❯♥✐✈❡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❡❧ ❋✐❧❤♦
❯♥✐✈❡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❡✳
Ficha de identificação da obra elaborada pelo autor,
através do Programa de Geração Automática da Biblioteca Universitária da UFSC.
Sehnem, Camila Fabre
Uma classificação de fibrados de Fell estáveis / Camila Fabre Sehnem ; orientador, Ruy Exel Filho - Florianópolis, SC, 2014.
137 p.
Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-Graduação em Matemática Pura e Aplicada. 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❛ ❞❡ B⊗K(l2(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✱ ✇❡ ❞❡✜♥❡
t❤❡ s♠❛s❤ ♣r♦❞✉❝t C∗✲❛❧❣❡❜r❛ B#C∗(G) ❛s ❛ ❝❡rt❛✐♥ s✉❜❛❧❣❡❜r❛ ♦❢ B⊗K(l2(G))✳
❲❡ ✉s❡ t❤❡ s♠❛s❤ ♣r♦❞✉❝tC∗✲❛❧❣❡❜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♦ ✻
✶ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ ✼
✶✳✶ ▼ó❞✉❧♦s ❞❡ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❖♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸ ❇✐♠ó❞✉❧♦s ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹ ▼♦r✐t❛ ❡q✉✐✈❛❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷ ❋✐❜r❛❞♦s ❞❡ ❋❡❧❧ ✸✽
✷✳✶ ❆C∗✲á❧❣❡❜r❛ s❡❝❝✐♦♥❛❧ ❝❤❡✐❛ ❞❡ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ✳ ✳ ✳ ✸✽
✷✳✷ ❆ r❡♣r❡s❡♥t❛çã♦ r❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✸ Pr♦❞✉t♦s ❈r✉③❛❞♦s ✺✷
✸✳✶ ❆çõ❡s ♣❛r❝✐❛✐s ❞❡ ❣r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✷ Pr♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✸ ❆çõ❡s ❞❡ ❣r✉♣♦s ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
✹ ❚❡♦r❡♠❛ ❞❡ ❇r♦✇♥✲●r❡❡♥✲❘✐❡✛❡❧ ✻✼
✹✳✶ C∗✲á❧❣❡❜r❛s ❡stá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✹✳✷ ❋✐❜r❛❞♦s ❞❡ ❋❡❧❧ ❡stá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✸ Pr♦❥❡çõ❡s ❝❤❡✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼
✺ Pr♦❞✉t♦s s♠❛s❤ ✾✺
✺✳✶ C∗✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺
✺✳✷ Pr♦❞✉t♦s s♠❛s❤ ❡ ❞✉❛❧✐❞❛❞❡ ❞❡ ❚❛❦❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✺✳✸ ❯♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ✜❜r❛❞♦s ❞❡ ❋❡❧❧ ❡ ❛çõ❡s ♣❛r❝✐❛✐s ✶✵✻
❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✶✶✻
❆ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ❛✉①✐❧✐❛r❡s ✶✶✽ ❆✳✶ C∗✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽
❆✳✷ ❊❧❡♠❡♥t♦s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✶ ❆✳✸ ➪❧❣❡❜r❛ ❞❡ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸ ❆✳✸✳✶ ❚♦♣♦❧♦❣✐❛ ❡str✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸ ❆✳✸✳✷ Pr♦❞✉t♦ t❡♥s♦r✐❛❧ ❡s♣❛❝✐❛❧ ❞❡ á❧❣❡❜r❛s ❞❡ ♠✉❧t✐✲
♣❧✐❝❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻
■♥tr♦❞✉çã♦
▼✉✐t♦ ❡♠❜♦r❛ ❛ t❡♦r✐❛ ❞❡ ✜❜r❛❞♦sC∗✲❛❧❣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
✉♠ ❣r✉♣♦Gs❡B=⊕g∈GBg✱ ❡♠ q✉❡✱ ♣❛r❛ ❝❛❞❛g✱Bgé ✉♠ s✉❜❡s♣❛ç♦ ❢❡❝❤❛❞♦ ❞❡B✱Bg∗=Bg−1✱ ❡BgBh⊆Bgh✱ ♣❛r❛ q✉❛✐sq✉❡rg, h∈G✳ ❊♠
