦✱ ′ ′ ′ π(µ) R (b)b = lim bπ(a n µ)π(a )b e n n ′ ′ ′ L π R (µ)) = lim bπ(a n )π(µa )b π(µ) L (b ). n n = be

      ▲♦❣♦✱ eπ(µ) = (eπ (µ), e é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ M(B)✳ ❆❧é♠ ❞✐ss♦✱ ♣r♦✈❛✲s❡ q✉❡ eπ : M(A) → M(B)✱ µ 7→ eπ(µ) é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ eπ é ✉♥✐t❛❧ ❡ ❡st❡♥❞❡ π✳

      ❙❡ ϕ : M(A) → M(B) é ♦✉tr♦ ∗✲❤♦♠♦♠♦r✜s♠♦ ✉♥✐t❛❧ ❡st❡♥❞❡♥❞♦ π

      ✱ ♣❛r❛ ❝❛❞❛ a ∈ A ❡ µ ∈ M(A)✱ t❡♠♦s q✉❡ π(µ)π(a).

      ϕ(µ)π(a) = ϕ(µa) = π(µa) = e ❯s❛♥❞♦ q✉❡ π é ♥ã♦✲❞❡❣❡♥❡r❛❞❛ ❡ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛✱ ♦❜t❡♠♦s q✉❡

      π(µ) ϕ(µ) = e ✱ ❞♦♥❞❡ ϕ = eπ✳

      ◆❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦✱ ✉s❛r❡♠♦s ❛ Pr♦♣♦s✐çã♦ ✷✳✸✸ ❞❡ q✉❡ ❞✐③ ♦ s❡❣✉✐♥t❡✿ s❡❥❛ X ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❇❛♥❛❝❤ ♥ã♦❞❡❣❡♥❡r❛❞♦✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ X é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ kaxk ≤ kakkxk✱ ❡ span{ax : a ∈

      A, x ∈ X} é ❞❡♥s♦ ❡♠ X✳ ❊♥tã♦ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ X é ❞❛ ❢♦r♠❛ ax✱

      ♣❛r❛ ❛❧❣✉♠ a ∈ A ❡ x ∈ X✳ Pr♦♣♦s✐çã♦ ❆✳✸✳✸✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s ❡ s❡❥❛ π : A → M(B) ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ♥ã♦❞❡❣❡♥❡r❛❞♦✳ ❊♥tã♦ ❛ ❡①t❡♥sã♦ ❞❡ π✱ eπ : M(A) → M (B)

      é ❝♦♥tí♥✉♦ ♥❛s r❡s♣❡❝t✐✈❛s t♦♣♦❧♦❣✐❛s ❡str✐t❛s ❞❡ M(A) ❡ M(B)✳ λ ) λ λ → µ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ (µ ∈Λ ✉♠ ♥❡t ❡♠ M(A) t❛❧ q✉❡ µ λ ) → π(µ) ❡str✐t❛♠❡♥t❡✳ ❚❡♠♦s ❞❡ ♠♦str❛r q✉❡ π(µ ❡str✐t❛♠❡♥t❡✳

      ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛ b ∈ B✳ ◆❡st❡ ❝❛s♦✱ B é ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❇❛♥❛❝❤ ❝♦♠ ❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❞❛❞❛ ♣♦r a 7→ π(a)b✳ ❆ss✐♠✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ q✉❡ ❡♥✉♥❝✐❛♠♦s ❛❝✐♠❛✱ t❡♠♦s q✉❡ b = π(a)c✱ ♣❛r❛ ❛❧❣✉♠ a ∈ A ❡ c ∈ B✳ ❙❡❣✉❡ q✉❡ kπ(µ)b − π(µ λ )bk = kπ(µa − µ λ a)ck ≤ kµa − µ λ akkck λ )k q✉❡ ❝♦♥✈❡r❣❡ 0✳ ❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ kbπ(µ) − bπ(µ ❝♦♥s✐❞❡r❛♥❞♦ B

      ❝♦♠♦ ✉♠ A✲♠ó❞✉❧♦ ❞❡ ❇❛♥❛❝❤ à ❞✐r❡✐t❛✳

      

