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Maximal left ideals of operators acting on a Banach space s str rst tr t t r t

  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❛✇✮ ✶

  2. ❙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♣❛❝❡ ❊ ❄ 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✐♦♥❛❧✳ ✷

  3. ▲❡t ✉s s♣❡❝✐❛❧✐s❡ t♦ ❊ t❤❡♥✳ ◗✉❡st✐♦♥ ■✳ ■s t❤✐s ❝♦♥❥❡❝t✉r❡ tr✉❡ ❢♦r ❊ ✱ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❜♦✉♥❞❡❞✱ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ❄ 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✰❑✳✮ ✷

  4. ◗✉❡st✐♦♥ ■✳ ■s t❤✐s ❝♦♥❥❡❝t✉r❡ tr✉❡ ❢♦r ❊ ✱ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❜♦✉♥❞❡❞✱ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ❄ 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❤❡♥✳ ✷

  5. 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 A = B ( ❊ ) ✱ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❜♦✉♥❞❡❞✱ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ❄ ✷

  6. t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ✳ ♣ ❡①❛♠♣❧❡s✿ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ t❤❡ ❝❧❛ss✐❝❛❧ ❇❛♥❛❝❤ s♣❛❝❡s ❝ ✵ ✱ ✶ ✱ ♣ ▲ ♣ ✵ ✶ ✶ ♣ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ❛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❛ ❛♥❞ t❤❡♥ ✇❡ ✜♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ✉❧tr❛✜❧t❡rs ♦❢ ✭t❤❡r❡ ❛r❡ ✷ ♦❢ t❤❡♠ ❜② P♦s♣í➨✐❧✬s t❤❡♦r❡♠✮ ❛♥❞ ❝❡rt❛✐♥ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✳ ❙♦♠❡ ❝❛s❡s ✇❤❡r❡ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s✿ 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 ✵ ✳ ✸

  7. t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ✳ ♣ ❡①❛♠♣❧❡s✿ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ t❤❡ ❝❧❛ss✐❝❛❧ ❇❛♥❛❝❤ s♣❛❝❡s ❝ ✵ ✱ ✶ ✱ ♣ ▲ ♣ ✵ ✶ ✶ ♣ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ■❞❡❛✳ ❲❡ ✐❞❡♥t✐❢② ♣r♦❥❡❝t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ♣♦✇❡r✲s❡t ❛❧❣❡❜r❛ ❛♥❞ t❤❡♥ ✇❡ ✜♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ✉❧tr❛✜❧t❡rs ♦❢ ✭t❤❡r❡ ❛r❡ ✷ ♦❢ t❤❡♠ ❜② P♦s♣í➨✐❧✬s t❤❡♦r❡♠✮ ❛♥❞ ❝❡rt❛✐♥ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✳ ❙♦♠❡ ❝❛s❡s ✇❤❡r❡ t❤❡ t❤❡♦r❡♠ ❛♣♣❧✐❡s✿ 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❡❞✦ ✸

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