❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛
❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
P✉r❛ ❡ ❆♣❧✐❝❛❞❛
❯♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛
s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ ❈✯✲á❧❣❡❜r❛s ❞❡
❣r✉♣ó✐❞❡s
❋❛❜✐♦ ❞❡ ❙❛❧❡s ❈❛s✉❧❛
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆❧❝✐❞❡s ❇✉ss
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛
❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
P✉r❛ ❡ ❆♣❧✐❝❛❞❛
❯♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡
❈✯✲á❧❣❡❜r❛s ❞❡ ❣r✉♣ó✐❞❡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✐❝❛✱ ❝♦♠ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ❆♥á❧✐s❡✳
Ficha de identificação da obra elaborada pelo autor,
através do Programa de Geração Automática da Biblioteca Universitária da UFSC.
Casula, Fabio de Sales
Uma caracterização da simplicidade de C*-álgebras de grupóides / Fabio de Sales Casula ; orientador, Alcides Buss - Florianópolis, SC, 2017.
156 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
❯♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡
❈✯✲á❧❣❡❜r❛s ❞❡ ❣r✉♣ó✐❞❡s
♣♦r❋❛❜✐♦ ❞❡ ❙❛❧❡s ❈❛s✉❧❛✶
❊st❛ ❉✐ss❡rt❛çã♦ ❢♦✐ ❥✉❧❣❛❞❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ❆♥á❧✐s❡✱ ❡ ❛♣r♦✈❛❞❛ ❡♠ s✉❛
❢♦r♠❛ ✜♥❛❧ ♣❡❧♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✳
Pr♦❢✳ ❉r✳ ❘✉② ❈♦✐♠❜r❛ ❈❤❛✲ rã♦
❈♦♦r❞❡♥❛❞♦r ❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛
Pr♦❢✳ ❉r✳ ❆❧❝✐❞❡s ❇✉ss ✭❖r✐❡♥t❛❞♦r ✲ ❯❋❙❈✮
Pr♦❢✳ ❉r✳ ▼✐s❤❛ ❉♦❦✉❝❤❛❡✈ ✭❯❙P✮
Pr♦❢✳ ❉r✳ ❘✉② ❊①❡❧ ✭❯❋❙❈✮
Pr♦❢✳ ❉r✳ ❊❧✐❡③❡r ❇❛t✐st❛ ✭❯❋❙❈✮
Pr♦❢✳ ❉r✳ ❏♦r❣❡ ●❛r❝és Pér❡③ ✭❯❋❙❈✮
✶❇♦❧s✐st❛ ❞♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❈✐❡♥tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦ ✲
❈◆Pq
❋❧♦r✐❛♥ó♣♦❧✐s✱ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✻✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❉❡♣♦✐s ❞❡ s❡✐s ❛♥♦s✱ ❞❡s❞❡ ♦ ✐♥í❝✐♦ ❞❡ ♠❡✉s ❡st✉❞♦s ❡♠ ♠❛t❡♠á✲ t✐❝❛✱ ❝♦♥❝❧✉♦ ❡st❛ ❡t❛♣❛ tã♦ ✐♠♣♦rt❛♥t❡ ❞❡ ♠✐♥❤❛ ✈✐❞❛✳ ▼✉✐t❛s ♣❡ss♦❛s ✜③❡r❛♠ ♣❛rt❡ ❞❡st❛ ❥♦r♥❛❞❛ ❡ sã♦ ❞✐❣♥❛s ❞❡ ❛❣r❛❞❡❝✐♠❡♥t♦✳
❈♦♠❡ç♦ ❛❣r❛❞❡❝❡♥❞♦ ♠❡✉s ♣❛✐s✱ q✉❡ s❡♠♣r❡ t✐✈❡r❛♠ t♦t❛❧ ❝♦♥✜❛♥ç❛ ❡♠ ♠✐♥❤❛s ❝❛♣❛❝✐❞❛❞❡s✱ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠ ❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ♦ ♠❡❧❤♦r✳ ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ✐r♠ã♦✱ ♠❡✉ ♠❡❧❤♦r ❛♠✐❣♦✱ q✉❡ s❡♠♣r❡ ❡st❡✈❡ ♣r❡s❡♥t❡ ♥❛s ❤♦r❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❡ ❞❡❝✐s✐✈❛s✳ ▼❡s♠♦ s❡♥❞♦ ♠❛✐s ♥♦✈♦ ❞♦ q✉❡ ❡✉✱ ❞❡ ♠♦❞♦ q✉❡ ❡✉ ❞❡✈❡ss❡ ♣❛ss❛r ♦ ❡①❡♠♣❧♦✱ ❡♠ ♠✉✐t❛s s✐t✉❛çõ❡s ♦ ❝♦♥trár✐♦ ♦❝♦rr❡✉ ❡ s♦✉ ❣r❛t♦ ♣♦r ♠✉✐t♦s ❝♦♥s❡❧❤♦s ❡ ❝♦♥✈❡rs❛s✳
❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♥❛♠♦r❛❞❛✱ ❏éss✐❦❛✱ q✉❡ ✈✐✈❡♥❝✐♦✉ t♦❞♦ ❡st❡ ♠❡s✲ tr❛❞♦ ❝♦♠✐❣♦✱ ♠❡ ❛♣♦✐❛♥❞♦✱ ♠❡ ❛❣✉❡♥t❛♥❞♦ ♥❛s ❤♦r❛s ♠❛✐s ❞✐❢í❝❡✐s ❡ ❝❡❧❡❜r❛♥❞♦ ❝♦♠✐❣♦ ❝❛❞❛ ✉♠❛ ❞❛s ❝♦♥q✉✐st❛s q✉❡ ♦❜t✐✈❡✳ ❈❡rt❛♠❡♥t❡ ✈♦❝ê ❢♦✐ ❛ ♣❡ss♦❛ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❡ ❋❧♦r✐❛♥ó♣♦❧✐s ❞✉r❛♥t❡ ❡st❡s ❛♥♦s✦ ❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ♠❡✉s ❛♠✐❣♦s✱ t❛♥t♦ ❞❡ P♦rt♦ ❆❧❡❣r❡ q✉❛♥t♦ ❞❡
❋❧♦r✐❛♥ó♣♦❧✐s✳ ❙ã♦ ♠✉✐t♦s ❡ ❝❛❞❛ ✉♠ ❞❡❧❡s s❛❜❡ q✉❡♠ é ❡ ♦ q✉❛♥t♦ ❢♦✐ ✐♠♣♦rt❛♥t❡ ❞✉r❛♥t❡ ❡st❛ ❥♦r♥❛❞❛✦
❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯❋❙❈✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦s ♣r♦❢❡ss♦r❡s ❘✉② ❊①❡❧ ❡ ❆❧❝✐❞❡s ❇✉ss✳ ❆♠❜♦s sã♦ ❡①❡♠♣❧♦s ❞❡ ♣❡sq✉✐s❛❞♦r❡s ❡ ♠❡ ❛❥✉❞❛r❛♠ ♠✉✐t♦ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦✳ ❊♠ ♣❛rt✐❝✉❧❛r ❛❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❆❧❝✐❞❡s✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ q✉❡ ❝♦♠ ♠❛❡str✐❛ ❧✐❞♦✉ ❝♦♠✐❣♦ ♣♦r ❡st❡s ❞♦✐s ❛♥♦s✳ ❋♦✐ ✉♠ ♣r❛③❡r ❞✐s❝✉t✐r t❛♥t❛ ▼❛t❡♠át✐❝❛ ❝♦♠ ✈♦❝ê✦
❋✐♥❛❧♠❡♥t❡✱ ❛❣r❛❞❡ç♦ à ❈❆P❊❙ ♣❡❧♦ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❡ ❛ t♦❞♦s ♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈✳
❘❡s✉♠♦
❉❛❞♦ ✉♠ ❣r✉♣ó✐❞❡ G ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦✱ ❤❛✉s❞♦r✛ ❡ ét❛❧❡✱ ❡s✲ t✉❞❛♠♦s r❡♣r❡s❡♥t❛çõ❡s ❞❛ ∗✲á❧❣❡❜r❛ Cc(G)❛ ✜♠ ❞❡ ❝♦♥str✉✐r ❛s C∗✲
á❧❣❡❜r❛s ❝❤❡✐❛ ❡ r❡❞✉③✐❞❛ ❞❡G✳
❈♦♥s❡❣✉✐♠♦s ❝❛r❛❝t❡r✐③❛r ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❛C∗✲á❧❣❡❜r❛ ❝❤❡✐❛✱ ❛ ♣❛r✲
t✐r ❞❡ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞❡ ●✳ ❋✐♥❛❧♠❡♥t❡✱ ❢❛③❡♠♦s ♦ ✉s♦ ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ♣❛r❛ ❞✐s❝✉t✐r ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ ❝❡rt❛s C∗✲á❧❣❡❜r❛s
❜❡♠ ❝♦♥❤❡❝✐❞❛s✳
❆❜str❛❝t
●✐✈❡♥ ❛ ❣r♦✉♣♦✐❞G❧♦❝❛❧❧② ❝♦♠♣❛❝t✱ ❤❛✉s❞♦r✛ ❛♥❞ ét❛❧❡✱ ✇❡ st✉❞② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡∗✲❛❧❣❡❜r❛Cc(G)✐r ♦r❞❡r t♦ ❜✉✐❧❞ t❤❡ ❢✉❧❧ ❛♥❞ t❤❡
r❡❞✉❝❡❞ C∗✲❛❧❣❡❜r❛s✳
❲❡ ❛❝❝♦♠♣❧✐s❤ ❛ ❝❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❢✉❧❧C∗✲❛❧❣❡❜r❛s✬ s✐♠♣❧✐❝✐t②✱
❢r♦♠ ❝❡rt❛✐♥ t♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ G✳ ❋✐♥❛❧❧②✱ ✇❡ ❛♣♣❧② t❤❡ ♠❛✐♥ t❤❡♦r❡♠ ✐♥ ♦r❞❡r t♦ ❞✐s❝✉ss t❤❡ s✐♠♣❧✐❝✐t② ♦❢ s♦♠❡ ✇❡❧❧ ❦♥♦✇♥ C∗✲
❛❧❣❡❜r❛s✳
❈♦♥t❡ú❞♦
■♥tr♦❞✉çã♦ ①✐
✶ Pr❡❧✐♠✐♥❛r❡s ✽
