[PPT] - Maximal left ideals of operators acting on a Banach space s PowerPoint Presentation

SLIDE 1

Maximal left ideals of operators acting on a Banach space

❚♦♠❛s③ ❑❛♥✐❛ ▲❛♥❝❛st❡r ❯♥✐✈❡rs✐t②

öt❡❜♦r❣✱ ✷✾t❤ ❏✉❧② ✷✵✶✸

❏♦✐♥t ✇♦r❦ ✇✐t❤ ●❛rt❤ ❉❛❧❡s ❛♥❞ ◆✐❡❧s ▲❛✉sts❡♥ ✭❜♦t❤ ▲❛♥❝❛st❡r✮✱ ❚♦♠❛s③ ❑♦❝❤❛♥❡❦ ✭❯♥✐✈❡rs✐t② ♦❢ ❙✐❧❡s✐❛✱ P♦❧❛♥❞✮ ❛♥❞ P✐♦tr ❑♦s③♠✐❞❡r ✭■▼P❆◆✱ ❲❛rs❛✇✮ ✶

SLIDE 2

Addressing the general question posed during the previous talk

❈♦♥❥❡❝t✉r❡ ✭❉❛❧❡s✕➏❡❧❛③❦♦ ✷✵✶✶✮✳ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡✈❡r② ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧✳ ❙t❛t✉s✿ ❲✐❞❡ ♦♣❡♥ ✐♥ t❤❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❝❛s❡ ❛♣❛rt ❢r♦♠ ❈✯✲❛❧❣❡❜r❛s ❛♥❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✖✐♥ ❛ s✉✐t❛❜❧❡ s❡♥s❡✖❍✐❧❜❡rt ❈✯✲♠♦❞✉❧❡s ✭❇❧❡❝❤❡r✰❑✳✮ ▲❡t ✉s s♣❡❝✐❛❧✐s❡ t♦ ❊ t❤❡♥✳ ◗✉❡st✐♦♥ ■✳ ■s t❤✐s ❝♦♥❥❡❝t✉r❡ tr✉❡ ❢♦r ❊ ✱ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❜♦✉♥❞❡❞✱ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊❄

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SLIDE 3

Addressing the general question posed during the previous talk

❈♦♥❥❡❝t✉r❡ ✭❉❛❧❡s✕➏❡❧❛③❦♦ ✷✵✶✶✮✳ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡✈❡r② ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧✳ ❙t❛t✉s✿ ❲✐❞❡ ♦♣❡♥ ✐♥ t❤❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❝❛s❡ ❛♣❛rt ❢r♦♠ ❈✯✲❛❧❣❡❜r❛s ❛♥❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✖✐♥ ❛ s✉✐t❛❜❧❡ s❡♥s❡✖❍✐❧❜❡rt ❈✯✲♠♦❞✉❧❡s ✭❇❧❡❝❤❡r✰❑✳✮ ▲❡t ✉s s♣❡❝✐❛❧✐s❡ t♦ ❊ t❤❡♥✳ ◗✉❡st✐♦♥ ■✳ ■s t❤✐s ❝♦♥❥❡❝t✉r❡ tr✉❡ ❢♦r ❊ ✱ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❜♦✉♥❞❡❞✱ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊❄

✷

SLIDE 4

Addressing the general question posed during the previous talk

❈♦♥❥❡❝t✉r❡ ✭❉❛❧❡s✕➏❡❧❛③❦♦ ✷✵✶✶✮✳ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡✈❡r② ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧✳ ❙t❛t✉s✿ ❲✐❞❡ ♦♣❡♥ ✐♥ t❤❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❝❛s❡ ❛♣❛rt ❢r♦♠ ❈✯✲❛❧❣❡❜r❛s ❛♥❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✖✐♥ ❛ s✉✐t❛❜❧❡ s❡♥s❡✖❍✐❧❜❡rt ❈✯✲♠♦❞✉❧❡s ✭❇❧❡❝❤❡r✰❑✳✮ ▲❡t ✉s s♣❡❝✐❛❧✐s❡ t♦ B(❊) t❤❡♥✳ ◗✉❡st✐♦♥ ■✳ ■s t❤✐s ❝♦♥❥❡❝t✉r❡ tr✉❡ ❢♦r ❊ ✱ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❜♦✉♥❞❡❞✱ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊❄

❙♦♠❡ ❝❛s❡s ✇❤❡r❡ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s✿

t❤❡ ❝❧❛ss✐❝❛❧ ❇❛♥❛❝❤ s♣❛❝❡s ❝✵✱ ℓ✶✱ ℓ♣, ▲♣[✵, ✶] (✶ < ♣ < ∞);

♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿

t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲✶ ✵ ✶ t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑❀ t❤❡ ♣t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏♣ ✶ ♣ ✳

✸

SLIDE 12

A partial answer to Question I

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✭♦r ❥✉st ✇✐t❤ |❊| = c✮ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B(❊) ❝♦♥t❛✐♥s ✷c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ℵ✵✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✦

■❞❡❛✳ ❲❡ ✐❞❡♥t✐❢② ♣r♦❥❡❝t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ♣♦✇❡r✲s❡t ❛❧❣❡❜r❛ P(N) ❛♥❞ t❤❡♥ ✇❡ ✜♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ✉❧tr❛✜❧t❡rs ♦❢ P(N) ✭t❤❡r❡ ❛r❡ ✷c ♦❢ t❤❡♠ ❜② P♦s♣í➨✐❧✬s t❤❡♦r❡♠✮ ❛♥❞ ❝❡rt❛✐♥ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊)✳

❙♦♠❡ ❝❛s❡s ✇❤❡r❡ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s✿

t❤❡ ❝❧❛ss✐❝❛❧ ❇❛♥❛❝❤ s♣❛❝❡s ❝✵✱ ℓ✶✱ ℓ♣, ▲♣[✵, ✶] (✶ < ♣ < ∞);
♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥

✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿

t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲✶ ✵ ✶ t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑❀ t❤❡ ♣t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏♣ ✶ ♣ ✳

✸

SLIDE 13

A partial answer to Question I

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✭♦r ❥✉st ✇✐t❤ |❊| = c✮ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B(❊) ❝♦♥t❛✐♥s ✷c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ℵ✵✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✦

■❞❡❛✳ ❲❡ ✐❞❡♥t✐❢② ♣r♦❥❡❝t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ♣♦✇❡r✲s❡t ❛❧❣❡❜r❛ P(N) ❛♥❞ t❤❡♥ ✇❡ ✜♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ✉❧tr❛✜❧t❡rs ♦❢ P(N) ✭t❤❡r❡ ❛r❡ ✷c ♦❢ t❤❡♠ ❜② P♦s♣í➨✐❧✬s t❤❡♦r❡♠✮ ❛♥❞ ❝❡rt❛✐♥ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊)✳

❙♦♠❡ ❝❛s❡s ✇❤❡r❡ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s✿

t❤❡ ❝❧❛ss✐❝❛❧ ❇❛♥❛❝❤ s♣❛❝❡s ❝✵✱ ℓ✶✱ ℓ♣, ▲♣[✵, ✶] (✶ < ♣ < ∞);
♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥

✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿

◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲✶[✵, ✶];

t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑❀ t❤❡ ♣t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏♣ ✶ ♣ ✳

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SLIDE 14

A partial answer to Question I

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✭♦r ❥✉st ✇✐t❤ |❊| = c✮ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B(❊) ❝♦♥t❛✐♥s ✷c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ℵ✵✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✦

■❞❡❛✳ ❲❡ ✐❞❡♥t✐❢② ♣r♦❥❡❝t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ♣♦✇❡r✲s❡t ❛❧❣❡❜r❛ P(N) ❛♥❞ t❤❡♥ ✇❡ ✜♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ✉❧tr❛✜❧t❡rs ♦❢ P(N) ✭t❤❡r❡ ❛r❡ ✷c ♦❢ t❤❡♠ ❜② P♦s♣í➨✐❧✬s t❤❡♦r❡♠✮ ❛♥❞ ❝❡rt❛✐♥ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊)✳