t❡r♠♦s ❣❡r❛✐s✱ ✉♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧B s♦❜r❡ ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦Gé ✉♠❛
❝♦❧❡çã♦ ❞❡ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤{Bt}t∈G ❝♦♠ ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦
· : Bt×Bs→Bts
❡ ✐♥✈♦❧✉çã♦
∗ : Bt→Bt−1
s❛t✐s❢❛③❡♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ s❡r✐❛♠ s❛t✐s❢❡✐t❛s s❡ ❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s✲ ♣❛ç♦s{Bt}t∈G ❢♦ss❡✱ ❞❡ ❢❛t♦✱ ✉♠❛ ❣r❛❞✉❛çã♦ ♣❛r❛ ❛❧❣✉♠❛C∗✲á❧❣❡❜r❛✳ ❯♠ ✜❜r❛❞♦ ❞❡ ❋❡❧❧ ❞á ♦r✐❣❡♠ ❛ C∗✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s ♣❡❧❛ ❝♦❧❡✲
çã♦ ❞❡ s✉❜❡s♣❛ç♦s{Bt}t∈G ✉♠ t❛♥t♦ ❡s♣❡❝✐❛✐s✱ ❛ s❛❜❡r ❛sC∗✲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❛
❣r❛❞✉❛❞❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s{Bt}t∈G✭✈❡❥❛ ❬✶✵❪✮✳ ■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❛✲ ♠❡♥t♦ ❞❛∗✲á❧❣❡❜r❛⊕g∈GBg❡♠ ✉♠❛ ❞❛❞❛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
❞á ♦r✐❣❡♠ ❛♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦A⋊αG✱ q✉❡ é ✉♠❛C∗✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛
♣♦r ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❡s♣❛ç♦s q✉❡ sã♦ ❝ó♣✐❛s ❞❡A✱ ♣❡❧♦ ♠❡♥♦s ❝♦♠♦
❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳ ◆♦ ❝❛s♦ ❡♠ q✉❡αé ✉♠❛ ❛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❛ Aé ✉♠ ♣❛r α= ({Dg}g∈G,{αg}g∈G)❡♠ q✉❡✱ ♣❛r❛ ❝❛❞❛ g∈G✱ Dg é ✉♠ ✐❞❡❛❧ ❞❡
A✱αg : Dg−1 →Dgé ✉♠∗✲✐s♦♠♦r✜s♠♦ ❡✱ ♣❡❧♦ ♠❡♥♦s q✉❛♥❞♦ ♣♦ssí✈❡❧✱
t❡♠♦s ✉♠❛ ❝❡rt❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♠♣♦s✐çã♦ ❞♦s
∗✲✐s♦♠♦r✜s♠♦sαg✬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 ❡♠ ❬✷✸❪ ❡ ❬✶✽❪✱ ❞❡✜♥✐♠♦sC∗✲♠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❡ àC∗✲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✉✐ ✈❛❧♦r0✳ 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
❞✐r❡✐t❛✮✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r XA✱ s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ X ×A →X✱
(x, a)7→xa s❛t✐s❢❛③❡♥❞♦✱ ♣❛r❛ q✉❛✐sq✉❡rx, 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✳
❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❙❡❥❛ XA ✉♠ A✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉✐③❡♠♦s q✉❡XA é ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ h·,·iA :
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✿
✭✐✮hx, λy+µziA=λhx, yiA+µhx, ziA❀ ✭✐✐✮hx, yaiA=hx, yiAa❀
✭✐✐✐✮hx, yi∗
A=hy, xiA❀
✭✐✈✮ hx, xiA ≥ 0✱ ✐✳❡✳✱ hx, xiA é ♣♦s✐t✐✈♦ ❝♦♠♦ ✉♠ ❡❧❡♠❡♥t♦ ❞❛ C∗✲ á❧❣❡❜r❛A❀
✭✈✮hx, xiA= 0✐♠♣❧✐❝❛ q✉❡x= 0✳
◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ h·,·iA : X ×X → A é ✉♠
A✲♣r♦❞✉t♦ ✐♥t❡r♥♦✳
❖❜s❡r✈❛çã♦ ✶✳✶✳✸✳ ❖s ❛①✐♦♠❛s ✭✐✮ ❡ ✭✐✐✐✮ ✐♠♣❧✐❝❛♠ q✉❡h·,·iAé ❝♦♥❥✉❣❛❞♦✲ ❧✐♥❡❛r ♥❛ ♣r✐♠❡✐r❛ ✈❛r✐á✈❡❧✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠ ❡❢❡✐t♦✱ t❡♠♦s
hλx+µy, ziA=hλz, λx+µyi∗A = (λhz, xiA+µhz, yiA)∗
= λ¯hz, xi∗A+ ¯µhz, yi∗A
= λ¯hx, ziA+ ¯µhy, ziA.