✶✳✶ ●r✉♣ó✐❞❡s ❡ ❈❛t❡❣♦r✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✶ ●r✉♣ó✐❞❡s ❝♦♠♦ ❝❛t❡❣♦r✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷ ●r✉♣ó✐❞❡s ❚♦♣♦❧ó❣✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷✳✶ ❖ ❝❛s♦ ➱t❛❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✸ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✸✳✶ ❊①❡♠♣❧♦s ❝❛♥ô♥✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✸✳✷ ❆çõ❡s ❞❡ ●r✉♣♦s ❡ ♦ ●r✉♣ó✐❞❡ ❞❡ tr❛♥s❢♦r♠❛çã♦ ✸✶ ✶✳✸✳✸ ❯♠ ❡①❡♠♣❧♦ ♠❛✐s ❡❧❛❜♦r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✷ ❆ ✯✲á❧❣❡❜r❛ Cc(G) ✺✻
✸ C∗✲á❧❣❡❜r❛s ❞❡ ✉♠ ❣r✉♣ó✐❞❡ ✻✼
✸✳✶ ❘❡♣r❡s❡♥t❛çõ❡s ❞❡Cc(G) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
✸✳✷ ❘❡♣r❡s❡♥t❛çõ❡s r❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✸✳✸ C∗✲á❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸
✹ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✾✵
✹✳✶ ❯♠ ♣♦✉❝♦ ♠❛✐s s♦❜r❡ ❣r✉♣ó✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ✹✳✷ ❘❡s✉❧t❛❞♦s Pr✐♥❝✐♣❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼ ✹✳✸ ❆♠❡♥❛❜✐❧✐❞❛❞❡ ❞❡ ●r✉♣ó✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✼ ✹✳✹ ❊①❡♠♣❧♦s ❡ ❛♣❧✐❝❛çõ❡s ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✹✳✹✳✶ ●r✉♣ó✐❞❡s ❞✐s❝r❡t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✹✳✹✳✷ ❈✯✲á❧❣❡❜r❛s ❞❡ ❣r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻ ✹✳✹✳✸ ❆çõ❡s ❞❡ ❚r❛♥s❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✽ ✹✳✹✳✹ ❆çõ❡s ❞❡ ❘♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✵ ✹✳✹✳✺ ❙✐♠♣❧✐❝✐❞❛❞❡ ❞❛s ➪❧❣❡❜r❛s ❞❡ ❘♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸ ✹✳✹✳✻ ●r✉♣ó✐❞❡ ❞❡ ❉❡❛❝♦♥✉✲❘❡♥❛✉❧t ❡ ➪❧❣❡❜r❛s ❞❡ ❈✉♥t③✶✸✺
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✶✹✷
■♥tr♦❞✉çã♦
❆ ❡ssê♥❝✐❛ ❞❡st❡ tr❛❜❛❧❤♦ é✱ ❛❧é♠ ❞❡ ❝❛r❛❝t❡r✐③❛r ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❛ C∗✲á❧❣❡❜r❛ ❞❡ ✉♠ ❣r✉♣ó✐❞❡✱ ❡st✉❞❛r ❣r✉♣ó✐❞❡s ❡♠ s✐ ❡ t❡♠❛s r❡❧❛❝✐✲ ♦♥❛❞♦s ❝♦♠ ♦s ♠❡s♠♦s✳ ❯♠ ❣r✉♣ó✐❞❡ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠❛ ❝❛t❡❣♦r✐❛ ♣❡q✉❡♥❛ ♥❛ q✉❛❧ t♦❞♦ ♠♦r✜s♠♦ é ✐s♦♠♦r✜s♠♦ ❡ é ✉♠❛ ❡str✉✲ t✉r❛ q✉❡ ♣❡r♠❡✐❛ ✈ár✐❛s ár❡❛s ❡♠ ▼❛t❡♠át✐❝❛✳ P❛r❛ r❡❢❡rê♥❝✐❛s s♦❜r❡ ♦ t❡♠❛✱ r❡❝♦♠❡♥❞❛♠♦s ❬✼❪✱ ❞❡ ♦♥❞❡ ♥♦s ❜❛s❡❛♠♦s ♣❛r❛ ❡s❝r❡✈❡r ♦ ♣r✐✲ ♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❈✐t❛♠♦s ❛✐♥❞❛ ♦ ❧✐✈r♦ ❞❡ ❏❡❛♥ ❘❡♥❛✉❧t ❬✶✸❪ q✉❡ é ✉♠❛ r❡❢❡rê♥❝✐❛ ❝❧áss✐❝❛ ♣❛r❛ ♦ ❛ss✉♥t♦✳
❈♦♠♦ é ❝♦♠✉♠ ❡♠ ➪❧❣❡❜r❛ ❞❡ ❖♣❡r❛❞♦r❡s✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❡str✉✲ t✉r❛ ❜❛s❡ t❡♥t❛✲s❡ ❝♦♥str✉✐r ❞❡ ♠❛♥❡✐r❛ ✐♥t❡❧✐❣❡♥t❡ C∗✲á❧❣❡❜r❛s✳ ◆❡st❡
❝❛s♦✱ ♦ tr❛❜❛❧❤♦ ❡♠ q✉❡stã♦ ❡st❛rá ✐♥t❡r❡ss❛❞♦ ❡♠✱ ❛ ♣❛rt✐r ❞❡ ✉♠ ❣r✉✲ ♣ó✐❞❡ t♦♣♦❧ó❣✐❝♦ ❝♦♠ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s ♣❛rt✐❝✉❧❛r❡s✱ ❝♦♥str✉✐r C∗✲
á❧❣❡❜r❛s ❞❡st❡s ❣r✉♣ó✐❞❡s✳ ❊st❛ t❡♦r✐❛ é ♠✉✐t♦ ❡st✉❞❛❞❛ ❡ ❥á s❡ ♠♦str♦✉ ♠✉✐t♦ út✐❧✱ ✉♠❛ ✈❡③ q✉❡ ✈ár✐❛s C∗✲á❧❣❡❜r❛s ❝♦♥❤❡❝✐❞❛s ❛❝❛❜❛♠ s❡♥❞♦
✐s♦♠♦r❢❛s à C∗✲á❧❣❡❜r❛ ❞❡ ✉♠ ❝❡rt♦ ❣r✉♣ó✐❞❡✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡
H ✉♠ ❣r✉♣♦ ❞✐s❝r❡t♦ ❛❣✐♥❞♦ ❡♠ ✉♠ ❡s♣❛ç♦ X ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s❞♦r✛✳ ❖ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ❞❡st❛ ❛çã♦ é ✐s♦♠♦r❢♦ à C∗✲á❧❣❡❜r❛ ❞♦
❣r✉♣ó✐❞❡ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❡ t❛❧ ❛çã♦✳ ❚❛❧ r❡s✉❧t❛❞♦ é ❜❡♠ ❝♦♥❤❡❝✐❞♦ ♥❛ ár❡❛ ❞❡ ➪❧❣❡❜r❛s ❞❡ ❖♣❡r❛❞♦r❡s ❡ ♣♦❞❡ s❡r ❞❡❞✉③✐❞♦ ❞❡ r❡s✉❧t❛❞♦s ♠❛✐s ❣❡r❛✐s✱ t❛✐s ❝♦♠♦ ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❡♠ ❬✶❪✳ ❖✉tr♦s ❡①❡♠♣❧♦s ❝❧áss✐❝♦s ❞❡ C∗✲á❧❣❡❜r❛s ✐s♦♠♦r❢❛s à C∗✲á❧❣❡❜r❛s ❞❡ ❣r✉♣ó✐❞❡s sã♦ ❛s
➪❧❣❡❜r❛s ❞❡ ❘♦t❛çã♦ ❡ ➪❧❣❡❜r❛s ❞❡ ❈✉♥t③✱ q✉❡ s❡rã♦ ❞✐s❝✉t✐❞❛s ♥♦ ✜♥❛❧ ❞❡st❡ tr❛❜❛❧❤♦✳
❆ ❝❧❛ss❡ ❞❡ ❣r✉♣ó✐❞❡s ❡♠ q✉❡ ❡st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ♥❡st❡ tr❛❜❛✲ ❧❤♦ é ❛ ❞♦s ❣r✉♣ó✐❞❡s ét❛❧❡✱ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s❞♦r✛✳ ■r❡♠♦s tr❛❜❛❧❤❛r ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❝♦♠ ❣r✉♣ó✐❞❡s ❡❢❡t✐✈♦s ❡ ♠✐♥✐♠❛✐s✳ ●r✉♣ó✐✲ ❞❡s ♣r✐♥❝✐♣❛✐s ❡ t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ♣r✐♥❝✐♣❛✐s ❢♦r♠❛♠ ✉♠❛ ❝❧❛ss❡ ❡s♣❡❝✐❛❧ ❞❡ ❣r✉♣ó✐❞❡s ❡❢❡t✐✈♦s ❡ t❛♠❜é♠ s❡rã♦ ❛❜♦r❞❛❞♦s ❡ ❡st✉❞❛❞♦s✳ ❆❧é♠ ❞❡st❡s✱ tr❛t❛r❡♠♦s r❛♣✐❞❛♠❡♥t❡ ❞❡ ❣r✉♣ó✐❞❡s ❛♠❡♥❛❜❧❡✱ ✉♠❛ ❝❧❛ss❡ ❞❡ ❣r✉♣ó✐❞❡s ♠✉✐t♦ ✐♥t❡r❡ss❛♥t❡ ❡ q✉❡ ❡stã♦ ❜❛st❛♥t❡ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞❡st❛ ❞✐ss❡rt❛çã♦✳
❆ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ ❞❡st❡ tr❛❜❛❧❤♦ é ❛ ❬✶✵❪ ❡ ♦ ❚❡♦r❡♠❛ ✺✳✶ ❞❡st❛ r❡❢❡rê♥❝✐❛✱ ❝✐t❛❞♦ ❛❜❛✐①♦✱ é ♦ t❡♦r❡♠❛ ❜❛s❡ ❞❛ ❞✐ss❡rt❛çã♦✳
❚❡♦r❡♠❛✳ ✭❚❡♦r❡♠❛ ✺✳✶ ❞❡ ❬✶✵❪✮ ❙❡❥❛ G✉♠ ❣r✉♣ó✐❞❡ ét❛❧❡✱ ❧♦✲ ❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s❞♦r✛ ❡ q✉❡ s❛t✐s❢❛③ ♦ s❡❣✉♥❞♦ ❛①✐♦♠❛ ❞❡ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡✳ ❊♥tã♦ C∗(G)é s✐♠♣❧❡s s❡ ❡ s♦♠❡♥t❡ s❡ ❛s ❝♦♥❞✐çõ❡s
❛ s❡❣✉✐r ❢♦r❡♠ s❛t✐s❢❡✐t❛s✿
1)C∗(G)≃C∗
r(G)✱
2)Gt♦♣♦❧♦❣✐❝❛♠❡♥t❡ ♣r✐♥❝✐♣❛❧✱
3)G♠✐♥✐♠❛❧✳