❙♦♠❡ ❝❛s❡s ✇❤❡r❡ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s✿

t❤❡ ❝❧❛ss✐❝❛❧ ❇❛♥❛❝❤ s♣❛❝❡s ❝✵✱ ℓ✶✱ ℓ♣, ▲♣[✵, ✶] (✶ < ♣ < ∞);
♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥

✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿

◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲✶[✵, ✶]; ◮ t❤❡ s♣❛❝❡ ❈(❑) ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝

s♣❛❝❡ ❑❀ t❤❡ ♣t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏♣ ✶ ♣ ✳

✸

SLIDE 15

A partial answer to Question I

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✭♦r ❥✉st ✇✐t❤ |❊| = c✮ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B(❊) ❝♦♥t❛✐♥s ✷c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ℵ✵✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✦

■❞❡❛✳ ❲❡ ✐❞❡♥t✐❢② ♣r♦❥❡❝t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ♣♦✇❡r✲s❡t ❛❧❣❡❜r❛ P(N) ❛♥❞ t❤❡♥ ✇❡ ✜♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ✉❧tr❛✜❧t❡rs ♦❢ P(N) ✭t❤❡r❡ ❛r❡ ✷c ♦❢ t❤❡♠ ❜② P♦s♣í➨✐❧✬s t❤❡♦r❡♠✮ ❛♥❞ ❝❡rt❛✐♥ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B(❊)✳

❙♦♠❡ ❝❛s❡s ✇❤❡r❡ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s✿

t❤❡ ❝❧❛ss✐❝❛❧ ❇❛♥❛❝❤ s♣❛❝❡s ❝✵✱ ℓ✶✱ ℓ♣, ▲♣[✵, ✶] (✶ < ♣ < ∞);
♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥

✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿

◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲✶[✵, ✶]; ◮ t❤❡ s♣❛❝❡ ❈(❑) ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝

A refinement of Question I

❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ {✵}✱ ML ① = {❚ ∈ B(❊) : ❚① = ✵} ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C①✳ ▼♦r❡♦✈❡r✱ ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ (①, ② ∈ ❊ \ {✵}). ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ML ① ❢♦r s♦♠❡ ① ∈ ❊ \ {✵} ✐s ✜①❡❞✳ ❚r✐✈✐❛❧✐t②✿

The Dichotomy Theorem

❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❚ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❞✐♠ ❦❡r(■ − ❙❚) < ∞ ❛♥❞ ❞✐♠ ❊ (■ − ❙❚)(❊) < ∞, ❢♦r ❡❛❝❤ ❙ ∈ B(❊)✳ ❚❤❡ s❡t E (❊) ♦❢ ✐♥❡ss❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ❊ ✐s ❛ ❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B(❊)✱ ❛♥❞ ♣r♦♣❡r ✇❤❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❊q✉✐✈❛❧❡♥t❧②✱ ❚ ∈ E (❊) ⇐ ⇒ ❚ ∈ π−✶ r❛❞ (B(❊)/K (❊))

.

❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♦♣❡r❛t♦r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❝♦♠♣❛❝t✳ ❚❤❡ s❛♠❡ ❢♦r ♦♣❡r❛t♦rs ♦♥ ℓ♣ ✭♣ ∈ [✶, ∞)✮ ❛♥❞ ❝✵✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ ✐s ✜①❡❞❀ ♦r ✭✐✐✮ ❝♦♥t❛✐♥s ❊ ✳

✻

SLIDE 36

The Dichotomy Theorem

❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❚ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❞✐♠ ❦❡r(■ − ❙❚) < ∞ ❛♥❞ ❞✐♠ ❊ (■ − ❙❚)(❊) < ∞, ❢♦r ❡❛❝❤ ❙ ∈ B(❊)✳ ❚❤❡ s❡t E (❊) ♦❢ ✐♥❡ss❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ❊ ✐s ❛ ❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B(❊)✱ ❛♥❞ ♣r♦♣❡r ✇❤❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❊q✉✐✈❛❧❡♥t❧②✱ ❚ ∈ E (❊) ⇐ ⇒ ❚ ∈ π−✶ r❛❞ (B(❊)/K (❊))

.

❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♦♣❡r❛t♦r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❝♦♠♣❛❝t✳ ❚❤❡ s❛♠❡ ❢♦r ♦♣❡r❛t♦rs ♦♥ ℓ♣ ✭♣ ∈ [✶, ∞)✮ ❛♥❞ ❝✵✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B(❊)✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ ❝♦♥t❛✐♥s ❊ ✳

✻

SLIDE 37

The Dichotomy Theorem

❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❚ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❞✐♠ ❦❡r(■ − ❙❚) < ∞ ❛♥❞ ❞✐♠ ❊ (■ − ❙❚)(❊) < ∞, ❢♦r ❡❛❝❤ ❙ ∈ B(❊)✳ ❚❤❡ s❡t E (❊) ♦❢ ✐♥❡ss❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ❊ ✐s ❛ ❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B(❊)✱ ❛♥❞ ♣r♦♣❡r ✇❤❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❊q✉✐✈❛❧❡♥t❧②✱ ❚ ∈ E (❊) ⇐ ⇒ ❚ ∈ π−✶ r❛❞ (B(❊)/K (❊))

.

❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♦♣❡r❛t♦r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❝♦♠♣❛❝t✳ ❚❤❡ s❛♠❡ ❢♦r ♦♣❡r❛t♦rs ♦♥ ℓ♣ ✭♣ ∈ [✶, ∞)✮ ❛♥❞ ❝✵✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B(❊)✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E (❊)✳

✻

SLIDE 38

Positive answers to Question II

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ s✉❝❤ t❤❛t✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭❡①❛♠♣❧❡s✿

♣ ❛♥❞ ▲♣ ✵ ✶ ❢♦r ✶

♣ ✮❀ ✭✐✐✮ ❊ ✐s ❛ ❞✉❛❧ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❜❛s✐s ✭❡①❛♠♣❧❡s✿ ❍ ❍ ❛♥❞ ❏♣ ❢♦r ✶ ♣ ✮❀ ✭✐✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭❡①❛♠♣❧❡s✿ ▲ ❢♦r s♦♠❡ ♠❡❛s✉r❡ ✮❀ ✭✐✈✮ ❊ ❝✵ ✱ ❊ ❍✱ ♦r ❊ ❝✵ ❍✱ ✇❤❡r❡ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✈✐✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✐s ✜①❡❞✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ❙① ✳

✼

SLIDE 39

Positive answers to Question II

❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ s✉❝❤ t❤❛t✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭❡①❛♠♣❧❡s✿

A negative answer to Question II: Argyros–Haydon’s Banach space

❚❤❡♦r❡♠ ✭❆r❣②r♦s✕❍❛②❞♦♥ ✷✵✶✶✮✳ ❚❤❡r❡ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳❆❍ ✇❤✐❝❤ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ r❡♠❛r❦❛❜❧❡ ♣r♦♣❡rt✐❡s✿ ✭✐✮ ❳❆❍ ❤❛s ✈❡r② ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❳❆❍ ✐s ❛ ❝♦♠♣❛❝t ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t②❀ ✭✐✐✮ ❳❆❍ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❳❆❍ ✐s ✐s♦♠♦r♣❤✐❝ t♦

✶✳

◆♦t❡ t❤❛t ❢♦r ❊ ❳❆❍ ❡✈❡r②t❤✐♥❣ ❣♦❡s ✇❡❧❧ s✐♥❝❡ ✐t ✐s ❛♥ ❍■ s♣❛❝❡ ❤❡♥❝❡ ❝♦♥❞✐t✐♦♥ ✭✈✮ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❛♣♣❧✐❡s✳

✽

SLIDE 52

A negative answer to Question II: Argyros–Haydon’s Banach space

❚❤❡♦r❡♠ ✭❆r❣②r♦s✕❍❛②❞♦♥ ✷✵✶✶✮✳ ❚❤❡r❡ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳❆❍ ✇❤✐❝❤ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ r❡♠❛r❦❛❜❧❡ ♣r♦♣❡rt✐❡s✿ ✭✐✮ ❳❆❍ ❤❛s ✈❡r② ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❳❆❍ ✐s ❛ ❝♦♠♣❛❝t ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t②❀ ✭✐✐✮ ❳❆❍ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❳❆❍ ✐s ✐s♦♠♦r♣❤✐❝ t♦

.