    ❆✳✸✳✷ Pr♦❞✉t♦ t❡♥s♦r✐❛❧ ❡s♣❛❝✐❛❧ ❞❡ á❧❣❡❜r❛s ❞❡ ♠✉❧✲

    t✐♣❧✐❝❛❞♦r❡s

      ▲❡♠❜r❡♠♦s q✉❡ s❡ A ❡ B sã♦ C ✲á❧❣❡❜r❛s✱ ❝♦♠ r❡♣r❡s❡♥t❛çõ❡s ✉♥✐✲ ✈❡rs❛✐s (ψ, H) ❡ (ϕ, K)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ∗✲ ❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦ π : A ⊗ B → B(H ˆ⊗K) t❛❧ q✉❡ π(a ⊗ b) = + ϕ(a) ⊗ π(b)

      ✱ ♣❛r❛ q✉❛✐sq✉❡r a ∈ A✱ b ∈ B✳ ❆ ❢✉♥çã♦ k · k : A ⊗ B → R ∗ ∗ c 7→ kπ(c)k é ✉♠❛ C ✲♥♦r♠❛ s♦❜r❡ A ⊗ B ❝❤❛♠❛❞❛ C ✲♥♦r♠❛ ❡s♣❛❝✐❛❧✳ ❖ ❝♦♠✲ ♣❧❡t❛♠❡♥t♦ ❞❡ A ⊗ B ❝♦♠ r❡❧❛çã♦ ❛ k · k ∗ é ❝❤❛♠❛❞♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧

      B ❡s♣❛❝✐❛❧ ❞❡ A ❡ B✱ ❡ é ❞❡♥♦t❛❞♦ ♣♦r A ⊗ ∗ ✳

      ❉✐③❡♠♦s q✉❡ A é ♥✉❝❧❡❛r✱ s❡ ♣❛r❛ t♦❞❛ C ✲á❧❣❡❜r❛ B✱ ❡①✐st❡ s♦♠❡♥t❡ ✉♠❛ C ✲♥♦r♠❛ s♦❜r❡ A ⊗ B✳ ◆❡st❡ ❝❛s♦✱ ❞❡♥♦t❛♠♦s ❛ C ✲á❧❣❡❜r❛ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ A ❡ B s✐♠♣❧❡s♠❡♥t❡ ♣♦r A⊗B✳ ❙❡❣✉❡ ❞♦ ❊①❡♠♣❧♦ ✻✳✷✳✸ ❞❡ q✉❡ K(H) é ♥✉❝❧❡❛r✱ ❡♠ q✉❡ H é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡ K(H)

      ❞❡♥♦t❛ ❛ C ✲á❧❣❡❜r❛ ❞❡ t♦❞♦s ♦s ♦♣❡r❛❞♦r❡s ❝♦♠♣❛❝t♦s s♦❜r❡ H✳ ❆ ✜♠ ❞❡ ♠♦str❛r q✉❡✱ s❡ A ❡ B sã♦ C ✲á❧❣❡❜r❛s✱ ❡♥tã♦ M(A) ⊗ ∗

      M (B) ⊆ M (A ⊗ ∗ ✱ ❢❛ç❛♠♦s ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❡ ♠♦str❛ q✉❡ ❛ á❧❣❡✲

      B) ❜r❛ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ✉♠❛ C ✲á❧❣❡❜r❛ A ♣♦ss✉✐ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✿ Pr♦♣♦s✐çã♦ ❆✳✸✳✹✳ ❙❡❥❛ B ✉♠❛ C ✲á❧❣❡❜r❛ ✉♥✐t❛❧ ❝♦♥t❡♥❞♦ A ❝♦♠♦ ✉♠ ✐❞❡❛❧✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ∗✲❤♦♠♦♠♦r✜s♠♦ ✉♥✐t❛❧ B → M(A) ❝✉❥❛ r❡str✐çã♦ ❛ A ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛ A → M(A)✳ ▼❛✐s ❛✐♥❞❛✱ s❡ A é ❡ss❡♥❝✐❛❧ ❡♠ B ✭bA = 0 ⇒ b = 0✮✱ ❡♥tã♦ ♦ ∗✲ ❤♦♠♦♠♦r✜s♠♦ ❛ss♦❝✐❛❞♦ B → M(A) é ✐♥❥❡t✐✈♦✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ B ✉♠❛ C ✲á❧❣❡❜r❛ ❝♦♥t❡♥❞♦ A ❝♦♠♦ ✐❞❡❛❧✳ ❉❛❞♦ b ∈ B b : A → A