❆✐♥❞❛ ♥❛ r❡❢❡rê♥❝✐❛ ❬✶✵❪✱ ♦ ▲❡♠❛ ✸✳✶ ❞❛ ♠❡s♠❛ ♥♦s ❛✜r♠❛ q✉❡ t♦✲ ♣♦❧♦❣✐❝❛♠❡♥t❡ ♣r✐♥❝✐♣❛❧ ✐♠♣❧✐❝❛ ❡♠ ❡❢❡t✐✈♦ ❡ q✉❡ ✈❛❧❡ ♦ ❝♦♥trár✐♦ ❝❛s♦ ♦ ❣r✉♣ó✐❞❡ s❛t✐s❢❛ç❛ ♦ s❡❣✉♥❞♦ ❛①✐♦♠❛ ❞❡ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡✳ ❆❞❡♠❛✐s✱ ❞✉r❛♥t❡ ❛ ❢❛s❡ ❞❡ ♣r❡♣❛r❛çã♦✱ ❝♦♥s❡❣✉✐♠♦s ❝♦♥❝❧✉✐r q✉❡ ♣♦❞❡rí❛♠♦s ♦♠✐t✐r ❛ ❤✐♣ót❡s❡ ❞❡ G s❛t✐s❢❛③❡r ♦ s❡❣✉♥❞♦ ❛①✐♦♠❛ ❞❡ ❡♥✉♠❡r❛❜✐❧✐✲ ❞❛❞❡ ❝❛s♦ tr♦❝áss❡♠♦s t♦♣♦❧♦❣✐❝❛♠❡♥t❡ ♣r✐♥❝✐♣❛❧ ♣♦r ❡❢❡t✐✈♦ ❡ ❛ss✐♠ ♦ ✜③❡♠♦s✱ ♦✉ s❡❥❛✱ ♠♦str❛♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛✳ ✭❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧✮ ❙❡❥❛ G✉♠ ❣r✉♣ó✐❞❡ ét❛❧❡✱ ❧♦❝❛❧✲ ♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s❞♦r✛✳ ❊♥tã♦ C∗(G)é s✐♠♣❧❡s s❡ ❡ s♦♠❡♥t❡ s❡
❛s ❝♦♥❞✐çõ❡s ❛ s❡❣✉✐r ❢♦r❡♠ s❛t✐s❢❡✐t❛s✿
1)C∗(G)≃C∗
r(G)✱
2)G❡❢❡t✐✈♦✱
3)G♠✐♥✐♠❛❧✳
❆❧é♠ ❞✐ss♦✱ ♥❛ ❙❡çã♦ ✹✳✸✱ s❡rá ❡①♣♦st♦ ❛♦ ❧❡✐t♦r ✉♠ r♦t❡✐r♦ ✐♥❞✐✲ ❝❛♥❞♦ q✉❡ ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❡♠ t❡r♠♦s ❞❡ ❣r✉♣ó✐❞❡s ❛♠❡♥❛❜❧❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❚❡♦r❡♠❛✳ ✭❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ❡♠ t❡r♠♦s ❞❡ ❆♠❡♥❛❜✐❧✐❞❛❞❡✮ ❙❡❥❛ G✉♠ ❣r✉♣ó✐❞❡ ét❛❧❡✱ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s❞♦r✛✳ ❊♥tã♦ C∗(G)
é s✐♠♣❧❡s s❡ ❡ s♦♠❡♥t❡ s❡ G❢♦r ❛♠❡♥❛❜❧❡✱ ♠✐♥✐♠❛❧ ❡ ❡❢❡t✐✈♦✳
❯♠❛ ✈❡③ q✉❡ ♦ tr❛❜❛❧❤♦ é ❜❛s❡❛❞♦ ❡♠ ✉♠ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❡ q✉❡ t❛❧ ❞❡♠♦♥str❛çã♦ é ❜❛st❛♥t❡ té❝♥✐❝❛✱ ❛ s❡❣✉✐r é ❡①♣♦st♦ ❛♦ ❧❡✐t♦r ✉♠❛
❜r❡✈❡ ❡str❛té❣✐❛ ❞❛ ❞❡♠♦♥str❛çã♦ ❡ ❞❛s ❝♦♥str✉çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳ ❈♦♥s✐❞❡r❡G✉♠ ❣r✉♣ó✐❞❡ ét❛❧❡✱ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s❞♦r✛✳ ❉❡✜♥✐♥❞♦ Cc(G)♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s f :G→
C❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱ ♦ ❚❡♦r❡♠❛ ✶ ❞♦ tr❛❜❛❧❤♦ ♠♦str❛ q✉❡ Cc(G)
♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ ∗✲á❧❣❡❜r❛✳ ❆ ✜♠ ❞❡ ❝♦♥str✉✐r ❛s C∗✲á❧❣❡❜r❛s
❝❤❡✐❛ ❡ r❡❞✉③✐❞❛ ❞❡ G✱ ❡st✉❞❛♠♦s r❡♣r❡s❡♥t❛çõ❡s ❞❡ Cc(G), ♦✉ s❡❥❛✱
∗✲❤♦♠♦♠♦r✜s♠♦s C
c(G) → B(H), ♦♥❞❡ H é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳
❊♠ ♣❛rt✐❝✉❧❛r✱ ♥❛ ❙❡çã♦ ✸✳✷ ❝♦♥str✉í♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ r❡❣✉❧❛r ❞❡ Cc(G), ❞❡♥♦t❛❞❛ ♣♦r πλ, ❡ ♠♦str❛♠♦s q✉❡ ❡st❛ r❡♣r❡s❡♥t❛çã♦ é ✜❡❧✳
❆ss✐♠✱ ❡st❛♠♦s ❡♠ ❝♦♥❞✐çõ❡s ❞❡ ❞❡✜♥✐r ❛sC∗✲á❧❣❡❜r❛s ❝❤❡✐❛ ❡ r❡❞✉③✐❞❛✳
❆♠❜❛s sã♦ ❝♦♠♣❧❡t❛♠❡♥t♦s ❞❡ Cc(G), ❝♦♠ r❡s♣❡✐t♦ às s❡❣✉✐♥t❡s C∗✲
♥♦r♠❛s✿
P❛r❛f ∈Cc(G),
kfku:= sup
πr❡♣kπ(f)k,
kfkr:=kπλ(f)k,
♦♥❞❡k.ku é ❛ ♥♦r♠❛ ❞❛C∗✲❝❤❡✐❛ ❡k.kr é ❞❛ r❡❞✉③✐❞❛✳
❚❡♥❞♦ ❝♦♥str✉í❞♦ ❛sC∗✲á❧❣❡❜r❛s✱ ❡st❛♠♦s ❡♠ ❝♦♥❞✐çõ❡s ❞❡ ❞✐s❝✉t✐r
♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧✳ ❆✐♥❞❛ ♥♦ ❝♦♥t❡①t♦ ❞❡ r❡♣r❡s❡♥t❛çõ❡s✱ ❛ Pr♦♣♦s✐çã♦ ✷✻ ❝♦♥stró✐✱ ♣❛r❛ ❝❛❞❛ u∈ G(0), ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ C∗(G), ❞❡♥♦✲
t❛❞❛ ♣♦rπ[u].❯s❛♠♦s ❡st❛s r❡♣r❡s❡♥t❛çõ❡s ♣❛r❛ ❡st✉❞❛r ❛ r❡♣r❡s❡♥t❛çã♦
s♦♠❛ ❞✐r❡t❛ ❞❡ π[u] ✭♣❛r❛ u∈G(0)✮ ❡ ❝♦♥❝❧✉✐r q✉❡ t❛❧ r❡♣r❡s❡♥t❛çã♦ é
✐♥❥❡t✐✈❛ q✉❛♥❞♦ r❡str✐t❛ àC0(G(0)).❈♦♠ ✐st♦✱ ❝♦♥s❡❣✉✐♠♦s ❞❡♠♦♥str❛r
❛ Pr♦♣♦s✐çã♦ ✷✼✱ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♦ tr❛❜❛❧❤♦✳ ❚❛❧ ♣r♦♣♦s✐çã♦ ❡st✉❞❛ ❣r✉♣ó✐❞❡s ❡❢❡t✐✈♦s ❡ ♦s r❡❧❛❝✐♦♥❛ ❝♦♠ ✐❞❡❛✐s ❞❛s C∗✬s ❝❤❡✐❛ ❡ r❡❞✉③✐❞❛✳
❆ s❡❣✉✐r✱ ❛♣ós ✉♠❛ sér✐❡ ❞❡ r❡s✉❧t❛❞♦s ❛✉①✐❧✐❛r❡s✱ ❝♦♥s❡❣✉✐♠♦s ❝♦♥✲ ❝❧✉✐r ❛ Pr♦♣♦s✐çã♦ ✷✽✱ q✉❡✱ ♥♦ ♠❡s♠♦ ❡st✐❧♦ ❞❛ Pr♦♣♦s✐çã♦ ✷✼✱ r❡❧❛❝✐♦♥❛ ❣r✉♣ó✐❞❡s ♠✐♥✐♠❛✐s ❝♦♠ ✐❞❡❛✐s ❞❛s C∗✲á❧❣❡❜r❛s✳
❯♠❛ ✈❡③ ❞❡♠♦♥str❛❞❛s ❛s ♣r♦♣♦s✐çõ❡s ✷✼ ❡ ✷✽✱ ♦ t❡♦r❡♠❛ ❢✉♥❞❛✲ ♠❡♥t❛❧✱ ❚❡♦r❡♠❛ ✹ ♥♦ t❡①t♦✱ s❡❣✉❡ ❞❡ ♠❛♥❡✐r❛ ♠✉✐t♦ s✐♠♣❧❡s✳
❖ tr❛❜❛❧❤♦ ❝♦♥té♠ ✹ ❝❛♣ít✉❧♦s✱ ♦r❣❛♥✐③❛❞♦s ❝♦♠♦ s❡❣✉❡✿
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❛❜♦r❞❛♠♦s ❛ t❡♦r✐❛ ❡❧❡♠❡♥t❛r ❞❡ ❣r✉♣ó✐❞❡s✳ ❈♦♠❡ç❛♠♦s ❡st✉❞❛♥❞♦ ❣r✉♣ó✐❞❡s ♥✉♠ ❝♦♥t❡①t♦ ❛❧❣é❜r✐❝♦✱ ♣❛r❛ ♣♦st❡✲ r✐♦r♠❡♥t❡ ❞❡✜♥✐r ❡ ❡st✉❞❛r ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❣r✉♣ó✐❞❡s t♦♣ó❧♦❣✐❝♦s✳ ❉❛✲ ♠♦s ✉♠ ❢♦❝♦ ❡s♣❡❝✐❛❧ à ❝❧❛ss❡ ❞♦s ❣r✉♣ó✐❞❡s ét❛❧❡✱ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s❞♦r✛✱ q✉❡ s❡rã♦ ❛❜♦r❞❛❞♦s ♣♦r t♦❞♦ ♦ t❡①t♦✳ ➱ ♥❡st❡ ❝❛♣ít✉❧♦ q✉❡ ♣r♦✈❛♠♦s ✐♥ú♠❡r❛s ♣r♦♣r✐❡❞❛❞❡s ❛ r❡s♣❡✐t♦ ❞❡ t❛✐s ❣r✉♣ó✐❞❡s ❡ q✉❡ s❡rã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❛s ❝♦♥str✉çõ❡s ❢❡✐t❛s ♥♦s ❝❛♣ít✉❧♦s ✷ ❡ ✸✳ ❚❡r✲ ♠✐♥❛♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ❡①❡♠♣❧♦s ✈❛r✐❛❞♦s✱ ❞❡s❞❡ ♦s ♠❛✐s ❜ás✐❝♦s ❛té ❡①❡♠♣❧♦s ♠❛✐s ❝♦♠♣❧❡①♦s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡st✉❞❛♠♦s ❛çõ❡s ❞❡ ❣r✉♣♦s✱ ❣r✉♣ó✐❞❡s ❞❡ tr❛♥s❢♦r♠❛çã♦ ❡ ♠♦str❛♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❝❡rt❛s ❛çõ❡s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❣r✉♣ó✐❞❡s ❞❡ tr❛♥s❢♦r♠❛çã♦✳ ❊st❡ ❝❛♣ít✉❧♦ ❢♦✐ ❜❛s❡❛❞♦ ♥❛s r❡❢❡rê♥❝✐❛s ❬✾❪ ❡ ❬✼❪ ♣❛r❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❣r✉♣ó✐❞❡s ❡ ♦s ❡①❡♠♣❧♦s ♠❛✐s ❜ás✐❝♦s✳ ❖ ❡①❡♠♣❧♦ ♠❛✐s ❡❧❛❜♦r❛❞♦ ❞♦
❝❛♣ít✉❧♦ ❡ ♣r♦✈❛✈❡❧♠❡♥t❡ ❞♦ t❡①t♦✱ ❢♦✐ r❡t✐r❛❞♦ ❞❡ ❬✶✽❪✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❥á ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❜❛s✐❝❛♠❡♥t❡ ❡♠ ❣r✉✲ ♣ó✐❞❡s t♦♣♦❧ó❣✐❝♦s✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ❢❛③❡r ❛ ❝♦♥str✉çã♦ ❞❛ ∗✲
á❧❣❡❜r❛Cc(G)♣❛r❛G✉♠ ❣r✉♣ó✐❞❡ ét❛❧❡✱ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s✲
❞♦r✛✳ ❯♠ ❝❛♣ít✉❧♦ ♣❡q✉❡♥♦ ♠❛s té❝♥✐❝♦✱ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❞❡✜♥✐r♠♦s ❛s C∗✲á❧❣❡❜r❛s ❞❡ ❣r✉♣ó✐❞❡s✱ ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✳ ❆s r❡❢❡rê♥❝✐❛s ❬✾❪ ❡
❬✶✼❪ ❢♦r❛♠ ❛s ♠❛✐s ✉s❛❞❛s ♥❡st❛ ♣❛rt❡ ❞♦ t❡①t♦✳
