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Locally finite-dimensional operator algebras and some transfinite - - PowerPoint PPT Presentation

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Locally finite-dimensional operator algebras and some transfinite combinatorial structures Piotr Koszmider Institute of Mathematics of the Polish Academy of Sciences, Warsaw Piotr Koszmider (IMPAN) LF algebras and transfinite structures

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Locally finite-dimensional operator algebras and some transfinite combinatorial structures

Piotr Koszmider

Institute of Mathematics of the Polish Academy of Sciences, Warsaw

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 1 / 13

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Outline of results

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 2 / 13

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Outline of results AF C*-algebras without big commutative subalgebras: Luzin almost disjoint families of N. (with T. Bice)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 2 / 13

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Outline of results AF C*-algebras without big commutative subalgebras: Luzin almost disjoint families of N. (with T. Bice) Counterexamples to permanence of stability of basic extensions and inductive limits of nonseparable AF algebras: Mr´

  • wka almost

disjoint families of N. (with S. Ghasemi)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 2 / 13

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Outline of results AF C*-algebras without big commutative subalgebras: Luzin almost disjoint families of N. (with T. Bice) Counterexamples to permanence of stability of basic extensions and inductive limits of nonseparable AF algebras: Mr´

  • wka almost

disjoint families of N. (with S. Ghasemi) C*-algebras without ≪-increasing approx. units; Scattered LF algebras, not AF: Canadian trees, Q-sets, embedding of ℘(ω1) into ℘(N)/Fin. (with T. Bice)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 2 / 13

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Outline of results AF C*-algebras without big commutative subalgebras: Luzin almost disjoint families of N. (with T. Bice) Counterexamples to permanence of stability of basic extensions and inductive limits of nonseparable AF algebras: Mr´

  • wka almost

disjoint families of N. (with S. Ghasemi) C*-algebras without ≪-increasing approx. units; Scattered LF algebras, not AF: Canadian trees, Q-sets, embedding of ℘(ω1) into ℘(N)/Fin. (with T. Bice)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 2 / 13

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Outline of results AF C*-algebras without big commutative subalgebras: Luzin almost disjoint families of N. (with T. Bice) Counterexamples to permanence of stability of basic extensions and inductive limits of nonseparable AF algebras: Mr´

  • wka almost

disjoint families of N. (with S. Ghasemi) C*-algebras without ≪-increasing approx. units; Scattered LF algebras, not AF: Canadian trees, Q-sets, embedding of ℘(ω1) into ℘(N)/Fin. (with T. Bice)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 2 / 13

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Scattered C*-algebras

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Scattered C*-algebras

Theorem (Jensen 78, Wojtaszczyk 74, Ghasemi+K)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Scattered C*-algebras

Theorem (Jensen 78, Wojtaszczyk 74, Ghasemi+K)

Suppose that A is a C*-algebra. The following conditions are equivalent:

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Scattered C*-algebras

Theorem (Jensen 78, Wojtaszczyk 74, Ghasemi+K)

Suppose that A is a C*-algebra. The following conditions are equivalent: Every non-zero ∗-homomorphic image of A has a minimal projection.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Scattered C*-algebras

Theorem (Jensen 78, Wojtaszczyk 74, Ghasemi+K)

Suppose that A is a C*-algebra. The following conditions are equivalent: Every non-zero ∗-homomorphic image of A has a minimal projection. There is an ordinal ht(A) and a continuous increasing sequence of closed ideals (Iα)α≤ht(A) such that I0 = {0}, Iht(A) = A and Iα+1/Iα is generated by all minimal projections in A/Iα, for every α < ht(A).