❖❜s❡r✈❛çã♦ ✶✳✶✳✹✳ ❆s ❝♦♥❞✐çõ❡s ✭✐✐✮ ❡ ✭✐✐✐✮ ✐♠♣❧✐❝❛♠ q✉❡ hxa, yiA=
a∗hx, yiA✱ ❞♦♥❞❡ s❡❣✉❡ q✉❡
hX, XiA:= span{hx, yiA:x, y∈X}
é ✉♠ ✐❞❡❛❧ ❡♠A✳
❖❜s❡r✈❛çã♦ ✶✳✶✳✺✳ ❙❡AX 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❡❥❛✱
Ahλx+µy, zi=λAhx, yi+µhy, zi ❡ Ahax, yi=aAhx, yi, ♣❛r❛ q✉❛✐sq✉❡rx, y∈X✱a∈A✱ ❡λ, µ∈C✳
❉❡✜♥✐çã♦ ✶✳✶✳✻✳ ❙❡❥❛X ✉♠A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳ ❉✐③❡♠♦s
q✉❡X é ❝❤❡✐♦ s❡ hX, XiA=A✳
❊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♦Aé ✉♠A✲♠ó❞✉❧♦ ❝♦♠
♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝❤❡✐♦ ❝♦♠ ❛ ❛çã♦ ❞❡ ♠ó❞✉❧♦ ❞❛❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣❡❧❛ ❞✐r❡✐t❛ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ha, biA = a∗b✱ ♣❛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✐♦♠❛✳
❏á ❛ ✐❣✉❛❧❞❛❞❡hA, AiA=A✱ s❡❣✉❡ ❞♦ ❢❛t♦ q✉❡
a= lim
λ uλa= limλ huλ, aiA,
❡♠ q✉❡a∈A ❡(uλ)λ∈Λ é ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛A✳
❋❛③❡♥❞♦ ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ❝❛s♦ ❡s❝❛❧❛r✱ ♣♦❞❡rí❛♠♦s ♥♦s ♣❡r❣✉♥✲ t❛r s❡ ❛ ❛♣❧✐❝❛çã♦ k · kA : X →R+✱ x7→ khx, xiAk
1
2 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∗Ahx, yiA≤ khx, xiAkhy, yiA
❝♦♠♦ ❡❧❡♠❡♥t♦s ❞❛ C∗✲á❧❣❡❜r❛A✳
❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ khx, xiAk = 1✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ a∈A
0≤ hxa−y, xa−yiA = a∗hx, xiAa− hy, xiAa−a∗hx, yiA+hy, yiA
≤ a∗a− hy, xi
Aa−a∗hx, yiA+hy, yiA,
❡♠ q✉❡ ♥❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡ a∗ba≤ kbka∗a✱ ♣❛r❛
t♦❞♦ b∈A+✳ ❈♦❧♦❝❛♥❞♦a=hx, yi
A ♦❜t❡♠♦s
hx, yi∗Ahx, yiA≤ hy, yiA,
❝♦♠♦ ❞❡s❡❥❛❞♦✳
P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡x= 0♥ã♦ ❤á ♥❛❞❛ ❛ ❢❛③❡r✳ P❛r❛ ♦ ❝❛s♦ ❣❡r❛❧✱
❜❛st❛ ❛♣❧✐❝❛r♠♦s ♦ q✉❡ ❥á ❢♦✐ ❢❡✐t♦ ♣❛r❛z=λx✱ ❡♠ q✉❡λ= 1
khx, xiAk
1 2
✳
❈♦r♦❧ár✐♦ ✶✳✶✳✶✵✳ ❙❡ X é ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ ❡♥tã♦ kxkA:=khx, xiAk
1 2
❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡♠X t❛❧ q✉❡kxakA≤ kakkxkA✳ ▼❛✐s ❛✐♥❞❛✱
XhX, XiA:= span{xhy, ziA:x, y, z∈X}
é ✉♠ s✉♣❡s♣❛ç♦ ❞❡♥s♦ ❡♠X✳
❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❡❥❛♠♦s q✉❡k · kA é ✉♠❛ ♥♦r♠❛ ❡♠
X✳
P❛r❛λ∈C❡x∈X t❡♠♦s
kλxkA=khλx, λxiAk
1
2 =k|λ|2hx, xi
Ak
1
2 =|λ|khx, xi
Ak
1
2 =|λ|kxk
A.