❖ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❝♦♥s❛❣r❛ ❛ ❝♦♥str✉çã♦ ❞❛s C∗✲á❧❣❡❜r❛s ❝❤❡✐❛ ❡
r❡❞✉③✐❞❛ ❞❡ ✉♠ ❣r✉♣ó✐❞❡ ét❛❧❡✱ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s❞♦r✛✳ ❆♠✲ ❜❛sC∗✲á❧❣❡❜r❛s sã♦ ❝♦♠♣❧❡t❛♠❡♥t♦s ❞❛∗✲á❧❣❡❜r❛C
c(G)❝♦♥str✉í❞❛ ♥♦
❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✳ P❛r❛ ♣♦❞❡r♠♦s ❞❡✜♥✐r ❛s ♥♦r♠❛s q✉❡ s❡rã♦ ✉s❛❞❛s ♥♦s ❝♦♠♣❧❡t❛♠❡♥t♦s✱ ❡st✉❞❛♠♦s r❡♣r❡s❡♥t❛çõ❡s ❞❡ Cc(G)✳ ❊♠ ♣❛rt✐✲
❝✉❧❛r✱ ❡st✉❞❛♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ r❡❣✉❧❛r ❞❡ Cc(G)✱ ❜❛s❡ ♣❛r❛ ❛ ❝♦♥s✲
tr✉çã♦ ❞❛ C∗✲r❡❞✉③✐❞❛✱ ❡ t❛♠❜é♠ ♠♦str❛♠♦s q✉❡ ❡st❛ r❡♣r❡s❡♥t❛çã♦
é ✐♥❥❡t✐✈❛✱ ❢❛t♦ ❡①tr❡♠❛♠❡♥t❡ út✐❧ ♣❛r❛ ❥✉st✐✜❝❛r♠♦s q✉❡ ❛ ❛té ❡♥tã♦ C∗✲s❡♠✐♥♦r♠❛ q✉❡ ❞á ♦r✐❣❡♠ à C∗✲❝❤❡✐❛ é✱ ❞❡ ❢❛t♦✱ ✉♠❛ C∗✲♥♦r♠❛✳
❈♦♥❝❧✉í♠♦s ♦ ❝❛♣ít✉❧♦ ♠♦str❛♥❞♦ q✉❡ C0(G(0)) ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦
s✉❜✲C∗✲á❧❣❡❜r❛ ❞❡C∗(G)✱ ❢❛t♦ ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ ♣❛r❛ ✉♠❛ ❝♦♥str✉çã♦ ♥♦
❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✳ ◆♦✈❛♠❡♥t❡✱ ❛s r❡❢❡rê♥❝✐❛s ❬✾❪ ❡ ❬✶✼❪ ❢♦r❛♠ ❛s ♠❛✐s ✉s❛❞❛s ♥❡st❡ ❝❛♣ít✉❧♦✳
❖ q✉❛rt♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦ tr❛t❛ ❞♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❞❡st❡ tr❛❜❛❧❤♦✳ ❈♦♠❡ç❛♠♦s ♦ ❝❛♣ít✉❧♦ ❡st✉❞❛♥❞♦ ✉♠ ♣♦✉❝♦ ♠❛✐s ❛ ❢✉♥❞♦ ❣r✉♣ó✐❞❡s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❣r✉♣ó✐❞❡s ♠✐♥✐♠❛✐s ❡ ❡❢❡t✐✈♦s sã♦ ❡st✉❞❛❞♦s
❝♦♠ ♠❛✐s ❞❡t❛❧❤❡s ❡ sã♦ ❛♣r❡s❡♥t❛❞❛s ♣r♦♣♦s✐çõ❡s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡ t❛✐s ❣r✉♣ó✐❞❡s✳ P♦st❡r✐♦r♠❡♥t❡ ❛♣r❡s❡♥t❛♠♦s ✈ár✐❛s ❝♦♥str✉çõ❡s ✐♠♣♦rt❛♥t❡s ♣♦r s✐ só✱ q✉❡ ♥♦s ❧❡✈❛♠ ❛ ❝♦♥❝❧✉✐r ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❡ ♠❛♥❡✐r❛ ❜❡♠ s✐♠♣❧❡s✱ ❛♣❡♥❛s ❝♦♠❜✐♥❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❛♥✲ t❡r✐♦r❡s✳ ❊♥tã♦ ❛❜♦r❞❛♠♦s ❞❡ ♠❛♥❡✐r❛ ❡①♣♦s✐t✐✈❛ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❛♠❡♥❛✲ ❜✐❧✐❞❛❞❡ ♣❛r❛ ❣r✉♣ó✐❞❡s✳ ❈✐t❛♠♦s ✈ár✐♦s ❢❛t♦s ❝♦♥❤❡❝✐❞♦s ❡ r❡❢❡rê♥❝✐❛s ♣❛r❛ ♦ t❡♠❛✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❞✐❛♥t❡ ❞♦s ❢❛t♦s ❝✐t❛❞♦s✱ ❝♦♥s❡❣✉✐r❡♠♦s r❡❡s❝r❡✈❡r ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❡♠ t❡r♠♦s ❞❡ ❛♠❡♥❛❜✐❧✐❞❛❞❡✳ ❋✐♥❛❧✲ ♠❡♥t❡ ❝♦♥❝❧✉í♠♦s ♦ tr❛❜❛❧❤♦ ❝♦♠ ✉♠❛ s❡çã♦ ❞❡ ❡①❡♠♣❧♦s ❡ ❛♣❧✐❝❛çõ❡s ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧✳ ◆❡st❛ s❡çã♦✱ ♦ ♦❜❥❡t✐✈♦ é ♣r♦✈❛r ❛ s✐♠♣❧✐❝✐❞❛❞❡ ♦✉ ❛ ♥ã♦ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ ❝❡rt❛s C∗✲á❧❣❡❜r❛s ❢❛③❡♥❞♦ ✉s♦ ❞♦ t❡♦r❡♠❛
♣r✐♥❝✐♣❛❧✳ ❊st✉❞❛♠♦s ❣r✉♣ó✐❞❡s ❞✐s❝r❡t♦s ❡ r❡t♦♠❛♠♦s ♦s ❛ss✉♥t♦s ❞❡ ❛çõ❡s ❞❡ ❣r✉♣♦s ❡ ❣r✉♣ó✐❞❡s ❞❡ tr❛♥s❢♦r♠❛çã♦✳ ❚❛♠❜é♠ tr❛t❛♠♦s ❞❡ C∗✲á❧❣❡❜r❛s ❞❡ ❣r✉♣♦s✱ ❛❧é♠ ❞❛s ➪❧❣❡❜r❛s ❞❡ ❘♦t❛çã♦ ❡ ❞❡ ❈✉♥t③✳ ❆
r❡❢❡rê♥❝✐❛ ♠❛✐s ✉s❛❞❛ ♥❡st❡ ❝❛♣ít✉❧♦ ❢♦✐ ❛ ❬✶✵❪✱ ❛rt✐❣♦ ❜❛s❡ ♣❛r❛ ❡st❛ ❞✐ss❡rt❛çã♦✳ ▼✉✐t❛s ♦✉tr❛s r❡❢❡rê♥❝✐❛s ❢♦r❛♠ ✉s❛❞❛s ♣❛r❛ ❝♦♥str✉✐r ❛ ❙❡çã♦ ✹✳✸ ❡ ❡st❛s sã♦ ❝✐t❛❞❛s ♥♦ ❞❡❝♦rr❡r ❞❛ s❡çã♦✳
❖s ♣ré✲r❡q✉✐s✐t♦s ♣❛r❛ ❡st❛ ❞✐ss❡rt❛çã♦ sã♦ ♣♦✉❝♦s✳ ❈♦♥s✐❞❡r❛♠♦s q✉❡ ♦ ❧❡✐t♦r ❝♦♥❤❡ç❛ ✉♠ ❜ás✐❝♦ ❞❡ ❚♦♣♦❧♦❣✐❛ ❡ ❞❡ ➪❧❣❡❜r❛ ❞❡ ❖♣❡✲ r❛❞♦r❡s✱ ♥♦ ♥í✈❡❧ ❞❛ r❡❢❡rê♥❝✐❛ ❬✽❪✱ ♣♦r ❡①❡♠♣❧♦✳ ◆❛ ú❧t✐♠❛ s❡çã♦ ❞♦ tr❛❜❛❧❤♦ t❡♠❛s ♠❛✐s ❛✈❛♥ç❛❞♦s sã♦ ❛❜♦r❞❛❞♦s✳ ❘❡❝♦♠❡♥❞❛♠♦s ❬✻❪✱ ❬✷❪ ❡ ❬✶✺❪ ♣❛r❛ t❡♠❛s ❝♦♠♦ C∗✲✉♥✐✈❡rs❛❧✱ ➪❧❣❡❜r❛s ❞❡ ❘♦t❛çã♦ ❡ Pr♦❞✉t♦s
❈r✉③❛❞♦s✳
❈❛♣ít✉❧♦ ✶
Pr❡❧✐♠✐♥❛r❡s
✶✳✶ ●r✉♣ó✐❞❡s ❡ ❈❛t❡❣♦r✐❛s
❊st❡ ❝❛♣ít✉❧♦ tr❛t❛ ❞❡ ❛♣r❡s❡♥t❛r ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❛ r❡s♣❡✐t♦ ❞❡ ❣r✉♣ó✐❞❡s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥❡st❛ s❡çã♦✱ ✐r❡♠♦s tr❛t❛r ❛♣❡♥❛s ♦ ❝❛s♦ ❛❧❣é❜r✐❝♦✱ ❛❧é♠ ❞❡ ❥✉st✐✜❝❛r q✉❡ ♣♦❞❡♠♦s ❡st✉❞❛r ❣r✉♣ó✐❞❡s ❝♦♠ ✉♠ ✈✐és ❝❛t❡❣ór✐❝♦✳
❉❡✜♥✐çã♦ ✶✳ ❙❡❥❛♠ G✉♠ ❝♦♥❥✉♥t♦ ❡ G(2) ⊂G×G✳ ❊♥tã♦ G é ✉♠
❣r✉♣ó✐❞❡ s❡ ❡①✐st✐r❡♠ ❛♣❧✐❝❛çõ❡s (γ, η)→γη ❞❡ G(2) ❡♠G❡ γ→γ−1
❞❡ G ❡♠ G ✭❝❤❛♠❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈❡rs❛✮ t❛✐s q✉❡✿
(a)✭❆ss♦❝✐❛t✐✈✐❞❛❞❡✮ ❙❡ (γ, η) ❡(η, ξ)❡stã♦ ❡♠ G(2)✱ ❡♥tã♦(γη, ξ)
❡ (γ, ηξ)❡stã♦ ❡♠ G(2) ❡ ✈❛❧❡ q✉❡ (γη)ξ=γ(ηξ)✳
(b) ✭Pr♦♣r✐❡❞❛❞❡ ✐♥✈♦❧✉t✐✈❛ ✭♦✉ ✐♥✈♦❧✉çã♦✮✮ P❛r❛ q✉❛❧q✉❡r γ ∈ G✱ ✈❛❧❡ q✉❡ (γ−1)−1=γ✳
(c)✭▲❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✮ P❛r❛ q✉❛❧q✉❡r γ∈G✱ t❡♠✲s❡ q✉❡
(γ−1, γ)∈G(2) ❡ s❡ (γ, η)∈G(2)✱ ❡♥tã♦γ−1(γη) =η ❡ (γη)η−1=γ✳
❖ ❝♦♥❥✉♥t♦G(2)é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥❥✉♥t♦ ❞♦s ♣❛r❡s ❝♦♠♣♦♥í✈❡✐s✳ ❯♠
❡❧❡♠❡♥t♦ (γ, η)∈G(2) é ❞✐t♦ ✉♠ ♣❛r ❝♦♠♣♦♥í✈❡❧ ❡γ−1é ❞✐t♦ ♦ ✐♥✈❡rs♦
❞❡γ✳
❉❡✜♥✐çã♦ ✷✳ ❙❡❥❛ G ✉♠ ❣r✉♣ó✐❞❡✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s γ ∈G t❛✐s q✉❡ γ = γ−1 =γ2 é ❞❡♥♦t❛❞♦ ♣♦r G(0) ❡ ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❞❡
✉♥✐❞❛❞❡s ❞❡ G✳ ❆ ❢✉♥çã♦ r : G → G(0) ❞❡✜♥✐❞❛ ♣♦r r(γ) = γγ−1 é
❝❤❛♠❛❞❛ ❞❡ r❛♥❣❡ ❡ ❛ ❢✉♥çã♦ s:G→G(0) ❞❡✜♥✐❞❛ ♣♦r s(γ) =γ−1γ é
❝❤❛♠❛❞❛ ❞❡ s♦✉r❝❡✳
P❛r❛ q✉❛❧q✉❡r γ∈G(0)✱ ❞❡✜♥✐♠♦s G
γ :=s−1(γ) ❡Gγ=r−1(γ)✳
❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ♠♦str❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞♦s ❝♦♥✲ ❥✉♥t♦s ❡ ♠❛♣❛s ❞❡✜♥✐❞♦s ❛❝✐♠❛✳
Pr♦♣♦s✐çã♦ ✶✳ ❙❡❥❛ G✉♠ ❣r✉♣ó✐❞❡✳ ❊♥tã♦ ✈❛❧❡ q✉❡✿
(a)❉❛❞♦sγ, η∈G,t❡♠♦s q✉❡(γ, η)∈G(2) s❡✱ ❡ s♦♠❡♥t❡ s❡s(γ) =
r(η)✳
(b)❙❡ (γ, η)∈G(2)✱ ❡♥tã♦r(γη) =r(γ)❡s(γη) =s(η)✳
(c)P❛r❛ q✉❛❧q✉❡rγ∈G,t❡♠✲s❡ q✉❡s(γ) =r(γ−1)❡r(γ) =s(γ−1).