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Scattered C*-algebras

Theorem (Jensen 78, Wojtaszczyk 74, Ghasemi+K)

Suppose that A is a C*-algebra. The following conditions are equivalent: Every non-zero ∗-homomorphic image of A has a minimal projection. There is an ordinal ht(A) and a continuous increasing sequence of closed ideals (Iα)α≤ht(A) such that I0 = {0}, Iht(A) = A and Iα+1/Iα is generated by all minimal projections in A/Iα, for every α < ht(A). Every subalgebra of A has a minimal projection.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Scattered C*-algebras

Theorem (Jensen 78, Wojtaszczyk 74, Ghasemi+K)

Suppose that A is a C*-algebra. The following conditions are equivalent: Every non-zero ∗-homomorphic image of A has a minimal projection. There is an ordinal ht(A) and a continuous increasing sequence of closed ideals (Iα)α≤ht(A) such that I0 = {0}, Iht(A) = A and Iα+1/Iα is generated by all minimal projections in A/Iα, for every α < ht(A). Every subalgebra of A has a minimal projection. A does not contain a copy of the C∗-algebra C([0, 1]).

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Scattered C*-algebras

Theorem (Jensen 78, Wojtaszczyk 74, Ghasemi+K)

Suppose that A is a C*-algebra. The following conditions are equivalent: Every non-zero ∗-homomorphic image of A has a minimal projection. There is an ordinal ht(A) and a continuous increasing sequence of closed ideals (Iα)α≤ht(A) such that I0 = {0}, Iht(A) = A and Iα+1/Iα is generated by all minimal projections in A/Iα, for every α < ht(A). Every subalgebra of A has a minimal projection. A does not contain a copy of the C∗-algebra C([0, 1]). The spectrum of every self-adjoint element is countable.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Scattered C*-algebras

Theorem (Jensen 78, Wojtaszczyk 74, Ghasemi+K)

Suppose that A is a C*-algebra. The following conditions are equivalent: Every non-zero ∗-homomorphic image of A has a minimal projection. There is an ordinal ht(A) and a continuous increasing sequence of closed ideals (Iα)α≤ht(A) such that I0 = {0}, Iht(A) = A and Iα+1/Iα is generated by all minimal projections in A/Iα, for every α < ht(A). Every subalgebra of A has a minimal projection. A does not contain a copy of the C∗-algebra C([0, 1]). The spectrum of every self-adjoint element is countable.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Scattered C*-algebras

Theorem (Jensen 78, Wojtaszczyk 74, Ghasemi+K)

Suppose that A is a C*-algebra. The following conditions are equivalent: Every non-zero ∗-homomorphic image of A has a minimal projection. There is an ordinal ht(A) and a continuous increasing sequence of closed ideals (Iα)α≤ht(A) such that I0 = {0}, Iht(A) = A and Iα+1/Iα is generated by all minimal projections in A/Iα, for every α < ht(A). Every subalgebra of A has a minimal projection. A does not contain a copy of the C∗-algebra C([0, 1]). The spectrum of every self-adjoint element is countable.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 3 / 13

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Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union;

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union; locally finite-dimensional (LF) if, for any finite subset F of A and any ε > 0, there is a finite-dimensional C*-subalgebra B of A such that ∀f ∈ F ∃b ∈ B f − b ≤ ε

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union; locally finite-dimensional (LF) if, for any finite subset F of A and any ε > 0, there is a finite-dimensional C*-subalgebra B of A such that ∀f ∈ F ∃b ∈ B f − b ≤ ε

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union; locally finite-dimensional (LF) if, for any finite subset F of A and any ε > 0, there is a finite-dimensional C*-subalgebra B of A such that ∀f ∈ F ∃b ∈ B f − b ≤ ε

Theorem

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union; locally finite-dimensional (LF) if, for any finite subset F of A and any ε > 0, there is a finite-dimensional C*-subalgebra B of A such that ∀f ∈ F ∃b ∈ B f − b ≤ ε

Theorem

(Bratteli, 1972) AF ⇔ LF for separable C*-algebras

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union; locally finite-dimensional (LF) if, for any finite subset F of A and any ε > 0, there is a finite-dimensional C*-subalgebra B of A such that ∀f ∈ F ∃b ∈ B f − b ≤ ε

Theorem

(Bratteli, 1972) AF ⇔ LF for separable C*-algebras (Farah-Katsura, 2010) AF ⇔ LF for C*-algebras of density ω1 but not bigger

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union; locally finite-dimensional (LF) if, for any finite subset F of A and any ε > 0, there is a finite-dimensional C*-subalgebra B of A such that ∀f ∈ F ∃b ∈ B f − b ≤ ε

Theorem

(Bratteli, 1972) AF ⇔ LF for separable C*-algebras (Farah-Katsura, 2010) AF ⇔ LF for C*-algebras of density ω1 but not bigger (Lin 1989, Kusuda, 2012) A C*-algebra is scattered if and only if all of its subalgebras are LF .