❑❡② ♣♦✐♥t✿ ❚✶,✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t K✶ = ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✶,✶ ✐s ❝♦♠♣❛❝t
✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B(❊)

✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳

✾

SLIDE 59

A negative answer to Question II: the example

▲❡t ❊ = ❳❆❍ ⊕ ℓ∞✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ (✷ × ✷)✲♠❛tr✐❝❡s ❚✶,✶ : ❳❆❍ → ❳❆❍ ❚✶,✷ : ℓ∞ → ❳❆❍ ❚✷,✶ : ❳❆❍ → ℓ∞ ❚✷,✷ : ℓ∞ → ℓ∞

.

❑❡② ♣♦✐♥t✿ ❚✶,✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t K✶ = ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✶,✶ ✐s ❝♦♠♣❛❝t
✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B(❊)✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛

♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳

✾

SLIDE 60

A negative answer to Question II: the example

▲❡t ❊ = ❳❆❍ ⊕ ℓ∞✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ (✷ × ✷)✲♠❛tr✐❝❡s ❚✶,✶ : ❳❆❍ → ❳❆❍ ❚✶,✷ : ℓ∞ → ❳❆❍ ❚✷,✶ : ❳❆❍ → ℓ∞ ❚✷,✷ : ℓ∞ → ℓ∞

.

❑❡② ♣♦✐♥t✿ ❚✶,✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t K✶ = ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✶,✶ ✐s ❝♦♠♣❛❝t
✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B(❊)✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛

♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞ ✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳

✾

SLIDE 61

A negative answer to Question II: the example

▲❡t ❊ = ❳❆❍ ⊕ ℓ∞✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ (✷ × ✷)✲♠❛tr✐❝❡s ❚✶,✶ : ❳❆❍ → ❳❆❍ ❚✶,✷ : ℓ∞ → ❳❆❍ ❚✷,✶ : ❳❆❍ → ℓ∞ ❚✷,✷ : ℓ∞ → ℓ∞

.

❑❡② ♣♦✐♥t✿ ❚✶,✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t K✶ = ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✶,✶ ✐s ❝♦♠♣❛❝t
✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B(❊)✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛

♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳

✾

SLIDE 62

A negative answer to Question II: the example continued

▼♦r❡ ♣r❡❝✐s❡❧②✱

✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ✭s✐♥❣❧❡✦✮ ♦♣❡r❛t♦r

▲ ✵ ✵ ❱❯ ❲ ✇❤❡r❡ ❳❆❍ ❳❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ❯

✶

❳❆❍✱ ❱

✶

✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❳❆❍

❱❯ ❲

✷ ✶ ✷

✶✵

SLIDE 63

A negative answer to Question II: the example continued

▼♦r❡ ♣r❡❝✐s❡❧②✱ K✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ✭s✐♥❣❧❡✦✮ ♦♣❡r❛t♦r ▲ =

✵

✵ ❱❯∗κ ❲

,

✇❤❡r❡ ❳❆❍ ❳❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ❯

✶

❳❆❍✱ ❱

✶

✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❳❆❍

❱❯ ❲

✷ ✶ ✷

✶✵

SLIDE 64

A negative answer to Question II: the example continued

▼♦r❡ ♣r❡❝✐s❡❧②✱ K✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ✭s✐♥❣❧❡✦✮ ♦♣❡r❛t♦r ▲ =

✵

✵ ❱❯∗κ ❲

,

✇❤❡r❡

κ: ❳❆❍ → ❳ ∗∗

❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱

❯

✶

❳❆❍✱ ❱

✶

✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❳❆❍

❱❯ ❲

✷ ✶ ✷

✶✵

SLIDE 65

A negative answer to Question II: the example continued

▼♦r❡ ♣r❡❝✐s❡❧②✱ K✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ✭s✐♥❣❧❡✦✮ ♦♣❡r❛t♦r ▲ =