      ✱ ♥♦t❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ L ✱ a 7→ ba é ❧✐♥❡❛r ❡ b (ac) = L b (a)c s❛t✐s❢❛③ L ✱ ♣❛r❛ q✉❛✐sq✉❡r a, c ∈ A✳ ❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ R b : A → A b (a)c = aL b (c)

      ✱ a 7→ ab ❡ ❛❧é♠ ❞✐ss♦✱ R ✳ ❉❡✜♥✐♠♦s π : B → M (A) b 7→ (L b , R b ). ❊♥tã♦ π é ✉♥✐t❛❧ ❡ ❡st❡♥❞❡ ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛ ❞❡ A ❡♠ M(A)✳ P❛r❛

      b, b ∈ B ❡ a ∈ A✱ t❡♠♦s ′ ′

      L b (L b (a)) = bb a = L bb (a)

      ❡ ′ ′ ′ ′ R b (R b (a)) = abb = R bb (a), ) = π(bb )

      ❞♦♥❞❡ π(b)π(b ✳ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ❆❧é♠ ❞✐ss♦✱ ∗ ∗ L (a) = (L b (a )) = ab = R b (a), R (a) = (R b (a )) = b a = L b (a). b b P♦rt❛♥t♦✱ ∗ ∗ ∗ ∗ ∗

      π(b) = (L b , R b ) = (R , L ) = (L b ∗ , R b ∗ ) = π(b ), b b ♦ q✉❡ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛ ❞❡ q✉❡ π é ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦✳

      P❛r❛ ✈❡r q✉❡ π é ú♥✐❝♦✱ s❡❥❛ ϕ : B → M(A) ✉♠ ∗✲❤♦♠♦♠♦r✜s♠♦ ✉♥✐t❛❧ ❡st❡♥❞❡♥❞♦ ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛ ι : A → M(A)✳ ❊♥tã♦✱ ♣❛r❛ a ∈ A

      ❡ b ∈ B✱ ϕ(b)ι(a) = ϕ(b)ϕ(a) = ϕ(ba) = π(ba) = π(b)ι(a). ▲♦❣♦✱

      (ϕ(b) − π(b)) ι(a) = 0, ♣❛r❛ t♦❞♦ a ∈ A. ❈♦♠♦ A é ✉♠ ✐❞❡❛❧ ❡ss❡♥❝✐❛❧ ❡♠ M(A)✱ s❡❣✉❡ q✉❡ ϕ(b) = π(b)✱ ♣❛r❛ t♦❞♦ b ∈ B✳

      ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ A s❡❥❛ ✉♠ ✐❞❡❛❧ ❡ss❡♥❝✐❛❧ ❡♠ B✳ ❆ss✐♠✱ s❡ b ∈ B é t❛❧ q✉❡ π(b) = 0 t❡♠♦s q✉❡ bA = 0 ❡ Ab = 0✱ ❞♦♥❞❡ b = 0✳ ❙❡❣✉❡ q✉❡ π

      é ✐♥❥❡t✐✈♦✳ ❖ s❡❣✉✐♥t❡ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ Pr♦♣♦s✐çã♦

      Pr♦♣♦s✐çã♦ ❆✳✸✳✺✳ ❙❡❥❛♠ A ❡ B C ✲á❧❣❡❜r❛s✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ∗✲ M (B) → M (A ⊗

      B) ❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t♦r ❡ ✉♥✐t❛❧ ❞❡ π : M(A) ⊗ ∗ ∗ ∗ B ∗