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union; locally finite-dimensional (LF) if, for any finite subset F of A and any ε > 0, there is a finite-dimensional C*-subalgebra B of A such that ∀f ∈ F ∃b ∈ B f − b ≤ ε

Theorem

(Bratteli, 1972) AF ⇔ LF for separable C*-algebras (Farah-Katsura, 2010) AF ⇔ LF for C*-algebras of density ω1 but not bigger (Lin 1989, Kusuda, 2012) A C*-algebra is scattered if and only if all of its subalgebras are LF . (Bice+K) There are scattered C*-algebras not AF.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union; locally finite-dimensional (LF) if, for any finite subset F of A and any ε > 0, there is a finite-dimensional C*-subalgebra B of A such that ∀f ∈ F ∃b ∈ B f − b ≤ ε

Theorem

(Bratteli, 1972) AF ⇔ LF for separable C*-algebras (Farah-Katsura, 2010) AF ⇔ LF for C*-algebras of density ω1 but not bigger (Lin 1989, Kusuda, 2012) A C*-algebra is scattered if and only if all of its subalgebras are LF . (Bice+K) There are scattered C*-algebras not AF.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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Definition

A C*-algebra A is called approximately finite-dimensional (AF) if it has a directed family of finite-dimensional subalgebras with dense union; locally finite-dimensional (LF) if, for any finite subset F of A and any ε > 0, there is a finite-dimensional C*-subalgebra B of A such that ∀f ∈ F ∃b ∈ B f − b ≤ ε

Theorem

(Bratteli, 1972) AF ⇔ LF for separable C*-algebras (Farah-Katsura, 2010) AF ⇔ LF for C*-algebras of density ω1 but not bigger (Lin 1989, Kusuda, 2012) A C*-algebra is scattered if and only if all of its subalgebras are LF . (Bice+K) There are scattered C*-algebras not AF.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

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C*-algebras without big commutative subalgebras

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

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C*-algebras without big commutative subalgebras

Results

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

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C*-algebras without big commutative subalgebras

Results

(Ogasawara, 1954) Every infinite dimensional C*-algebra has an infinite dimensional commutative C*-subalgebra

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

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C*-algebras without big commutative subalgebras

Results

(Ogasawara, 1954) Every infinite dimensional C*-algebra has an infinite dimensional commutative C*-subalgebra (Dixmier, 1970): Does every nonseparable C*-agebra has a nonseparable commutative C*-subalgebra?

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

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C*-algebras without big commutative subalgebras

Results

(Ogasawara, 1954) Every infinite dimensional C*-algebra has an infinite dimensional commutative C*-subalgebra (Dixmier, 1970): Does every nonseparable C*-agebra has a nonseparable commutative C*-subalgebra? (Akemann-Donner, 1979): CH implies: No.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

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C*-algebras without big commutative subalgebras

Results

(Ogasawara, 1954) Every infinite dimensional C*-algebra has an infinite dimensional commutative C*-subalgebra (Dixmier, 1970): Does every nonseparable C*-agebra has a nonseparable commutative C*-subalgebra? (Akemann-Donner, 1979): CH implies: No. (Popa, 1983): No in ZFC.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

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C*-algebras without big commutative subalgebras

Results

(Ogasawara, 1954) Every infinite dimensional C*-algebra has an infinite dimensional commutative C*-subalgebra (Dixmier, 1970): Does every nonseparable C*-agebra has a nonseparable commutative C*-subalgebra? (Akemann-Donner, 1979): CH implies: No. (Popa, 1983): No in ZFC. (Bice+K; 2017) Akemann-Donner algebras can be done without CH, in ZFC.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

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C*-algebras without big commutative subalgebras