❙❡kxkA= 0✱ ❡♥tã♦hx, xiA= 0❡ ❞❛ ❝♦♥❞✐çã♦ ✭✈✮ ❞❛ ❉❡✜♥✐çã♦ ✶✳✶✳✷ ✈❡♠ q✉❡x= 0✳
❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ k · kA s❛t✐s❢❛③ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✳ ❖ ▲❡♠❛ ✶✳✶✳✾ ❡ ♦C∗✲❛①✐♦♠❛ ♥♦s ❞✐③❡♠ q✉❡khx, yi
Ak ≤ kxkAkykA✱ ♣❛r❛ q✉❛✐sq✉❡rx, y∈X✳ ❆ss✐♠✱
kx+yk2A ≤ khx, xiAk+khx, yiAk+khy, xiAk+khy, yiAk
≤ kxk2
A+ 2kxkAkykA+kyk2A
= (kxkA+kykA)2.
P♦rt❛♥t♦✱k · kAé ✉♠❛ ♥♦r♠❛ ❡♠ X✳ ❆❧é♠ ❞✐ss♦✱
kxak2A=khxa, xaiAk=ka∗hx, xiAak, ❡♠ q✉❡ a ∈ A✳ ❯♠❛ ✈❡③ q✉❡ a∗hx, xi
Aa ≤ khx, xiAka∗a✱ ♦❜t❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡kxakA≤ kakkxkA✳
P♦r ✜♠✱ ♠♦str❡♠♦s q✉❡ XhX, XiA é ❞❡♥s♦ ❡♠ X✳ ❙❡♥❞♦ (uλ)λ∈Λ
✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ ♦ ✐❞❡❛❧ ❢❡❝❤❛❞♦hX, XiA✱ t❡♠♦s kx−xuλk2A=khx, xiA− hx, xiAuλ−uλhx, xiA+uλhx, xiAuλk. ❉❛í✱ ❞❛❞♦ε >0✱ ❡①✐st❡λ0t❛❧ q✉❡ kx−xuλ0kA<
ε
2✳
❆❣♦r❛✱ s❡❥❛y ❡♠hX, XiA t❛❧ q✉❡
kuλ0−yk< ε
2.
❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ♣❛r❛ k · kA ❡ ♦ ❢❛t♦ q✉❡ kx(uλ0 − y)kA≤ kxkAkuλ0−yk✱ ❝♦♥❝❧✉í♠♦s q✉❡
kx−xykA< ε.
❉♦♥❞❡XhX, XiA é ❞❡♥s♦A✳
■ss♦ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛ ❞♦ ❝♦r♦❧ár✐♦✳
❉❡✜♥✐çã♦ ✶✳✶✳✶✶✳ ❯♠ A✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt é ✉♠A✲♠ó❞✉❧♦ ❝♦♠ ♣r♦✲
❞✉t♦ ✐♥t❡r♥♦X q✉❡ é ❝♦♠♣❧❡t♦ ♥❛ ♥♦r♠❛ k · kA✳
❊①❡♠♣❧♦ ✶✳✶✳✶✷✳ ❚♦❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❧✐♥❡❛r ♥❛ s❡❣✉♥❞❛ ✈❛r✐á✈❡❧ é ✉♠ C✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt✳
❊①❡♠♣❧♦ ✶✳✶✳✶✸✳ ❙❡❥❛A✉♠❛C∗✲á❧❣❡❜r❛✳ ❊♥tã♦Aé ✉♠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❡♠♣❧♦ ✶✳✶✳✽✳ ❉❡♠♦♥str❛çã♦✿ ■ss♦ s❡❣✉❡ ❞♦ ❢❛t♦ q✉❡ ❛ ♥♦r♠❛k · kA ❝♦✐♥❝✐❞❡ ❝♦♠ ❛
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♦ ♣❡❧❛ ❞✐r❡✐t❛ ❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡✜♥✐❞♦ ♣♦r hap, bpipAp =pa∗bp✱ ♣❛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✳
◆♦✈❛♠❡♥t❡✱ ❛ ♥♦r♠❛k · kpAp ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ♥♦r♠❛ ❞❡Ap❤❡r❞❛❞❛ ❞❡A✱ ♣♦✐s
kapk2pAp =khpa∗apipApk=kapk2.