      B) ❡st❡♥❞❡♥❞♦ ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛ ❞❡ A ⊗ ❡♠ M(A ⊗ ✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ (ϕ, H) ❡ (ψ, K) r❡♣r❡s❡♥t❛çõ❡s ✐♥❥❡t✐✈❛s ❡ ♥ã♦✲ ❞❡❣❡♥❡r❛❞❛s ❞❡ A ❡ B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P❡❧♦ ❚❡♦r❡♠❛ ✻✳✸✳✸ ❞❡ ✱ ∗ B → B(H ˆ ⊗K) ❡①✐st❡ ✉♠ ú♥✐❝♦ ∗✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t♦r ϕ ˆ⊗ψ : A ⊗ t❛❧ q✉❡

      ϕ ˆ ⊗ψ(a ⊗ b) = ϕ(a) ˆ ⊗ψ(b), ♣❛r❛ q✉❛✐sq✉❡r a ∈ A ❡ b ∈ B✳ ❆❧é♠ ❞✐ss♦✱ ϕ ˆ⊗ψ é ♥ã♦✲❞❡❣❡♥❡r❛❞❛✳

      ϕ : M (A) → B(H) ψ : M (B) → B(K) P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛♠ e ❡ e

      ❛s ❡①t❡♥sõ❡s ❞❡ ϕ ❡ ψ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ M(A) ❡ M(B)✱ ❝♦♠♦ ♥❛ ϕ ψ

      Pr♦♣♦s✐çã♦ ❚❡♠♦s q✉❡ e ❡ e sã♦ ✐♥❥❡t✐✈♦s ❡✱ ❛ss✐♠✱ ♦❜t❡♠♦s ✉♠

      ∗ ϕ ˆ ⊗ e ψ : M (A) ⊗ M (B) → B(H ˆ ⊗K) ✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦ e ∗ t❛❧ q✉❡

      ϕ ˆ ⊗ e ψ ϕ(µ) ˆ ⊗ e ψ(ν), e (µ ⊗ ν) = e ♣❛r❛ q✉❛✐sq✉❡r µ ∈ M(A) ❡ ν ∈ M(B)✳

      ϕ ˆ ⊗ e ψ B ❖❜s❡r✈❛♠♦s q✉❡ ❛ r❡str✐çã♦ ❞❡ e ❛ A ⊗ ∗ é ❡①❛t❛♠❡♥t❡ ϕ ˆ⊗ψ✳

      M (B)

      B) = 0 ϕ ˆ ⊗ e ψ(c) = ❉♦♥❞❡ s❡ c ∈ M(A)⊗ ∗ é t❛❧ q✉❡ c(A⊗ ∗ ✱ t❡♠♦s e

      ✱ ♣♦✐s ϕ ˆ⊗ψ é ♥ã♦✲❞❡❣❡♥❡r❛❞❛✳ P♦rt❛♥t♦✱ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ❞❛ Pr♦♣♦✲ s✐çã♦

      ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

    ❬✶❪ ❆❜❛❞✐❡ ✱ ❋✳✱ ❊♥✈❡❧♦♣✐♥❣ ❛❝t✐♦♥s ❛♥❞ ❚❛❦❛✐ ❞✉❛❧✐t② ❢♦r ♣❛rt✐❛❧ ❛❝t✐♦♥s✱ ❛r✲

    ❳✐✈✿♠❛t❤✴✵✵✵✼✶✵✾✈✶✳

    ❬✷❪ ❆❦❡♠❛♥♥✱ ❈✳ ❆✳✱ P❡❞❡rs❡♥✱ ●✳ ❑✳✱ ❛♥❞ ❚♦♠✐②❛♠❛✱ ❏✳✱ ▼✉❧t✐♣❧✐❡rs ♦❢ C ✲

      ❛❧❣❡❜r❛s✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧②s✐s ✶✸ ✭✶✾✼✸✮✱ ✷✼✼✕✸✵✶✳

    ❬✸❪ ❇♦❛✈❛✱ ●✳✱ ❈❛r❛❝t❡r✐③❛çõ❡s ❞❛ C ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r ✉♠❛ ❝♦♠♣r❡ssã♦ ❛♣❧✐❝❛✲