✵

✵ ❱❯∗κ ❲

,

✇❤❡r❡

κ: ❳❆❍ → ❳ ∗∗

❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱

❯ : ℓ✶ → ❳ ∗

❆❍✱ ❱ : ℓ∗ ✶ = ℓ∞ → ℓ∞(✷N − ✶) ❛♥❞ ❲ : ℓ∞ → ℓ∞(✷N) ❛r❡

✐s♦♠♦r♣❤✐s♠s✳ ❳❆❍

❱❯ ❲

✷ ✶ ✷

✶✵

SLIDE 66

A negative answer to Question II: the example continued

▼♦r❡ ♣r❡❝✐s❡❧②✱ K✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ✭s✐♥❣❧❡✦✮ ♦♣❡r❛t♦r ▲ =

✵

✵ ❱❯∗κ ❲

,

✇❤❡r❡

κ: ❳❆❍ → ❳ ∗∗

❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱

❯ : ℓ✶ → ❳ ∗

❆❍✱ ❱ : ℓ∗ ✶ = ℓ∞ → ℓ∞(✷N − ✶) ❛♥❞ ❲ : ℓ∞ → ℓ∞(✷N) ❛r❡

✐s♦♠♦r♣❤✐s♠s✳ ❳ ∗∗

❆❍ ⊕ ℓ∞ ❱❯∗

❲
ℓ∞(✷N − ✶)

⊕ ℓ∞(✷N) = ℓ∞

✶✵

SLIDE 67

How about Question I?

❘❡❝❛❧❧✿ ❊ = ❳❆❍ ⊕ ℓ∞✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ ✐❞❡❛❧ K✶ ✐s t❤❡ ✉♥✐q✉❡ ♥♦♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ❍❡♥❝❡ ❚✶ ✶ ❚✶ ✷ ❚✷ ✶ ❚✷ ✷ ❊ ❚✷ ✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✇❤✐❝❤ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❊ ✱ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ✐s ♣♦s✐t✐✈❡ ❢♦r ❊✳

✶✶

SLIDE 68

How about Question I?

❘❡❝❛❧❧✿ ❊ = ❳❆❍ ⊕ ℓ∞✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ ✐❞❡❛❧ K✶ ✐s t❤❡ ✉♥✐q✉❡ ♥♦♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ❍❡♥❝❡ ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✷,✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r
,

✇❤✐❝❤ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B(❊)✱ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ✐s ♣♦s✐t✐✈❡ ❢♦r ❊✳

✶✶

SLIDE 69

How about Question I?

❘❡❝❛❧❧✿ ❊ = ❳❆❍ ⊕ ℓ∞✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ ✐❞❡❛❧ K✶ ✐s t❤❡ ✉♥✐q✉❡ ♥♦♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ❍❡♥❝❡ ❚✶,✶ ❚✶,✷ ❚✷,✶ ❚✷,✷

∈ B(❊) : ❚✷,✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r
,

✇❤✐❝❤ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B(❊)✱ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ✐s ♣♦s✐t✐✈❡ ❢♦r ❊✳