(d)❙❡ (γ, η)∈G(2)✱ ❡♥tã♦ (η−1, γ−1)∈G(2) ❡ t❡♠♦s q✉❡
(γη)−1=η−1γ−1.
(e) P❛r❛ q✉❛❧q✉❡rγ∈G, t❡♠✲s❡ q✉❡ s(γ), r(γ)∈G(0). ❆❞❡♠❛✐s✱r
❡ ssã♦ r❡tr❛çõ❡s ❡♠ G(0).
(f)P❛r❛ q✉❛❧q✉❡rγ∈G,t❡♠♦s q✉❡(r(γ), γ),(γ, s(γ))∈G(2) ❡ ✈❛❧❡
q✉❡r(γ)γ=γ=γs(γ).
❉❡♠♦♥str❛çã♦✿ ✭❛✮ ❙❡❥❛♠ γ, η ∈Gt❛✐s q✉❡ (γ, η)∈G(2). ❆ ❧❡✐ ❞♦
❝❛♥❝❡❧❛♠❡♥t♦ ❣❛r❛♥t❡ q✉❡ (γ−1, γ) ∈ G(2). ❯s❛♥❞♦ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡✱
t❡♠♦s q✉❡(γ−1γ, η)∈G(2),❞❡ ♠♦❞♦ q✉❡
(γ−1γ)η=γ−1(γη) =η,
♦♥❞❡ ❛ ✐❣✉❛❧❞❛❞❡ ❞❛ ❞✐r❡✐t❛ s❡❣✉❡ ❞❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✳
◆♦✈❛♠❡♥t❡ ♣❡❧❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✱ t❡♠♦s q✉❡ (η, η−1) ∈ G(2).
❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
ηη−1= ((γ−1γ)η)η−1=γ−1γ,
♦♥❞❡ ❛ ✐❣✉❛❧❞❛❞❡ ❞❛ ❞✐r❡✐t❛ é ♥♦✈❛♠❡♥t❡ ❥✉st✐✜❝❛❞❛ ♣❡❧❛ ❧❡✐ ❞♦ ❝❛♥❝❡✲ ❧❛♠❡♥t♦✳ P♦rt❛♥t♦✱ ❝♦♥❝❧✉✐♠♦s q✉❡ γ−1γ=ηη−1,♦✉ s❡❥❛✱ s(γ) =r(η).
❆❣♦r❛ s✉♣♦♥❤❛ q✉❡s(γ) =r(η).❚❡♠♦s q✉❡(γ, γ−1)∈G(2), ❞♦♥❞❡ s❡❣✉❡ q✉❡ (γ, γ−1γ) ∈ G(2), ✉s❛♥❞♦ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡✳ P♦r ❤✐♣ót❡s❡✱
γ−1γ =ηη−1, ❞❡ ♠♦❞♦ q✉❡ (γ, ηη−1)∈ G(2). ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲
s❡ q✉❡ (ηη−1, η) ∈ G(2). ❆ss✐♠✱ ✉s❛♥❞♦ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡✱ s❡❣✉❡ q✉❡
(γ,(ηη−1)η)∈G(2).▼❛s ❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦ ❣❛r❛♥t❡ q✉❡(ηη−1)η=
η,❞♦♥❞❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳
✭❝✮ ❙❡❥❛γ∈G.❯s❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦✱ t❡♠♦s r(γ−1) =γ−1(γ−1)−1=γ−1γ=s(γ).
❆ ♦✉tr❛ ✐❣✉❛❧❞❛❞❡ é ❛♥á❧♦❣❛✳
✭❜✮ ❙❡ (γ, η) ∈ G(2), ❡♥tã♦ γη ∈ G, ❞❡ ♠♦❞♦ q✉❡ ((γη)−1, γη) ∈
G(2). ❆❞❡♠❛✐s✱ (γ, η) ∈ G(2) t❛♠❜é♠ ❣❛r❛♥t❡ q✉❡ (γη, η−1) ∈ G(2).
❉❛í✱ ♣❡❧❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡✱ t❡♠♦s q✉❡ ((γη)−1,(γη)η−1)∈G(2),♦✉ s❡❥❛✱
((γη)−1, γ) ∈ G(2), ♣❡❧❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✳ ❆❣♦r❛✱ ✉s❛♥❞♦ ♦ ✐t❡♠
(a), t❡♠♦ss((γη)−1) =r(γ).P❡❧♦ ✐t❡♠ (c),t❡♠♦s s((γη)−1) =r(γη), ❞♦♥❞❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳ ❖ ♦✉tr♦ ❝❛s♦ é ❛♥á❧♦❣♦✳
✭❞✮ ❈♦♠❜✐♥❛♥❞♦ ♦s ✐t❡♥s ❥á ❞❡♠♦♥str❛❞♦s✱ t❡♠♦s q✉❡✱ s❡ (γ, η) ∈
G(2), ❡♥tã♦s(γ) =r(η), ❞❡ ♠♦❞♦ q✉❡ r(γ−1) =s(η−1), ♦ q✉❡ ❣❛r❛♥t❡
q✉❡(η−1, γ−1)∈G(2).
❆❞❡♠❛✐s✱s(η) =s(γη) =r((γη)−1)❣❛r❛♥t❡ q✉❡ (η,(γη)−1)∈G(2),
❞❡ ♠♦❞♦ q✉❡ (η−1η)(γη)−1 = (γη)−1, ♣❡❧❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✳ P♦r
♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✱ ♦❜t❡♠♦s
η−1γ−1 = (η−1γ−1(γη))(γη)−1
= (η−1(γ−1(γη)))(γη)−1= (η−1η)(γη)−1,
❞♦♥❞❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳
s(γ)−1= (γ−1γ)−1=γ−1γ=s(γ).
❆❧é♠ ❞✐ss♦✱ r(s(γ)) = r(s(γ)−1) = s(s(γ)), ♦ q✉❡ ❣❛r❛♥t❡ q✉❡
(s(γ), s(γ))∈G(2).P❡❧❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✱ s❡❣✉❡ q✉❡
s(γ)s(γ) = (γ−1(γγ−1))γ=γ−1γ=s(γ),
♦✉ s❡❥❛✱ s(γ) ∈ G(0). ❯s❛♥❞♦ q✉❡ r(γ) = s(γ−1), s❡❣✉❡ t❛♠❜é♠ q✉❡
r(γ)∈G(0).
❆❣♦r❛✱ ♣❛r❛ q✉❛❧q✉❡ru∈G(0), t❡♠♦s q✉❡
s(u) =u−1u=u2=u
❡
r(u) =uu−1=u2=u,
♦ q✉❡ ♠♦str❛ q✉❡r❡ssã♦ r❡tr❛çõ❡s ❡♠ G(0).
✭❢✮ ❙❡❥❛ γ ∈ G.▼❛✐s ✉♠❛ ✈❡③ ❢❛③❡♥❞♦ ♦ ✉s♦ ❞♦s ✐t❡♥s ❛♥t❡r✐♦r❡s✱ t❡♠♦s q✉❡
s(r(γ)) =s(γγ−1) =s(γ−1) =r(γ),
♦ q✉❡ ♠♦str❛ q✉❡(r(γ), γ)∈G(2).❉❛í✱ ♣❡❧❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✱ s❡❣✉❡ q✉❡
r(γ)γ= (γγ−1)γ=γ.
❖ ♦✉tr♦ ❝❛s♦ é ❛♥á❧♦❣♦✳ ❋✐❝❛✱ ♣♦rt❛♥t♦✱ ❞❡♠♦♥str❛❞❛ ❛ ♣r♦♣♦s✐çã♦✳
✶✳✶✳✶ ●r✉♣ó✐❞❡s ❝♦♠♦ ❝❛t❡❣♦r✐❛s
●r✉♣ó✐❞❡s t❛♠❜é♠ ♣♦❞❡♠ s❡r ❞❡✜♥✐❞♦s ❡♠ t❡r♠♦s ❞❡ ❝❛t❡❣♦r✐❛s✳ ▲❡♠❜r❛♠♦s q✉❡ ✉♠❛ ❝❛t❡❣♦r✐❛ C❝♦♥s✐st❡ ❞❡✿
(i) ✉♠❛ ❝❧❛ss❡ ❞❡ ♦❜❥❡t♦s ❖❜(C);
(ii) ♣❛r❛ t♦❞♦ ♣❛r (u, v) ❞❡ ♦❜❥❡t♦s ❡♠ C, ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠♦r✜s♠♦s ❍♦♠C(u, v)❞❡u♣❛r❛v❀
(iii) ♣❛r❛ q✉❛❧q✉❡r ♦❜❥❡t♦w ❡♠ ❖❜(C), ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ Iw ❡♠ ❍♦♠C(w, w)❝❤❛♠❛❞♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡❀
(iv)♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦s u, v, w❡♠ ❖❜(C),❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❍♦♠C(u, v)×❍♦♠C(v, w)→❍♦♠C(u, w)
(f, g)→g◦f
❝❤❛♠❛❞❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s✱ q✉❡ s❛t✐s❢❛③ ♦ s❡❣✉✐♥t❡s ❛①✐♦♠❛s✿
(a)♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦su❡v✱ ♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡Iu s❛t✐s❢❛③
f◦Iu=f❡Iu◦g=g,♣❛r❛ q✉❛✐sq✉❡rf ∈❍♦♠C(u, v)❡g∈❍♦♠C(v, u);
(b)❛ ❝♦♠♣♦s✐çã♦ é ❛ss♦❝✐❛t✐✈❛✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡rf ∈❍♦♠C(u, v),
g∈❍♦♠C(v, w)❡h∈❍♦♠C(w, z),t❡♠♦s q✉❡
h◦(g◦f) = (h◦g)◦f. ❊s❝r❡✈❡♠♦s u f /
/v ♦✉ f : u→ v, ♣❛r❛ ✐♥❞✐❝❛r q✉❡ t❡♠♦s ✉♠
♠♦r✜s♠♦f ∈❍♦♠C(u, v).❆❞❡♠❛✐s✱ué ❞❡♥♦♠✐♥❛❞♦ ♣♦r ❞♦♠í♥✐♦ ❞❡ f
❡ v ❝♦❞♦♠í♥✐♦ ❞❡ f✳ ❆ ❝♦❧❡çã♦ ❞❡ t♦❞♦s ♦ ♠♦r✜s♠♦s é ❞❡♥♦t❛❞♦ ♣♦r ❍♦♠(C).