    ❞❛s ❛ ❝r✐st❛✐s ❡ q✉❛s✐❝r✐st❛✐s✱ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ❋❧♦r✐❛♥ó♣♦❧✐s✱ ✷✵✵✼✳

      

    ❬✹❪ ❇♦✛✱ P✳ ❘✱ ❈♦❛çõ❡s ❞❡ ❣r✉♣♦s ❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧✱ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✱

    ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ❋❧♦r✐✲ ❛♥ó♣♦❧✐s✱ ✷✵✶✸✳

    ❬✺❪ ❇r♦✇♥✱ ▲✳ ●✳✱ ■s♦♠♦r♣❤✐s♠ ♦❢ ❤❡r❡❞✐t❛r② s✉❜❛❧❣❡❜r❛s ♦❢ C ✲❛❧❣❡❜r❛s✱ P❛❝✐✜❝

    ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s ✼✶ ✭✶✾✼✼✮✱ ✸✸✺✕✸✹✽✳

    ❬✻❪ ❇r♦✇♥✱ ▲✳ ●✳✱ ●r❡❡♥✱ P✳✱ ❛♥❞ ❘✐❡✛❡❧✱ ▼✳ ❆✱ ❙t❛❜❧❡ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ str♦♥❣

    ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ C ✲❛❧❣❡❜r❛s✱ P❛❝✐✜❝ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s ✼✶ ✭✶✾✼✼✮✱

      ✸✹✾✕✸✻✸✳

    ❬✼❪ ❇✉ss✱ ❆✳✱ ▼❡②❡r✱ ❘✳✱ ❛♥❞ ❩❤✉✱ ❈✳✱ ❆ ❍✐❣❤❡r ❈❛t❡❣♦r② ❆♣♣r♦❛❝❤ t♦ ❚✇✐st❡❞

    ❆❝t✐♦♥s ♦♥ C ✲❛❧❣❡❜r❛s✱ ❊❞✐♥❜✉r❣❤ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ✺✻ ✭✷✵✶✸✮✱ ✸✽✼✕ ✹✷✻✳

    ❬✽❪ ❉❛✈✐❞s♦♥✱ ❑✳❘✳✱ ❈✯✲❛❧❣❡❜r❛s ❜② ❊①❛♠♣❧❡✱ ❋✐❡❧❞s ■♥st✐t✉t❡ ❢♦r ❘❡s❡❛r❝❤

    ✐♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s ❚♦r♦♥t♦✿ ❋✐❡❧❞s ■♥st✐t✉t❡ ♠♦♥♦❣r❛♣❤s✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✶✾✾✻✳

    ❬✾❪ ❉✐①♠✐❡r✱ ❏✳✱ C ✲❛❧❣❡❜r❛s✱ ◆♦rt❤✲❍♦❧❧❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ ❧✐❜r❛r②✱ ◆♦rt❤✲❍♦❧❧❛♥❞✱

      ✶✾✽✷✳

    ❬✶✵❪ ❊①❡❧✱ ❘✳✱ ❆♠❡♥❛❜✐❧✐t② ❢♦r ❋❡❧❧ ❜✉♥❞❧❡s✱ ❏✳ r❡✐♥❡ ❛♥❣❡✇✳ ▼❛t❤✳ ✹✾✷ ✭✶✾✾✼✮✱ ✹✶✕

    ✼✸✳

    ❬✶✶❪ ✱ ❚✇✐st❡❞ ♣❛rt✐❛❧ ❛❝t✐♦♥s✱ ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ r❡❣✉❧❛r C ✲❛❧❣❡❜r❛✐❝ ❜✉♥✲

      ❞❧❡s✱ Pr♦❝✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✼✹ ✭✶✾✾✼✮✱ ✹✶✼✕✹✹✸✳

    ❬✶✷❪ ✱ ❈✐r❝❧❡ ❛❝t✐♦♥s ♦♥ C ✲❛❧❣❡❜r❛s✱ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❛♥❞ ❛ ❣❡♥❡r❛✲