✶✶

SLIDE 70

Open problems

■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B(❊) ❝♦♥t❛✐♥s ❛

✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡✱ ❊ ❳❆❍ ✳✮ ▲❡t ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t ❈ ❑ ❝✵✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✜①❡❞❄ ■s t❤❡r❡ ❛ ❝♦✉♥t❡r❡①❛♠♣❧❡ ❢♦r ◗✳ ■❄ ❚❤❡r❡ ❛r❡ ✈❡r② ❢❡✇ ✭s❡♣❛r❛❜❧❡✮ ❇❛♥❛❝❤ s♣❛❝❡s ❢♦r ✇❤✐❝❤ ✇❡ ❞♦ ♥♦t ❦♥♦✇ t❤❡ ❛♥s✇❡r t♦ t❤❛t q✉❡st✐♦♥✳ ❆♥ ❡①❛♠♣❧❡✿ P✐s✐❡r✬s s♣❛❝❡ P ✇❤✐❝❤ s❛t✐s✜❡s P P P P✳ ❘❡❢❡r❡♥❝❡s ❉✳ ❇❧❡❝❤❡r ❛♥❞ ❚✳ ❑❛♥✐❛✱ ❋✐♥✐t❡ ❣❡♥❡r❛t✐♦♥ ✐♥ ❈✯✲❛❧❣❡❜r❛s ❛♥❞ ❍✐❧❜❡rt ❈✯✲♠♦❞✉❧❡s✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ s✉❜♠✐tt❡❞❀ ♣r❡♣r✐♥t ❛✈❛✐❧❛❜❧❡ t❤r♦✉❣❤ t❤❡ ❛r❳✐✈✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✼✸✕✶✾✸✳

✶✷

SLIDE 71

Open problems

■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B(❊) ❝♦♥t❛✐♥s ❛

✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡✱ ❊ = ❳❆❍ ⊕ ℓ∞✳✮ ▲❡t ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t ❈ ❑ ❝✵✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✜①❡❞❄ ■s t❤❡r❡ ❛ ❝♦✉♥t❡r❡①❛♠♣❧❡ ❢♦r ◗✳ ■❄ ❚❤❡r❡ ❛r❡ ✈❡r② ❢❡✇ ✭s❡♣❛r❛❜❧❡✮ ❇❛♥❛❝❤ s♣❛❝❡s ❢♦r ✇❤✐❝❤ ✇❡ ❞♦ ♥♦t ❦♥♦✇ t❤❡ ❛♥s✇❡r t♦ t❤❛t q✉❡st✐♦♥✳ ❆♥ ❡①❛♠♣❧❡✿ P✐s✐❡r✬s s♣❛❝❡ P ✇❤✐❝❤ s❛t✐s✜❡s P P P P✳ ❘❡❢❡r❡♥❝❡s ❉✳ ❇❧❡❝❤❡r ❛♥❞ ❚✳ ❑❛♥✐❛✱ ❋✐♥✐t❡ ❣❡♥❡r❛t✐♦♥ ✐♥ ❈✯✲❛❧❣❡❜r❛s ❛♥❞ ❍✐❧❜❡rt ❈✯✲♠♦❞✉❧❡s✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ s✉❜♠✐tt❡❞❀ ♣r❡♣r✐♥t ❛✈❛✐❧❛❜❧❡ t❤r♦✉❣❤ t❤❡ ❛r❳✐✈✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✼✸✕✶✾✸✳

✶✷

SLIDE 72

Open problems

■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B(❊) ❝♦♥t❛✐♥s ❛

✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡✱ ❊ = ❳❆❍ ⊕ ℓ∞✳✮

▲❡t ❊ = ❈(❑)✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t

❈(❑) ∼ = ❝✵✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✜①❡❞❄ ■s t❤❡r❡ ❛ ❝♦✉♥t❡r❡①❛♠♣❧❡ ❢♦r ◗✳ ■❄ ❚❤❡r❡ ❛r❡ ✈❡r② ❢❡✇ ✭s❡♣❛r❛❜❧❡✮ ❇❛♥❛❝❤ s♣❛❝❡s ❢♦r ✇❤✐❝❤ ✇❡ ❞♦ ♥♦t ❦♥♦✇ t❤❡ ❛♥s✇❡r t♦ t❤❛t q✉❡st✐♦♥✳ ❆♥ ❡①❛♠♣❧❡✿ P✐s✐❡r✬s s♣❛❝❡ P ✇❤✐❝❤ s❛t✐s✜❡s P P P P✳ ❘❡❢❡r❡♥❝❡s ❉✳ ❇❧❡❝❤❡r ❛♥❞ ❚✳ ❑❛♥✐❛✱ ❋✐♥✐t❡ ❣❡♥❡r❛t✐♦♥ ✐♥ ❈✯✲❛❧❣❡❜r❛s ❛♥❞ ❍✐❧❜❡rt ❈✯✲♠♦❞✉❧❡s✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ s✉❜♠✐tt❡❞❀ ♣r❡♣r✐♥t ❛✈❛✐❧❛❜❧❡ t❤r♦✉❣❤ t❤❡ ❛r❳✐✈✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✼✸✕✶✾✸✳