❚❛♠❜é♠ ❧❡♠❜r❛♠♦s q✉❡ ✉♠❛ ❝❛t❡❣♦r✐❛ é ❞✐t❛ ♣❡q✉❡♥❛ s❡ ♦ ❖❜(C)
❡ ❍♦♠(C) sã♦ ❝♦♥❥✉♥t♦s✳ ❆❞❡♠❛✐s✱ ✉♠ ♠♦r✜s♠♦ f : u → v é ✉♠ ✐s♦♠♦r✜s♠♦ s❡ ❡①✐st✐r ✉♠ ♠♦r✜s♠♦ g : v → u t❛❧ q✉❡ f ◦g = Iv ❡
g◦f =Iu,t❛❧ ♠♦r✜s♠♦ gé ❞❡♥♦t❛❞♦ ♣♦r f−1.
❈♦♠ ✐ss♦✱ ❡st❛♠♦s ❡♠ ❝♦♥❞✐çõ❡s ❞❡ ❞❡✜♥✐r ✉♠ ❣r✉♣ó✐❞❡ ❡♠ t❡r♠♦s ❞❡ ❝❛t❡❣♦r✐❛s✿
❉❡✜♥✐çã♦ ✸✳ ❯♠ ❣r✉♣ó✐❞❡ é ✉♠❛ ❝❛t❡❣♦r✐❛ ♣❡q✉❡♥❛ ♥❛ q✉❛❧ t♦❞♦s ♦s ♠♦r✜s♠♦s sã♦ ✐s♦♠♦r✜s♠♦s✳
❆ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ❛❧é♠ ❞❡ ❡❧❡❣❛♥t❡✱ é ♠✉✐t♦ út✐❧✱ ✈✐st♦ q✉❡ ♣♦ss✐✲ ❜✐❧✐t❛ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ❣r✉♣ó✐❞❡✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ✉s❛✲ r❡♠♦s ❡ss❡♥❝✐❛❧♠❡♥t❡ ❛ ♣r✐♠❡✐r❛ ❞❡✜♥✐çã♦✱ ♠❛s✱ ❛ ✜♠ ❞❡ ❝♦♠♣❧❡t✉❞❡✱ ✈❛♠♦s ♠♦str❛r q✉❡ t❛✐s ❞❡✜♥✐çõ❡s ❝♦✐♥❝✐❞❡♠ ❡ ❝♦♠♦ ♣♦❞❡♠♦s ❢❛③❡r t❛❧ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛✳
❙❡❥❛G ✉♠❛ ❝❛t❡❣♦r✐❛ ♣❡q✉❡♥❛ ♥❛ q✉❛❧ t♦❞♦s ♦s ♠♦r✜s♠♦s sã♦ ✐s♦✲
♠♦r✜s♠♦s✳ ❉❡✜♥❛G:=❍♦♠(G).➱ ✐♠❡❞✐❛t♦ ✈❡r q✉❡ ❡①✐st❡ ✉♠❛ ❝♦rr❡s✲ ♣❡♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ♦❜❥❡t♦s ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ ♦s ♠♦r✜s♠♦s ✐❞❡♥t✐❞❛❞❡✱ ♦✉ s❡❥❛✱
{u∈❖❜(G)} ↔ {Iu∈❍♦♠(G)|u∈❖❜(G)}
u←→Iu
❞❡ ♠♦❞♦ q✉❡ ♣♦❞❡♠♦s ❞❡✜♥✐r G(0):=❖❜(G)❡✱ ✈✐❛ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛✱
t❡r♠♦s q✉❡G(0)⊂G.
❯♠❛ ✈❡③ q✉❡ t♦❞♦ ♠♦r✜s♠♦ é ✐s♦♠♦r✜s♠♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ✐♥✲ ✈❡rsã♦ ❡♠ G ❞❛ ♠❛♥❡✐r❛ ó❜✈✐❛✳ ❆❞❡♠❛✐s✱ ♣❛r❛ q✉❛❧q✉❡r ♠♦r✜s♠♦ γ:u→v,❞❡✜♥✐♠♦s ❛s ❢✉♥çõ❡s s♦✉r❝❡ ❡ r❛♥❣❡ ♣♦r
s:G→G(0)γ7→u
r:G→G(0)γ7→v,
♦✉ s❡❥❛✱ ❡st❛♠♦s ✐♥t❡r♣r❡t❛♥❞♦ γ ❝♦♠♦ ✉♠❛ ✢❡❝❤❛ ❞❡ u = s(γ) ♣❛r❛
v=r(γ)
u=s(γ)
γ
-r(γ) =v
❆ss✐♠✱ ❞❛❞♦s q✉❛✐sq✉❡r ♠♦r✜s♠♦s γ:v→w❡η:u→v,❛ ❢✉♥çã♦ ❝♦♠♣♦s✐çã♦ ❣❛r❛♥t❡ q✉❡
u
γ◦η
3
3
η
(
(v
γ
(
(w ,
♦♥❞❡v=s(γ) =r(η),♦ q✉❡ ♥♦s ♠♦t✐✈❛ ❛ ❞❡✜♥✐r
G(2) :={(γ, η)∈❍♦♠(G)×❍♦♠(G)|s(γ) =r(η)},
❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ Gé ❞❡✜♥✐❞❛ ♣♦r
G(2)→G(γ, η)7→γ◦η:=γη.
❖❜s❡r✈❛♠♦s q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r ♠♦r✜s♠♦ γ:u→v,t❡♠♦s γγ−1=
Iv ❡γ−1γ =Iu. ❉❛í✱ ✉s❛♥❞♦ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❛❝✐♠❛✱ ♣♦❞❡♠♦s ✐❞❡♥✲
t✐✜❝❛r Iv =v =r(γ) ❡Iu =u =s(γ). ❚❛♠❜é♠ é ❢á❝✐❧ ♣❡r❝❡❜❡r q✉❡✱
♥♦✈❛♠❡♥t❡ ✈✐❛ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛✱ G(0) é ❡①❛t❛♠❡♥t❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s
♠♦r✜s♠♦s γ t❛✐s q✉❡γ =γ−1 =γ2. ❉❡ ❢❛t♦✱ é ❝❧❛r♦ q✉❡ q✉❛❧q✉❡r I
u
s❛t✐s❢❛③Iu=Iu−1=Iu2.P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ ✉♠ ♠♦r✜s♠♦ γ:u→v
t❛❧ q✉❡ γ = γ−1 = γ2, t❡♠♦s u = v, ✈✐st♦ q✉❡ γ = γ−1, ❡ γ = γ2
❣❛r❛♥t❡ q✉❡ γγ−1=I
u=γ2γ−1=γ,❞❡ ♠♦❞♦ q✉❡γ∈G(0).
❋✐♥❛❧♠❡♥t❡✱ ❛ ♣❛rt✐r ❞❛s ❞❡✜♥✐çõ❡s ❞❛ ❝❛t❡❣♦r✐❛G,é ✐♠❡❞✐❛t♦ ✈❡r✐✜✲ ❝❛r ❛s ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s r❡st❛♥t❡s ♣❛r❛ ❣❛r❛♥t✐r q✉❡Gé ✉♠ ❣r✉♣ó✐❞❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ✶✳
❆❣♦r❛✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡✱ ❛ ♣❛rt✐r ❞❡ ✉♠ ❣r✉♣ó✐❞❡ G ✜①❛❞♦✱ ♣♦❞❡♠♦s ❡♥①❡r❣❛r G ❝♦♠♦ ✉♠❛ ❝❛t❡❣♦r✐❛ ♣❡q✉❡♥❛ G ♦♥❞❡ ♦s ♠♦r✜s✲
♠♦s sã♦ ✐s♦♠♦r✜s♠♦s✳ ❉❡ ❢❛t♦✱ ❞❡✜♥❛ ❖❜(G) :=G(0) ❡ ❍♦♠(G) :=G.