Results

(Ogasawara, 1954) Every infinite dimensional C*-algebra has an infinite dimensional commutative C*-subalgebra (Dixmier, 1970): Does every nonseparable C*-agebra has a nonseparable commutative C*-subalgebra? (Akemann-Donner, 1979): CH implies: No. (Popa, 1983): No in ZFC. (Bice+K; 2017) Akemann-Donner algebras can be done without CH, in ZFC.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

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C*-algebras without big commutative subalgebras

Results

(Ogasawara, 1954) Every infinite dimensional C*-algebra has an infinite dimensional commutative C*-subalgebra (Dixmier, 1970): Does every nonseparable C*-agebra has a nonseparable commutative C*-subalgebra? (Akemann-Donner, 1979): CH implies: No. (Popa, 1983): No in ZFC. (Bice+K; 2017) Akemann-Donner algebras can be done without CH, in ZFC.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

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Akemann-Donner algebras:

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2) If B ⊆ A is nonseparable and commutative, there are (q(n) : n ∈ N) ⊆ P1 such that b(n)q(n) = q(n)b(n) and all b ∈ B

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2) If B ⊆ A is nonseparable and commutative, there are (q(n) : n ∈ N) ⊆ P1 such that b(n)q(n) = q(n)b(n) and all b ∈ B One proves that there is an uncountable F′ such that for every F ∈ F′ lim

n∈F q(n) = pF.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2) If B ⊆ A is nonseparable and commutative, there are (q(n) : n ∈ N) ⊆ P1 such that b(n)q(n) = q(n)b(n) and all b ∈ B One proves that there is an uncountable F′ such that for every F ∈ F′ lim

n∈F q(n) = pF.

For two uncountable G, H ⊆ F′ there are s, r ∈ P1 such that pF − s < r − s/2 for all F ∈ G and pF − r < r − s/2 for all F ∈ H and

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2) If B ⊆ A is nonseparable and commutative, there are (q(n) : n ∈ N) ⊆ P1 such that b(n)q(n) = q(n)b(n) and all b ∈ B One proves that there is an uncountable F′ such that for every F ∈ F′ lim

n∈F q(n) = pF.

For two uncountable G, H ⊆ F′ there are s, r ∈ P1 such that pF − s < r − s/2 for all F ∈ G and pF − r < r − s/2 for all F ∈ H and X = {n ∈ N : q(n) − r < r − s/2} and Y = {n ∈ N : q(n) − s < r − s/2}

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2) If B ⊆ A is nonseparable and commutative, there are (q(n) : n ∈ N) ⊆ P1 such that b(n)q(n) = q(n)b(n) and all b ∈ B One proves that there is an uncountable F′ such that for every F ∈ F′ lim

n∈F q(n) = pF.

For two uncountable G, H ⊆ F′ there are s, r ∈ P1 such that pF − s < r − s/2 for all F ∈ G and pF − r < r − s/2 for all F ∈ H and X = {n ∈ N : q(n) − r < r − s/2} and Y = {n ∈ N : q(n) − s < r − s/2}

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2) If B ⊆ A is nonseparable and commutative, there are (q(n) : n ∈ N) ⊆ P1 such that b(n)q(n) = q(n)b(n) and all b ∈ B One proves that there is an uncountable F′ such that for every F ∈ F′ lim

n∈F q(n) = pF.

For two uncountable G, H ⊆ F′ there are s, r ∈ P1 such that pF − s < r − s/2 for all F ∈ G and pF − r < r − s/2 for all F ∈ H and X = {n ∈ N : q(n) − r < r − s/2} and Y = {n ∈ N : q(n) − s < r − s/2} satisfy F ⊆Fin X and F ′ ⊆Fin Y for all F ∈ G and all F ′ ∈ H

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2) If B ⊆ A is nonseparable and commutative, there are (q(n) : n ∈ N) ⊆ P1 such that b(n)q(n) = q(n)b(n) and all b ∈ B One proves that there is an uncountable F′ such that for every F ∈ F′ lim

n∈F q(n) = pF.