❙❡❣✉❡ q✉❡Apé ❝♦♠♣❧❡t♦✱ ❥á q✉❡Ap= (Ap)p❡✱ ♣♦rt❛♥t♦✱ q✉❛❧q✉❡r
s❡q✉ê♥❝✐❛ ❡♠Ap❝♦♥✈❡r❣❡♥t❡ ❡♠A✱ ♣♦ss✉✐ ♦ ❧✐♠✐t❡ ❡♠Ap✳ ❆❧é♠ ❞✐ss♦✱
s❡(uλ)λ∈Λé ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ♣❛r❛A❡a∈A✱ ♦❜s❡r✈❛♠♦s q✉❡
pap= lim
λ puλap= limλ huλp, apipAp,
❞♦♥❞❡hAp, ApipAp é ❞❡♥s♦ ❡♠pAp✳
▲♦❣♦✱Apé ✉♠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′)iA:=hx, x′iA+hy, y′iA.
❉❡♠♦♥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✳
❱❛♠♦s ♠♦str❛r q✉❡Z é ❝♦♠♣❧❡t♦ ❝♦♠ ❛ ♥♦r♠❛k · kA✳
❈♦♠ ❡❢❡✐t♦✱
hx, xiA≤ hx, xiA+hy, yiA, ❡ ✐ss♦ ♥♦s ❞✐③ q✉❡
kxk2A≤ khx, xiA+hy, yiAk=k(x, y)k2A≤ kxk
2
A+kyk
2
A.
❙✐♠✐❧❛r♠❡♥t❡✱
kyk2A≤ k(x, y)k2A≤ kxk2A+kyk2A. ❖✉ s❡❥❛✱
max{kxkA,kykA} ≤ k(x, y)kA≤
q
kxk2
A+kyk2A. ✭†✮ ❙❡❥❛ (zn)n∈N ✉♠ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ Z✳ ❊s❝r❡✈❡♠♦s zn =
(xn, yn)✱ ♣❛r❛ ❝❛❞❛n✳ ❈♦♠♦X ❡Y sã♦ ❝♦♠♣❧❡t♦s✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❡♠ ✭†✮ ♥♦s ❞✐③ q✉❡ ❡①✐st❡x∈X ❡y∈Y t❛✐s q✉❡xn→x ❡ yn → y✳ ❆❣♦r❛✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭†✮ ✐♠♣❧✐❝❛ q✉❡
zn= (xn, yn)→(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 é ❛❞❥✉♥tá✈❡❧ s❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦T∗ : Y →X t❛❧ q✉❡
hT(x), yiA=hx, T∗(y)iA,
♣❛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❡Zé ✉♠A✲♠ó❞✉❧♦
❞❡ ❍✐❧❜❡rt ❡ x∈ Z é t❛❧ q✉❡ hx, ziA = 0✱ ♣❛r❛ t♦❞♦ z ∈ Z✱ ❡♥tã♦ ❛♦ ❡s❝♦❧❤❡rz=x❝♦♥❝❧✉í♠♦s q✉❡x= 0✳
❉❡st❛ ❢♦r♠❛✱ s❡♥❞♦ x ∈ X ❡ y ✉♠ ❡❧❡♠❡♥t♦ ❡s❝♦❧❤✐❞♦ ❛r❜✐tr❛r✐❛✲
♠❡♥t❡ ❡♠ Y✱ t❡♠♦s
hT(xa), yiA = hxa, T∗(y)iA=a∗hx, T∗(y)iA
= a∗hT(x), yiA=hT(x)a, yiA.