    ❧✐③❡❞ P✐♠s♥❡r✲❱♦✐❝✉❧❡s❝✉ ❡①❛❝t s❡q✉❡♥❝❡✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧②s✐s ✶✷✷ ✭✶✾✾✹✮✱ ✸✻✶✕ ✹✵✶✳

      

    ❬✶✸❪ ❊①❡❧✱ ❘✳ ❛♥❞ ◆❣✱ ❈✳✱ ❆♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt② ♦❢ ❈✯✲❛❧❣❡❜r❛✐❝ ❜✉♥❞❧❡s✱ ▼❛t❤✳

    Pr♦❝✳ ❈❛♠❜r✐❞❣❡ P❤✐❧♦s✳ ❙♦❝✳ ✶✸✷ ✭✷✵✵✷✮✱ ✺✵✾✕✺✷✷✳

      

    ❬✶✹❪ ❋❡❧❧✱ ❏✳▼✳●✳ ❛♥❞ ❉♦r❛♥✱ ❘✳❙✳✱ ❘❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ∗✲❆❧❣❡❜r❛s✱ ▲♦❝❛❧❧② ❈♦♠♣❛❝t

    ●r♦✉♣s✱ ❛♥❞ ❇❛♥❛❝❤ ∗✲❆❧❣❡❜r❛✐❝ ❇✉♥❞❧❡s✿ ❇❛♥❛❝❤ ∗✲❆❧❣❡❜r❛✐❝ ❇✉♥❞❧❡s✱ ■♥❞✉✲ ❝❡❞ ❘❡♣r❡s❡♥t❛t✐♦♥s✱ ❛♥❞ t❤❡ ●❡♥❡r❛❧✐③❡❞ ▼❛❝❦❡② ❆♥❛❧②s✐s✱ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❊❧s❡✈✐❡r ❙❝✐❡♥❝❡✱ ✶✾✽✽✳

      

    ❬✶✺❪ ❋❡❧❧✱ ❏✳ ▼✳ ●✳ ❛♥❞ ❉♦r❛♥✱ ❘✳ ❙✳✱ ❘❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ∗✲❛❧❣❡❜r❛s✱ ▲♦❝❛❧❧② ❈♦♠✲

    ♣❛❝t ●r♦✉♣s✱ ❛♥❞ ❇❛♥❛❝❤ ✯✲❛❧❣❡❜r❛✐❝ ❇✉♥❞❧❡s✱ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ✶✾✽✽✳

    ❬✶✻❪ ❋♦❧❧❛♥❞✱ ●✳ ❇✳✱ ❆ ❈♦✉rs❡ ✐♥ ❆❜str❛❝t ❍❛r♠♦♥✐❝ ❆♥❛❧②s✐s✱ ❙t✉❞✐❡s ✐♥ ❆❞✈❛♥❝❡❞

    ▼❛t❤❡♠❛t✐❝s✱ ❚❛②❧♦r ✫ ❋r❛♥❝✐s✱ ✶✾✾✹✳

    ❬✶✼❪ ❍❥❡❧♠❜♦r❣✱ ❏✳ ✈✳❇✳ ❛♥❞ ❘ør❞❛♠✱ ▼✳✱ ❖♥ st❛❜✐❧✐t② ♦❢ C ✲❛❧❣❡❜r❛s✱ ❏♦✉r♥❛❧ ❋✉♥❝✲

    t✐♦♥❛❧ ❆♥❛❧②s✐s ✶✺✺ ✭✶✾✾✽✮✱ ✶✺✸✕✶✼✵✳

    ❬✶✽❪ ▲❛♥❝❡✱ ❊✳❈✳✱ ❍✐❧❜❡rt C ✲▼♦❞✉❧❡s✿ ❆ ❚♦♦❧❦✐t ❢♦r ❖♣❡r❛t♦r ❆❧❣❡❜r❛✐sts✱ ▲❡❝t✉r❡

    ♥♦t❡ s❡r✐❡s✿ ▲♦♥❞♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✺✳

    ❬✶✾❪ ▼❝❈❧❛♥❛❤❛♥✱ ❑✳✱ K✲❚❤❡♦r② ❢♦r ♣❛rt✐❛❧ ❝r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❞✐s❝r❡t❡ ❣r♦✉♣s✱ ❏✳