For two uncountable G, H ⊆ F′ there are s, r ∈ P1 such that pF − s < r − s/2 for all F ∈ G and pF − r < r − s/2 for all F ∈ H and X = {n ∈ N : q(n) − r < r − s/2} and Y = {n ∈ N : q(n) − s < r − s/2} satisfy F ⊆Fin X and F ′ ⊆Fin Y for all F ∈ G and all F ′ ∈ H This is impossibe if F is a Luzin family.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2) If B ⊆ A is nonseparable and commutative, there are (q(n) : n ∈ N) ⊆ P1 such that b(n)q(n) = q(n)b(n) and all b ∈ B One proves that there is an uncountable F′ such that for every F ∈ F′ lim

n∈F q(n) = pF.

For two uncountable G, H ⊆ F′ there are s, r ∈ P1 such that pF − s < r − s/2 for all F ∈ G and pF − r < r − s/2 for all F ∈ H and X = {n ∈ N : q(n) − r < r − s/2} and Y = {n ∈ N : q(n) − s < r − s/2} satisfy F ⊆Fin X and F ′ ⊆Fin Y for all F ∈ G and all F ′ ∈ H This is impossibe if F is a Luzin family.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = (PF : F ∈ F) ⊆ P1 ⊆ M2 such that pF − p0 < 1/4 for some p0 ∈ P1 A = AF,P ⊆ ℓ∞(M2) generated by (pFχF : F ∈ F) and c0(M2) If B ⊆ A is nonseparable and commutative, there are (q(n) : n ∈ N) ⊆ P1 such that b(n)q(n) = q(n)b(n) and all b ∈ B One proves that there is an uncountable F′ such that for every F ∈ F′ lim

n∈F q(n) = pF.

For two uncountable G, H ⊆ F′ there are s, r ∈ P1 such that pF − s < r − s/2 for all F ∈ G and pF − r < r − s/2 for all F ∈ H and X = {n ∈ N : q(n) − r < r − s/2} and Y = {n ∈ N : q(n) − s < r − s/2} satisfy F ⊆Fin X and F ′ ⊆Fin Y for all F ∈ G and all F ′ ∈ H This is impossibe if F is a Luzin family.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

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SLIDE 52

Counterexamples to permanence of stability:

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

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SLIDE 53

Counterexamples to permanence of stability: A C*-algebra is stable iff A ≡ A ⊗ K

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

slide-54
SLIDE 54

Counterexamples to permanence of stability: A C*-algebra is stable iff A ≡ A ⊗ K (Blackadar 1980) Extensions of stable separable AF algebras by stable separable AF algebras are stable.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

slide-55
SLIDE 55

Counterexamples to permanence of stability: A C*-algebra is stable iff A ≡ A ⊗ K (Blackadar 1980) Extensions of stable separable AF algebras by stable separable AF algebras are stable. (Hjelmborg-Rordam, 1998) Countable inductive limits of stable σ-unital C*-algebras are stable.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

slide-56
SLIDE 56

Counterexamples to permanence of stability: A C*-algebra is stable iff A ≡ A ⊗ K (Blackadar 1980) Extensions of stable separable AF algebras by stable separable AF algebras are stable. (Hjelmborg-Rordam, 1998) Countable inductive limits of stable σ-unital C*-algebras are stable.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

slide-57
SLIDE 57

Counterexamples to permanence of stability: A C*-algebra is stable iff A ≡ A ⊗ K (Blackadar 1980) Extensions of stable separable AF algebras by stable separable AF algebras are stable. (Hjelmborg-Rordam, 1998) Countable inductive limits of stable σ-unital C*-algebras are stable.

Theorem (Ghasemi+K)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

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SLIDE 58

Counterexamples to permanence of stability: A C*-algebra is stable iff A ≡ A ⊗ K (Blackadar 1980) Extensions of stable separable AF algebras by stable separable AF algebras are stable. (Hjelmborg-Rordam, 1998) Countable inductive limits of stable σ-unital C*-algebras are stable.