❈♦♠❡ç❛♠♦s ♣♦r ♦❜s❡r✈❛r q✉❡✱ ✉♠❛ ✈❡③ q✉❡ Gé ✉♠ ❝♦♥❥✉♥t♦✱ ♥❛t✉r❛❧✲ ♠❡♥t❡ ❛ ❝❛t❡❣♦r✐❛ G s❡rá ♣❡q✉❡♥❛✳
◗✉❛❧q✉❡r γ ∈ G é ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ γ ∈ ❍♦♠G(s(γ), r(γ)), ❞❡
♠♦❞♦ q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ ❣r✉♣ó✐❞❡ ❣❛r❛♥t❡ ❛ ❜♦❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ ❝♦♠♣♦s✐❝ã♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ s❡❥❛♠ γ, η∈Gt❛✐s q✉❡ (γ, η)∈G(2),
♦✉ s❡❥❛✱ s(γ) = r(η). ❙❡❣✉❡ q✉❡ γη := ξ ∈ G s❛t✐s❢❛③ s(ξ) = s(η) ❡
r(ξ) =r(γ).❆ss✐♠✱ ❛ ❢✉♥çã♦ ❝♦♠♣♦s✐çã♦
❍♦♠G(s(η), r(η))×❍♦♠G(s(γ), r(γ))→❍♦♠G(s(ξ), r(ξ))
(η, γ)→γ◦η :=γη=ξ
❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❆❞❡♠❛✐s✱ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞♦ ❣r✉♣ó✐❞❡ ❣❛r❛♥t❡ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ❝♦♠♣♦s✐çã♦ ❞♦s ♠♦r✜s♠♦s✳
❚❛♠❜é♠ é ❝❧❛r♦ q✉❡ ❛ ✐♥✈❡rsã♦ ❞♦ ❣r✉♣ó✐❞❡ ❣❛r❛♥t❡ q✉❡ t♦❞♦γ∈G ✈✐st♦ ❝♦♠♦ ♠♦r✜s♠♦ s❡rá ✉♠ ✐s♦♠♦r✜s♠♦✳ ❋✐♥❛❧♠❡♥t❡✱ ✈❛♠♦s ❝♦♥str✉✐r ♦s ♠♦r✜s♠♦s ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ q✉❛❧q✉❡r u ∈ G(0). ❯♠❛ ✈❡③ q✉❡ u =
s(u) =r(u)❡ q✉❡G(0)⊂G,♣♦❞❡♠♦s ❞❡✜♥✐rI
u s❡♥❞♦ ❡①❛t❛♠❡♥t❡u∈
G. ❉❛í✱ é ❝❧❛r♦ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r γ, η∈Gt❛✐s q✉❡ s(γ) =u=r(η), t❡♠♦s q✉❡γ◦u=γu=γs(γ) =γ❡u◦η=uη=r(η)η=η,❣❛r❛♥t✐♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ♠♦r✜s♠♦s ✐❞❡♥t✐❞❛❞❡✳ ❋✐❝❛ ❛ss✐♠ ❞❡♠♦♥str❛❞❛ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛s ❞❡✜♥✐çõ❡s ❛♣r❡s❡♥t❛❞❛s✳
✶✳✷ ●r✉♣ó✐❞❡s ❚♦♣♦❧ó❣✐❝♦s
❆té ❡♥tã♦✱ tr❛t❛♠♦s ❞❡ ❣r✉♣ó✐❞❡s ❛♣❡♥❛s ♥♦ ❝♦♥t❡①t♦ ❛❧❣é❜r✐❝♦✳ ◆♦ ❡♥t❛♥t♦✱ ♣♦❞❡♠♦s ♠✉♥✐✲❧♦s ❝♦♠ t♦♣♦❧♦❣✐❛s✱ ❡♥r✐q✉❡❝❡♥❞♦ ❜❛st❛♥t❡ s✉❛ ❡str✉t✉r❛✳ ❉❡ ❢❛t♦✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ❡st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❣r✉♣ó✐❞❡s ❝♦♠ t♦♣♦❧♦❣✐❛✱ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❣r✉♣ó✐❞❡s t♦♣♦❧ó❣✐❝♦s✳ ▼❛✐s ♣r❡❝✐s❛✲
♠❡♥t❡✿
❉❡✜♥✐çã♦ ✹✳ ❙❡❥❛ G✉♠ ❣r✉♣ó✐❞❡ ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ t♦♣♦✲ ❧ó❣✐❝♦✳ ❈♦♥s✐❞❡r❡ G(2) ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ❞❛ t♦♣♦❧♦❣✐❛ ♣r♦❞✉t♦ ❡
G(0) ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛✳ ❊♥tã♦ Gs❡rá ❞✐t♦ ✉♠ ❣r✉♣ó✐❞❡ t♦♣♦❧ó✲
❣✐❝♦ s❡ ❛s ❛♣❧✐❝❛çõ❡s ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈❡rs❛ ❢♦r❡♠ ❝♦♥tí♥✉❛s✳
❖❇❙✳✿ ❙❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐ç❛♦ ❛♥t❡r✐♦r q✉❡✱ ♥✉♠ ❣r✉♣ó✐❞❡ t♦♣♦❧ó❣✐❝♦✱ ❛s ❢✉♥çõ❡s r❛♥❣❡ ❡ s♦✉r❝❡ t❛♠❜é♠ s❡rã♦ ❝♦♥tí♥✉❛s✳
❱❛♠♦s ❛❣♦r❛ ❡st✉❞❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❞❡✜♥✐çõ❡s ❜ás✐❝❛s ❞♦s ❣r✉♣ó✐❞❡s t♦♣♦❧ó❣✐❝♦s✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❡ G✉♠ ❣r✉♣ó✐❞❡ t♦♣♦❧ó❣✐❝♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ❍❛✉s❞♦r✛✳
Pr♦♣♦s✐çã♦ ✷✳ ❖s ❝♦♥❥✉♥t♦s G(0) ❡ G(2) sã♦ ❢❡❝❤❛❞♦s ❞❡ G❡G×G✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ✉♠❛ ♥❡tui❡♠G(0)❝♦♥✈❡r❣✐♥❞♦ ♣❛r❛u❡♠
G✳ ❈♦♠♦ ❛s ❢✉♥çõ❡s r❛♥❣❡ ❡ s♦✉r❝❡ sã♦ ❝♦♥tí♥✉❛s✱ t❡♠♦s q✉❡ r(ui)→
r(u)❡s(ui)→s(u). ▼❛sui ∈G(0),❞❡ ♠♦❞♦ q✉❡ r(ui) =s(ui) =ui,
❣❛r❛♥t✐♥❞♦ q✉❡u=s(u) =r(u)∈G(0),♦✉ s❡❥❛✱ G(0) é ❢❡❝❤❛❞♦ ❞❡G✳
❙❡❥❛ ❛❣♦r❛ ✉♠❛ ♥❡t (γi, ηi) ❡♠ G(2) ❝♦♥✈❡r❣✐♥❞♦ ♣❛r❛ (γ, η) ❡♠
G×G. P♦rt❛♥t♦ t❡♠♦s q✉❡ γi → γ ❡ ηi → η, ♦ q✉❡ ❣❛r❛♥t❡ q✉❡
s(γi)→s(γ)❡r(ηi)→r(η).▼❛ss(γi) =r(ηi), ❞❡ ♠♦❞♦ q✉❡
s(γ) =r(η),✐st♦ é✱(γ, η)∈G(2),❝♦♠♦ ❣♦st❛rí❛♠♦s✳
❆ ✜♠ ❞❡ ❥✉st✐✜❝❛r ♠❛✐s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❣r✉✲
♣ó✐❞❡s t♦♣♦❧ó❣✐❝♦s✱ ✈❛♠♦s ✉s❛r ❞♦✐s ❢❛t♦s ❞❡ t♦♣♦❧♦❣✐❛ ❣❡r❛❧✱ ❛ s❛❜❡r✿
(1)❙❡❥❛X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❊♥tã♦ ✉♠ s✉❜❝♦♥❥✉♥t♦Aé ❛❜❡rt♦ s❡✱ ❡ s♦♠❡♥t❡ s❡ ♣❛r❛ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦a∈A❡ ✉♠❛ ♥❡txi❝♦♥✈❡r❣✐♥❞♦
♣❛r❛ a✱ ❡①✐st❡ ✉♠ í♥❞✐❝❡i0 t❛❧ q✉❡ xi∈A ♣❛r❛ t♦❞♦i0i✭✐♥❢♦r♠❛❧✲
♠❡♥t❡✱ ❞✐r❡♠♦s q✉❡✱ ♥❡ss❡ ❝❛s♦✱ xi ∈ A ♣❛r❛ í♥❞✐❝❡s s✉✜❝✐❡♥t❡♠❡♥t❡
❣r❛♥❞❡s ♦✉ q✉❡ xi∈A❡✈❡♥t✉❛❧♠❡♥t❡✮✳
(2) ❙❡❥❛ f : X → Y ✉♠❛ ❛♣❧✐❝❛çã♦ s♦❜r❡❥❡t✐✈❛ ❡ ❝♦♥tí♥✉❛ ❡♥tr❡ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦sX ❡Y✳
❊♥tã♦f é ✉♠❛ ❛♣❧✐❝❛çã♦ ❛❜❡rt❛ s❡✱ ❡ s♦♠❡♥t❡ s❡ ♣❛r❛ q✉❛❧q✉❡r ♥❡t
(yi) ∈ Y ❝♦♥✈❡r❣❡♥t❡✱ ❞✐❣❛♠♦s✱ yi → f(x), ❡①✐st❡ ✉♠❛ s✉❜♥❡t (yj) ❡
❡❧❡♠❡♥t♦s (xj)∈X t❛✐s q✉❡xj→x❡♠X ❡yj =f(xj), ♣❛r❛ t♦❞♦j.
Pr♦♣♦s✐çã♦ ✸✳ ❙❡❥❛ G ✉♠ ❣r✉♣ó✐❞❡ t❛❧ q✉❡ G(0) é ✉♠ ❛❜❡rt♦ ❡♠ G✳
❊♥tã♦ ♣❛r❛ q✉❛❧q✉❡r g∈G(0), t❡♠♦s q✉❡ r−1(g)❡ s−1(g)sã♦ ❡s♣❛ç♦s
❞✐s❝r❡t♦s✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛❧q✉❡r ♥❡t ❝♦♥✈❡r❣❡♥t❡ hi → h ❡♠ r−1(g),
t❡♠♦s q✉❡ hi =h ♣❛r❛ í♥❞✐❝❡s i′s s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s ✭❡ ❛♥á❧♦❣♦
♣❛r❛s−1(g)✮✳
❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s hi → h ❝♦♠ r(hi) = r(h) = s(h−1). ❙❡❣✉❡
q✉❡h−1h
i→h−1h=s(h)∈G(0).
❚❡♠♦s q✉❡G(0)é ❛❜❡rt♦✱ ❞❡ ♠♦❞♦ q✉❡h−1h
i∈G(0) ❡✈❡♥t✉❛❧♠❡♥t❡
✭♣❡❧♦ ❢❛t♦ ✭✶✮ ❝✐t❛❞♦ ❛❝✐♠❛✮✳ P♦rt❛♥t♦ h−1h
i = s(h−1hi) = s(hi) =
h−1i hi ♣❛r❛ t❛✐s í♥❞✐❝❡si′s✳
P❡❧❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✱ s❡❣✉❡ q✉❡✱ ♥♦✈❛♠❡♥t❡ ♣❛r❛ t❛✐s í♥❞✐❝❡s✱
h=hi,❣❛r❛♥t✐♥❞♦ q✉❡r−1(g)é ❞✐s❝r❡t♦✳ ❆ ❞❡♠♦♥str❛çã♦ ♣❛r❛s−1(g)
é ❛♥á❧♦❣❛✳
✶✳✷✳✶ ❖ ❝❛s♦ ➱t❛❧❡
❱❛♠♦s ❛❣♦r❛ ❞❡✜♥✐r ❡ ❡st✉❞❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❣r✉♣ó✐❞❡s ét❛❧❡✳ ❊st❛ é ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝❧❛ss❡ ❞❡ ❣r✉♣ó✐❞❡s ❡ ❣r❛♥❞❡ ♣❛rt❡ ❞♦s t❡♦r❡♠❛s ❡ ♣r♦♣♦s✐çõ❡s ❞❡st❡ tr❛❜❛❧❤♦ s❡rã♦ ❢❡✐t❛s s♦❜r❡ ❡st❛ ❝❧❛ss❡✳