■ss♦ ♥♦s ❞✐③ q✉❡
hT(xa)−T(x)a, yiA= 0,
♣❛r❛ ❝❛❞❛y∈Y✱ ❞♦♥❞❡T(xa) =T(x)a✳
❙✐♠✐❧❛r♠❡♥t❡✱ ♣r♦✈❛✲s❡ q✉❡T(λx) =λT(x)❡T(x1+x2) =T(x1) +
T(x2)✱ ❡♠ q✉❡x1, x2∈X ❡λ∈C✳
❘❡st❛ ♣r♦✈❛r♠♦s q✉❡T é ❧✐♠✐t❛❞♦✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s ✉s❛r ♦ t❡♦r❡♠❛
❞♦ ❣rá✜❝♦ ❢❡❝❤❛❞♦✳
❙❡❥❛(xn)n∈N✉♠❛ s❡q✉ê♥❝✐❛ ❡♠X❝♦♥✈❡r❣✐♥❞♦ ❛x❡ t❛❧ q✉❡T(xn)→
y ❡♠Y✳ ❙❡❥❛z∈Y✳ P♦r ✉♠ ❧❛❞♦✱
hT(xn), ziA→ hy, ziA
❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱
hT(xn), ziA=hxn, T∗(z)iA→ hx, T∗(z)iA=hT(x), ziA,
❡♠ q✉❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❛♣❧✐❝❛çã♦ x→ hx, ziA 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❛❞♦rA✲❧✐♥❡❛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 éA✲❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦✱ ♠❛s ♥ã♦ é ❛❞❥✉♥tá✈❡❧✳
❉❡♠♦♥str❛çã♦✿ ➱ ❢á❝✐❧ ✈❡r q✉❡T éA✲❧✐♥❡❛r✳ P❛r❛ ✈❡r q✉❡T é ❝♦♥tí✲
♥✉♦✱ ♥♦t❡♠♦s q✉❡
kT(f, g)kA = k(g,0)kA=khg, giAk
1 2
≤ khf, fiA+hg, giAk
1
2 =k(f, g)k
A.
❚♦♠❛♥❞♦ g ∈ J ❝♦♠kgkA = 1✱ ❝♦♥❝❧✉í♠♦s q✉❡kTk = 1✳ ▲♦❣♦✱ T é
A✲❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦✳
❙✉♣♦♥❤❛ q✉❡T s❡❥❛ ❛❞❥✉♥tá✈❡❧ ❡ s❡❥❛ (f, g) :=T∗(1,0)✳ P❛r❛ t♦❞♦
(h, k)∈X t❡♠♦s
¯
k=hT(h, k),(1,0)iA=h(h, k),(f, g)iA= ¯hf+ ¯kg. ✭†✮ ❉❛í✱ s❡❣✉❡ q✉❡f(0) = 0✳
❆❣♦r❛✱ ♣❛r❛k∈J ❛r❜✐trár✐♦✱ ❛ ✐❣✉❛❧❞❛❞❡ ✭†✮ ✐♠♣❧✐❝❛ q✉❡
¯
k(1−f−g) = 0.
▲♦❣♦✱ ❞❡✈❡♠♦s t❡rf+g= 1✳
❆ss✐♠✱1 =f(0) +g(0) = 0 +g(0) =g(0)✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱
♣♦✐sg∈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✳ ◗✉❛♥❞♦ X = Y✱ ❡s❝r❡✈❡♠♦s L(X)✱ ♦✉ ❛✐♥❞❛ L(XA)✱ ❡♠ ✈❡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∗ é ❛❞❥✉♥tá✈❡❧ ❡T∗∗=T❀
✭✐✐✐✮ P❛r❛ ❝❛❞❛λ∈C✱λT +S é ❛❞❥✉♥tá✈❡❧ ❡(λT +S)∗= ¯λT∗+S∗❀
✭✐✈✮RT é ❛❞❥✉♥tá✈❡❧ ❡(RT)∗=T∗R∗✳
❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙❡❥❛U : Y →X t❛❧ q✉❡ hT(x), yiA=hx, U(y)iA,
♣❛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)iA=hx, U(y)iA.
P♦r ✉♠ ❛r❣✉♠❡♥t♦ ❥á ✉t✐❧✐③❛❞♦✱ ✐ss♦ ✐♠♣❧✐❝❛ T∗(y) =U(y)✳
✭✐✐✮ P❛r❛ x∈X ❡y∈Y✱ ✈❛❧❡ q✉❡
hT∗(y), xiA=hx, T∗(y)i∗A=hT(x), yi∗A=hy, T(x)iA.
❉♦♥❞❡T∗∗=T✳
✭✐✐✐✮ ◆♦✈❛♠❡♥t❡✱
h(λT +S)(x), yiA = λ¯hT(x), yiA+hS(x), yiA
= λ¯hx, T∗(x)iA+hx, S∗(y)iA
= hx,λ¯(T∗+S∗)(y)iA.