    ❋✉♥❝t✳ ❆♥❛❧②s✐s ✶✸✵ ✭✶✾✾✺✮✱ ✼✼✕✶✶✼✳

    ❬✷✵❪ ▼✉r♣❤②✱ ●✳ ❏✳✱ C ✲❛❧❣❡❜r❛s ❛♥❞ ♦♣❡r❛t♦r t❤❡♦r②✱ ❆❈❆❉❊▼■❈ Pr❡ss■◆❈✱ ✶✾✾✵✳

      

    ❬✷✶❪ P✐❡r❝❡✱ ❘✳ ❙✳✱ ❆ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛s✱ ●r❛❞✉❛t❡ t❡①ts ✐♥ ♠❛t❤❡♠❛t✐❝s✱ ❙♣r✐♥❣❡r✲

    ❱❡r❧❛❣✱ ✶✾✽✷✳

    ∗ ∗

    ❬✷✷❪ ◗✉✐❣❣✱ ❏✳ ❈✳✱ ❉✐s❝r❡t❡ C ✲❝♦❛❝t✐♦♥s ❛♥❞ C ✲❛❧❣❡❜r❛✐❝ ❜✉♥❞❧❡s✱ ❏✳ ❆✉str❛❧✳

    ▼❛t❤✳ ❙♦❝✳✭❙❡r✐❡s ❆✮ ✻✵ ✭✶✾✾✻✮✱ ✷✵✹✕✷✷✶✳

    ❬✷✸❪ ❘❛❡❜✉r♥✱ ■✳ ❛♥❞ ❲✐❧❧✐❛♠s✱ ❉✳P✳✱ ▼♦r✐t❛ ❊q✉✐✈❛❧❡♥❝❡ ❛♥❞ ❈♦♥t✐♥✉♦✉s✲tr❛❝❡ C ✲

    ❛❧❣❡❜r❛s✱ ▼❛t❤❡♠❛t✐❝❛❧ s✉r✈❡②s ❛♥❞ ♠♦♥♦❣r❛♣❤s✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦✲ ❝✐❡t②✱ ✶✾✾✽✳

      

    ❬✷✹❪ ❘✉❞✐♥✱ ❲✳✱ ❋♦✉r✐❡r ❛♥❛❧②s✐s ♦♥ ❣r♦✉♣s✱ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❙❡r✐❡s✱

    ■♥t❡rs❝✐❡♥❝❡ P✉❜❧✐s❤❡rs✱ ✶✾✻✷✳

    ❬✷✺❪ ❚❛❦❛✐✱ ❍✳✱ ❖♥ ❛ ❉✉❛❧✐t② ❢♦r ❈r♦ss❡❞ Pr♦❞✉❝ts ♦❢ C ✲❆❧❣❡❜r❛s✱ ❏♦✉r♥❛❧ ❋✉♥❝t✐✲

    ♦♥❛❧ ❆♥❛❧②s✐s ✶✾ ✭✶✾✼✺✮✱ ✷✺✕✸✾✳

    ❬✷✻❪ ❯❣❣✐♦♥✐✱ ❇✳ ❇✳✱ ❙♦❜r❡ ♣r♦❞✉t♦s ❝r✉③❛❞♦s ❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ▼♦r✐t❛✱ ❉✐ss❡rt❛✲

    çã♦ ❞❡ ▼❡str❛❞♦✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ❋❧♦r✐❛♥ó♣♦❧✐s✱ ✷✵✶✸✳

    ❬✷✼❪ ❲✐❧❧✐❛♠s✱ ❉✳P✳✱ ❈r♦ss❡❞ Pr♦❞✉❝ts ♦❢ C ✲❛❧❣❡❜r❛s✱ ▼❛t❤❡♠❛t✐❝❛❧ s✉r✈❡②s ❛♥❞

      ♠♦♥♦❣r❛♣❤s✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✷✵✵✼✳

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