✶✷

SLIDE 73

Open problems

■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B(❊) ❝♦♥t❛✐♥s ❛

✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡✱ ❊ = ❳❆❍ ⊕ ℓ∞✳✮

▲❡t ❊ = ❈(❑)✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t

❈(❑) ∼ = ❝✵✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✜①❡❞❄

■s t❤❡r❡ ❛ ❝♦✉♥t❡r❡①❛♠♣❧❡ ❢♦r ◗✳ ■❄ ❚❤❡r❡ ❛r❡ ✈❡r② ❢❡✇ ✭s❡♣❛r❛❜❧❡✮ ❇❛♥❛❝❤

s♣❛❝❡s ❢♦r ✇❤✐❝❤ ✇❡ ❞♦ ♥♦t ❦♥♦✇ t❤❡ ❛♥s✇❡r t♦ t❤❛t q✉❡st✐♦♥✳ ❆♥ ❡①❛♠♣❧❡✿ P✐s✐❡r✬s s♣❛❝❡ P ✇❤✐❝❤ s❛t✐s✜❡s P ⊗π P = P ⊗ε P✳ ❘❡❢❡r❡♥❝❡s ❉✳ ❇❧❡❝❤❡r ❛♥❞ ❚✳ ❑❛♥✐❛✱ ❋✐♥✐t❡ ❣❡♥❡r❛t✐♦♥ ✐♥ ❈✯✲❛❧❣❡❜r❛s ❛♥❞ ❍✐❧❜❡rt ❈✯✲♠♦❞✉❧❡s✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ s✉❜♠✐tt❡❞❀ ♣r❡♣r✐♥t ❛✈❛✐❧❛❜❧❡ t❤r♦✉❣❤ t❤❡ ❛r❳✐✈✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✼✸✕✶✾✸✳

✶✷

SLIDE 74

Open problems and references

■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B(❊) ❝♦♥t❛✐♥s ❛

✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡ ❛❜♦✈❡✱ ❊ = ❳❆❍ ⊕ ℓ∞✳✮

▲❡t ❊ = ❈(❑)✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t

❈(❑) ∼ = ❝✵✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B(❊) ✜①❡❞❄

■s t❤❡r❡ ❛ ❝♦✉♥t❡r❡①❛♠♣❧❡ ❢♦r ◗✳ ■❄ ❚❤❡r❡ ❛r❡ ✈❡r② ❢❡✇ ✭s❡♣❛r❛❜❧❡✮ ❇❛♥❛❝❤

s♣❛❝❡s ❢♦r ✇❤✐❝❤ ✇❡ ❞♦ ♥♦t ❦♥♦✇ t❤❡ ❛♥s✇❡r t♦ t❤❛t q✉❡st✐♦♥✳ ❆♥ ❡①❛♠♣❧❡✿ P✐s✐❡r✬s s♣❛❝❡ P ✇❤✐❝❤ s❛t✐s✜❡s P ⊗π P = P ⊗ε P✳ ❘❡❢❡r❡♥❝❡s

❉✳ ❇❧❡❝❤❡r ❛♥❞ ❚✳ ❑❛♥✐❛✱ ❋✐♥✐t❡ ❣❡♥❡r❛t✐♦♥ ✐♥ ❈✯✲❛❧❣❡❜r❛s ❛♥❞ ❍✐❧❜❡rt

❈✯✲♠♦❞✉❧❡s✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳

❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱

▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ s✉❜♠✐tt❡❞❀ ♣r❡♣r✐♥t ❛✈❛✐❧❛❜❧❡ t❤r♦✉❣❤ t❤❡ ❛r❳✐✈✳

❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