❈♦♠♦xé ❛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∗Tk ≤ kT∗kkTk,
♣❛r❛ t♦❞♦T ❡♠L(X)✳
❆❧é♠ ❞✐ss♦✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱
kT∗Tk ≥ sup
kxk≤1
khT∗T(x), xiAk= sup
kxk≤1
khT(x), T(x)iAk=kTk2 ❡ ♦❜t❡♠♦s q✉❡kTk ≤ kT∗k✳ ❉❡ ♠❡s♠❛ ❢♦r♠❛✱ ♦❜s❡r✈❛♥❞♦ q✉❡T∗∗=T✱
❝♦♥❝❧✉í♠♦s q✉❡kT∗k ≤ kTk✳ ▲♦❣♦✱kT∗k=kTk✳
❆ss✐♠✱ kTk2 ≤ kT∗Tk ≤ kT∗kkTk = kTk2✱ ❞♦♥❞❡ 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♦
hT(x), T(x)iA≤ kTk2hx, xiA.
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ kTk2−T∗T é ✉♠ ❡❧❡♠❡♥t♦ ♣♦s✐t✐✈♦ ❞❛ C∗✲
á❧❣❡❜r❛ L(X)✱ ❡①✐st❡ S ∈ L(X) t❛❧ q✉❡ kTk2−T∗T = S∗S✳ ❉❡st❛
❢♦r♠❛✱
kTk2hx, xiA− hT(x), T(x)iA = kTk2hx, xiA− hT∗T(x), xiA
= h(kTk2−T∗T)(x), xiA=hS∗S(x), xiA
= hS(x), S(x)iA≥0.
❉♦♥❞❡ s❡❣✉❡hT(x), T(x)iA≤ kTk2hx, xiA.
✶✳✸ ❇✐♠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✉❡rx, y, z∈X✱
Ahx, yiz=xhy, ziB.
❊①❡♠♣❧♦ ✶✳✸✳✷✳ ❯♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rtH 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(H)hh, ki:=h⊗k,
❡♠ q✉❡ (h⊗k)(z) =hk, zih,♣❛r❛ t♦❞♦z∈H✱ ❡ hh, kiC:=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❛❞♦✲
r❡s ❞❛ ❢♦r♠❛h⊗k✱h, k∈H✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡K(H)hH, Hié ❞❡♥s♦ ❡♠
K(H)✳
▼❛✐s ❛✐♥❞❛✱ ♦❜s❡r✈❛♠♦s
khkK(H)=k(h⊗h)k
1
2 = (khk2) 1 2 =khk,
❞♦♥❞❡H é ✉♠K(H)✲♠ó❞✉❧♦ ❞❡ ❍✐❧❜❡rt ❝❤❡✐♦✳
❱❛♠♦s ✈❡r✐✜❝❛r ❛ ❝♦♥❞✐çã♦ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ❙❡❥❛♠x, y, z∈H✳ ❊♥tã♦✱
K(H)hx, yiz=hy, zix=xhy, ziC.
▲♦❣♦✱H é ✉♠ K(H)✲C❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡✳
❊①❡♠♣❧♦ ✶✳✸✳✸✳ ❙❡❥❛ A✉♠C∗✲á❧❣❡❜r❛✳ ❊♥tã♦Aé ✉♠A✲A❜✐♠ó❞✉❧♦
❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ ❜✐♠ó❞✉❧♦ ❞❛❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ A✱ ❡ ❝♦♠ ♣r♦❞✉t♦s ✐♥t❡r♥♦s Aha, bi = ab∗ ❡ ha, biA = a∗b✱ ♣❛r❛
a, b∈A✳
❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ✈❡r✐✜❝❛r ♦ ✐t❡♠ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ■st♦ s❡❣✉❡ ❞♦ s❡❣✉✐♥t❡ ❝á❧❝✉❧♦✿
Aha, bic=ab∗c=ahb, ciA,
❡♠ 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
pAphpaq, pbqi=paqb∗p
❡
hpaq, pbqiqAq =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❧❝✉❧♦✿
pAphpaq, pbqipcq=paqb∗pcq=paqhpbq, pcqiqAq,
❡♠ q✉❡a, b, c∈ A✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛♦ ❡s❝♦❧❤❡r q= 1✱ s❡❣✉❡ q✉❡ pAé
✉♠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✉❡ra∈A✱b∈B✱ ❡x, y∈X✱
Ahxb, yi=Ahx, yb∗i ❡ hax, yiB =hx, a∗yiB; ✭✐✐✮ P❛r❛ q✉❛✐sq✉❡ra∈A✱b∈B✱ ❡x∈X✱
hax, axiB ≤ kak2hx, xiB ❡ Ahxb, xbi ≤ kbk2Ahx, xi.
❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙✉♣♦♥❤❛ q✉❡X s❡❥❛ ✉♠A✲B ❜✐♠ó❞✉❧♦ ❞❡ ✐♠♣r✐✲
♠✐t✐✈✐❞❛❞❡✳ P❡❧♦ ✐t❡♠ ✭✐✐✐✮✱ ❞❛❞♦sx, y, z∈X✱ t❡♠♦s q✉❡ xhay, ziB= Ahx, ayiz=Ahx, yia∗z=xhy, a∗ziB.
❆ss✐♠✱ s❡wé ♦✉tr♦ ❡❧❡♠❡♥t♦ ❞❡X✱ ✈❛❧❡ q✉❡ hw, xiBhay, ziB =hw, xiBhy, a∗ziB.
❯♠❛ ✈❡③ q✉❡hX, XiB é ❞❡♥s♦ ❡♠B✱ ❝♦♥❝❧✉í♠♦s q✉❡
bhay, ziB =bhy, a∗ziB,
♣❛r❛ t♦❞♦b∈B✳ P♦r ♠❡✐♦ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛✱ ♦❜t❡♠♦s q✉❡ hay, ziB =hy, a∗ziB,
♣❛r❛ q✉❛✐sq✉❡r y, z∈X✳
❙✐♠✐❧❛r♠❡♥t❡✱ ♣r♦✈❛✲s❡ q✉❡Ahyb, zi= Ahy, zbi✱ ♣❛r❛ q✉❛✐sq✉❡ry, z ∈
X✳
✭✐✐✮ P❡❧♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♥♦ ✐t❡♠ ✭✐✮✱A❛❣❡ ♣♦r ♦♣❡r❛❞♦r❡s ❛❞❥✉♥tá✈❡✐s ❡♠ XB✳ ❉❡st❛ ❢♦r♠❛✱ ✈❛♠♦s ♠♦str❛r q✉❡ ❛ ❛♣❧✐❝❛çã♦ ϕ : A → L(XB)✱
a7→ϕ(a)é ✉♠∗✲❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦✱ ❡♠ q✉❡ ϕ(a)(x) =ax,
♣❛r❛ t♦❞♦x∈X✳
❉❡ ❢❛t♦✱ ❞♦ ✐t❡♠ ✭✐✮✱ ❝♦♥❝❧✉í♠♦s q✉❡ ϕ(a)∗ = ϕ(a∗) ❡ ϕ é ✉♠ ∗✲
❤♦♠♦♠♦r✜s♠♦✳ P❛r❛ ✈❡r q✉❡ϕé ✐♥❥❡t✐✈♦✱ s❡❥❛a∈At❛❧ q✉❡ϕ(a) = 0✳
❊♥tã♦ s❡❣✉❡ q✉❡ax= 0✱ ♣❛r❛ t♦❞♦x∈X✳ ▼❛s ✐ss♦ s✐❣♥✐✜❝❛ q✉❡✱ ♣❛r❛
❝❛❞❛y∈X✱
0 =Ahax, yi=aAhx, yi.
❯♠❛ ✈❡③ q✉❡ ♦ ✐❞❡❛❧AhX, Xié ❞❡♥s♦ ❡♠A✱ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡a= 0✳ ❆ss✐♠✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✳✼✱
hax, axiB=hϕ(a)(x), ϕ(a)(x)iB ≤ kϕ(a)k2hx, xiB=kak2hx, xiB. ❯♠ ❛r❣✉♠❡♥t♦ ❛♥á❧♦❣♦ ♠♦str❛ q✉❡
Ahxb, xbi ≤ kbk2Ahx, xi.
❖❜s❡r✈❛çã♦ ✶✳✸✳✻✳ ❉♦ ▲❡♠❛ ✶✳✷✳✷ ❡ ❞♦ ✐t❡♠ ✭✐✮ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✸✳✺✱