❉❡✜♥✐çã♦ ✺✳ ❯♠ ❣r✉♣ó✐❞❡ t♦♣♦❧ó❣✐❝♦ G é ❞✐t♦ ét❛❧❡ s❡ ❛s ❢✉♥çõ❡s r, s : G → G(0) ❢♦r❡♠ ❤♦♠❡♦♠♦r✜s♠♦s ❧♦❝❛✐s✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛❧q✉❡r
g ∈ G, ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ U ❞❡ G t❛❧ q✉❡ s(U) é ❛❜❡rt♦ ❡
s|U :U →s(U)é ❤♦♠❡♦♠♦r✜s♠♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡❣✉❡ q✉❡ r ❡ ssã♦
❛♣❧✐❝❛çõ❡s ❛❜❡rt❛s✳
Pr♦♣♦s✐çã♦ ✹✳ ❙✉♣♦♥❤❛ q✉❡ G é ✉♠ ❣r✉♣ó✐❞❡ t♦♣♦❧ó❣✐❝♦ t❛❧ q✉❡ ❛s ❢✉♥çõ❡s r❛♥❣❡ ❡ s♦✉r❝❡ sã♦ ❛♣❧✐❝❛çõ❡s ❛❜❡rt❛s✳ ✭❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ G é ét❛❧❡✱ ❥á q✉❡ t♦❞♦ ❤♦♠❡♦♠♦r✜s♠♦ ❧♦❝❛❧ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❛❜❡rt❛✳✮ ❊♥tã♦ ❛ ❛♣❧✐❝❛çã♦ ♠✉❧t✐♣❧✐❝❛çã♦ G(2)→Gt❛♠❜é♠ é ❛❜❡rt❛✳
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ A, B⊂G❛❜❡rt♦s✳ ❱❛♠♦s ♠♦str❛r q✉❡ AB:={αβ|(α, β)∈G(2), α∈A, β∈B}
é ❛❜❡rt♦✳ ❈♦♥s✐❞❡r❡ ✉♠ ❡❧❡♠❡♥t♦ αβ∈AB❡ ✉♠❛ ♥❡t yi→αβ.❇❛st❛
♠♦str❛r q✉❡yi ∈AB❡✈❡♥t✉❛❧♠❡♥t❡✱ ❝♦♠♦ ❥á ❢♦✐ ❞✐s❝✉t✐❞♦✳ ❚❡♠♦s q✉❡
r(yi) → r(αβ) ∈ r(A). ❈♦♠♦ ❛ ❢✉♥çã♦ r❛♥❣❡ é ❛♣❧✐❝❛çã♦ ❛❜❡rt❛✱ s❡✲
❣✉❡ ✭❞❛ ❝❛r❛❝t❡r✐③❛ç❛♦ ❞❡ ❛♣❧✐❝❛çã♦ ❛❜❡rt❛ ❛♥t❡r✐♦r♠❡♥t❡ ❝✐t❛❞❛✮ q✉❡✱ ♣❛ss❛♥❞♦ ❛ ✉♠❛ s✉❜♥❡t s❡ ♥❡❝❡ssár✐♦✱ ❡①✐st❡ ✉♠❛ ♥❡t αi ❡♠ At❛❧ q✉❡
αi → α❡ r(αi) = r(yi). P♦rt❛♥t♦ t❡♠♦s q✉❡ (α−1i , yi) ∈G(2) ❡ ❛ ♥❡t
α−1i yi → β ∈ B. ▼❛s ❝♦♠♦ B é ❛❜❡rt♦✱ s❡❣✉❡ q✉❡ αi−1yi ∈ B ❡✈❡♥✲
t✉❛❧♠❡♥t❡✱ ❞❡ ♠♦❞♦ q✉❡ αiαi−1yi = yi ∈ AB ❡✈❡♥t✉❛❧♠❡♥t❡✱ ❝♦♠♦
❣♦st❛rí❛♠♦s✳
❉❡✜♥✐çã♦ ✻✳ ❙❡❥❛ G✉♠ ❣r✉♣ó✐❞❡ t♦♣♦❧ó❣✐❝♦✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ U ⊂G é ❞✐t♦ ✉♠❛ ❜✐ss❡çã♦ s❡ ❛s r❡str✐çõ❡s r|U, s|U ❞❛s ❢✉♥çõ❡s r❛♥❣❡ ❡ s♦✉r❝❡
❛U ❢♦r❡♠ ❤♦♠❡♦♠♦r✜s♠♦s s♦❜r❡ ❛ ✐♠❛❣❡♠✳
Pr♦♣♦s✐çã♦ ✺✳ ❙❡ G❢♦r ét❛❧❡✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ Bis(G)❞❛s ❜✐ss❡çõ❡s
❛❜❡rt❛s ❢♦r♠❛ ✉♠❛ ❜❛s❡ ♣❛r❛ ❛ t♦♣♦❧♦❣✐❛ ❞❡ G✳
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ A ✉♠ ❛❜❡rt♦ ❞❡ G ❡ x ∈ A. ❈♦♠♦ G é ét❛❧❡✱ ❡①✐st❡♠ ✈✐③✐♥❤❛♥ç❛s ❛❜❡rt❛s U1 ❡ U2 ❞❡ x t❛✐s q✉❡ ❛s ❢✉♥çõ❡s
r❛♥❣❡ ❡ s♦✉r❝❡ sã♦ ❤♦♠❡♦♠♦r✜s♠♦s ❡♠ t❛✐s r❡s♣❡❝t✐✈❛s ✈✐③✐♥❤❛♥ç❛s✳ ❆ss✐♠✱ ❞❡✜♥❛ V := A∩U1∩U2. ❙❡❣✉❡ q✉❡ V é ❛❜❡rt♦✱ x ∈ V ⊂A
❡ q✉❡ ❛s ❢✉♥çõ❡s r❛♥❣❡ ❡ s♦✉r❝❡ sã♦ ❤♦♠❡♦♠♦r✜s♠♦s ❡♠ V, ♦✉ s❡❥❛✱ V ∈Bis(G).■ss♦ ❣❛r❛♥t❡ q✉❡Bis(G)é ❜❛s❡ ♣❛r❛ ❛ t♦♣♦❧♦❣✐❛ ❞❡G✳
❈♦r♦❧ár✐♦ ✶✳ ❙❡ Gé ét❛❧❡✱ ❡♥tã♦ éG(0) é ❛❜❡rt♦ ❡♠ G✳
❉❡♠♦♥str❛çã♦✿ ❇❛st❛ ♦❜s❡r✈❛r q✉❡ ♣❛r❛ q✉❛❧q✉❡r u∈ G(0), ❡①✐st❡
g ∈ Gt❛❧ q✉❡ r(g) =u ❡ q✉❡✱ ♣❛r❛ t❛❧ g, ❡①✐st❡ ✉♠❛ ❜✐ss❡çã♦ ❛❜❡rt❛ U ❣❛r❛♥t✐♥❞♦ q✉❡ r(U)s❡❥❛ ✉♠ ❛❜❡rt♦ ❝♦♥t❡♥❞♦ u✳ ❆ss✐♠✱ s❡❣✉❡ q✉❡
G(0) é ❡s❝r✐t♦ ❝♦♠♦ ✉♥✐ã♦ ❞❡ t❛✐s ❛❜❡rt♦s✳
❙❡❣✉✐♥❞♦ ❝♦♠ ❛ ♥♦t❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ✈❛♠♦s ♣r♦✈❛r ❞♦✐s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ❛ r❡s♣❡✐t♦ ❞❛ ❡str✉t✉r❛ ❞❡ Bis(G), q✉❡ s❡rã♦ ❢✉♥❞❛♠❡♥t❛✐s ❛❞✐❛♥t❡✳
Pr♦♣♦s✐çã♦ ✻✳ ❙❡❥❛ Gét❛❧❡✳ ❚❡♠♦s q✉❡✿
(1)❙❡ U ∈Bis(G),❡♥tã♦ U−1:={γ−1|γ∈U} ∈Bis(G).
(2)❙❡ U, V ∈Bis(G),❡♥tã♦
U V :={γη|(γ, η)∈U×V∩G(2)∈Bis(G)×Bis(G)}=p(U×V∩G(2)),
♦♥❞❡ pé ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ ❣r✉♣ó✐❞❡✳ ❆❞❡♠❛✐s✱ U V ∈Bis(G).
❉❡♠♦♥str❛çã♦✿ P❛r❛ ❞❡♠♦♥str❛r (1), ❜❛st❛ ♦❜s❡r✈❛r q✉❡ r|U−1 : U−1 → r(U−1) = s(U) é ♣r❡❝✐s❛♠❡♥t❡ ❛ ❝♦♠♣♦s✐çã♦ s◦ι, ♦♥❞❡ ι :
U−1 →U é ❛ ✐♥✈❡rsã♦ ❡♠ ●✳ ❈♦♠♦ ❛ ❢✉♥çã♦ s♦✉r❝❡ é ✉♠ ❤♦♠❡♦♠♦r✲
✜s♠♦ ❡♠U ❡ ❛ ❢✉♥çã♦ ✐♥✈❡rsã♦ é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ✭❥á q✉❡ é ❜✐❥❡t✐✈♦✱ ❝♦♥tí♥✉♦ ❡ ❝♦♠ ✐♥✈❡rs❛ ι−1 =ι, ♣♦✐s ♣❛r❛ t♦❞♦ γ ∈ G,(γ−1)−1 =γ✮✱
t❡♠♦s q✉❡ U−1 é ❛❜❡rt♦ ❡r|
U−1 é ❤♦♠❡♦ ♥❛ ✐♠❛❣❡♠✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ s|U−1 s❡rá ❤♦♠❡♦ s♦❜r❡ ❛ ✐♠❛❣❡♠✱ ♦ q✉❡ ❞❡♠♦♥str❛ (1).
❆❣♦r❛ ❝♦♥s✐❞❡r❡ U ❡V ❜✐ss❡çõ❡s ❛❜❡rt❛s✳ ❱❛♠♦s ♠♦str❛r q✉❡ U V é t❛♠❜é♠ ✉♠❛ ❜✐ss❡çã♦ ❛❜❡rt❛✳
❆♥t❡s ❞❡ ♠❛✐s ♥❛❞❛✱ ♦❜s❡r✈❡♠♦s q✉❡✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ s(U) =r(V). ❉❡ ❢❛t♦✱ ❞❡✜♥❛ ♦s ❛❜❡rt♦sE :=
s(U)∩r(V), U′ := s|
U−1(E) ❡ V′ := r|V−1(E). ❈♦♠♦ U ❡ V sã♦
❜✐ss❡çõ❡s ❛❜❡rt❛s✱ s❡❣✉❡ q✉❡ E = s(U′) = r(V′) ❡ q✉❡ U′ ❡ V′ sã♦
t❛♠❜é♠ ❜✐ss❡çõ❡s ❛❜❡rt❛s✳
❆❞❡♠❛✐s✱ t❡♠♦s q✉❡U×V ∩G(2)=U′×V′∩G(2),♣♦✐s s❡
(g, h) ∈ U×V ∩G(2), ❡♥tã♦ s(g) = r(h)∈ E, ❞❡ ♠♦❞♦ q✉❡ ❡①✐st❡♠
g′ ∈ U′, h′ ∈ V′ t❛✐s q✉❡ s(g) = s(g′) ❡ r(h) = r(h′), ♠❛s✱ ❝♦♠♦
❛s ❢✉♥çõ❡s r❛♥❣❡ ❡ s♦✉r❝❡ sã♦ ❤♦♠❡♦♠♦r✜s♠♦s ❡♠ U ❡ V✱ s❡❣✉❡ q✉❡ g = g′, h = h′. P♦rt❛♥t♦ t❡♠♦s q✉❡ U V = U′V′ ❡ ❛ss✐♠ ♣♦❞❡♠♦s
❝♦♥s✐❞❡r❛r ❛s ❜✐ss❡çõ❡s ❛❜❡rt❛s U ❡V t❡♥❞♦ s(U) =r(V).
❆❣♦r❛ ❞❡✜♥❛ ♣♦r φ❛ ❝♦♠♣♦s✐çã♦ U →s(U) =r(V)→V, ♦✉ s❡❥❛✱ φ:=r|V−1◦s|U ❡ ❡♥tã♦ ❝♦♥s✐❞❡r❡f :U →U×V∩G(2), x7→(x, φ(x)).
➱ ❝❧❛r♦ q✉❡φé ❤♦♠❡♦♠♦r✜s♠♦ ❡ q✉❡f é ❝♦♥tí♥✉❛ ❡ ✐♥❥❡t✐✈❛✳ ❆❞❡♠❛✐s✱ f é s♦❜r❡❥❡t✐✈❛✱ ♣♦✐s ♣❛r❛ q✉❛❧q✉❡r (g, h) ∈U ×V ∩G(2), t❡♠♦s q✉❡
s(g) = r(h) ❣❛r❛♥t❡ q✉❡ h = r|V−1◦s|U(g) = φ(g). ❉❡♥♦t❛♥❞♦ ♣♦r
π1 ❛ ♣r♦❥❡çã♦ ❞❡ U×V ∩G(2) ❡♠ U, ♦❜s❡r✈❛♠♦s q✉❡ π1 é ✐♥❥❡t✐✈❛ ❡
s♦❜r❡❥❡t✐✈❛ ❡♠U✳
❉❡ ❢❛t♦✱ ❝♦♠♦π1◦f =IdU ❡f s♦❜r❡❥❡t✐✈❛✱ s❡❣✉❡ q✉❡π1é ✐♥❥❡t✐✈❛✳
❆❞❡♠❛✐s✱ ♣❛r❛ q✉❛❧q✉❡ru∈U,t❡♠♦s q✉❡s(u)∈s(U) =r(V),❞❡ ♠♦❞♦ q✉❡ s(u) = r(h), ♣❛r❛ ❝❡rt♦ h∈V, ❧♦❣♦ ❡①✐st❡ (u, h)∈ U×V ∩G(2)
q✉❡ ❣❛r❛♥t❡ ❛ s♦❜r❡❥❡t✐✈✐❞❛❞❡ ❞❡π1❡♠U✳ ❈♦♠ ✐ss♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r
q✉❡✱ ❝♦♠♦π1é ❝♦♥tí♥✉❛✱ f ❡π1 sã♦ ❤♦♠❡♦♠♦r✜s♠♦s✳