Theorem (Ghasemi+K)

There is a scattered C*-subalgebra A of B(ℓ2) satisfying the following short exact sequence 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0, where ι[K(ℓ2)] is an essential ideal of A and A is not stable. In fact the algebra of multipliers M(A) of A is equal to the unitization of A.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

slide-59
SLIDE 59

Counterexamples to permanence of stability: A C*-algebra is stable iff A ≡ A ⊗ K (Blackadar 1980) Extensions of stable separable AF algebras by stable separable AF algebras are stable. (Hjelmborg-Rordam, 1998) Countable inductive limits of stable σ-unital C*-algebras are stable.

Theorem (Ghasemi+K)

There is a scattered C*-subalgebra A of B(ℓ2) satisfying the following short exact sequence 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0, where ι[K(ℓ2)] is an essential ideal of A and A is not stable. In fact the algebra of multipliers M(A) of A is equal to the unitization of A. There is a nonstable scattered algebra A such that A =

α<ω1 Iα

where the sequence is increasing and Iα+1/Iα ≡ K(ℓ2) and so each Iα is stable.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

slide-60
SLIDE 60

Counterexamples to permanence of stability: A C*-algebra is stable iff A ≡ A ⊗ K (Blackadar 1980) Extensions of stable separable AF algebras by stable separable AF algebras are stable. (Hjelmborg-Rordam, 1998) Countable inductive limits of stable σ-unital C*-algebras are stable.

Theorem (Ghasemi+K)

There is a scattered C*-subalgebra A of B(ℓ2) satisfying the following short exact sequence 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0, where ι[K(ℓ2)] is an essential ideal of A and A is not stable. In fact the algebra of multipliers M(A) of A is equal to the unitization of A. There is a nonstable scattered algebra A such that A =

α<ω1 Iα

where the sequence is increasing and Iα+1/Iα ≡ K(ℓ2) and so each Iα is stable.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

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SLIDE 61

Counterexamples to permanence of stability: A C*-algebra is stable iff A ≡ A ⊗ K (Blackadar 1980) Extensions of stable separable AF algebras by stable separable AF algebras are stable. (Hjelmborg-Rordam, 1998) Countable inductive limits of stable σ-unital C*-algebras are stable.

Theorem (Ghasemi+K)

There is a scattered C*-subalgebra A of B(ℓ2) satisfying the following short exact sequence 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0, where ι[K(ℓ2)] is an essential ideal of A and A is not stable. In fact the algebra of multipliers M(A) of A is equal to the unitization of A. There is a nonstable scattered algebra A such that A =

α<ω1 Iα

where the sequence is increasing and Iα+1/Iα ≡ K(ℓ2) and so each Iα is stable.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

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SLIDE 62

An extension of compact operators by compact operators:

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 63

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

holds for every α, β, ξ, η < κ,

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 64

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

holds for every α, β, ξ, η < κ,

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 65

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

2

Tβ,α Tη,ξ =K δα,ηTβ,ξ.

holds for every α, β, ξ, η < κ,

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 66

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

2

Tβ,α Tη,ξ =K δα,ηTβ,ξ.

holds for every α, β, ξ, η < κ, Let A = A(T ) be the C∗-subalgebra of B(ℓ2) generated by K and T = (Tη,ξ : ξ, η ∈ 2ω)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 67

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

2

Tβ,α Tη,ξ =K δα,ηTβ,ξ.

holds for every α, β, ξ, η < κ, Let A = A(T ) be the C∗-subalgebra of B(ℓ2) generated by K and T = (Tη,ξ : ξ, η ∈ 2ω) Every multiplier R ∈ M(A) ⊆ B(ℓ2) defines a 2ω × 2ω-matrix Λ(R) satisfying Tξ,ξRTη,η =K ΛT (R)ξ,ηTξ,η.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 68

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

2

Tβ,α Tη,ξ =K δα,ηTβ,ξ.

holds for every α, β, ξ, η < κ, Let A = A(T ) be the C∗-subalgebra of B(ℓ2) generated by K and T = (Tη,ξ : ξ, η ∈ 2ω) Every multiplier R ∈ M(A) ⊆ B(ℓ2) defines a 2ω × 2ω-matrix Λ(R) satisfying Tξ,ξRTη,η =K ΛT (R)ξ,ηTξ,η.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 69

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

2

Tβ,α Tη,ξ =K δα,ηTβ,ξ.

holds for every α, β, ξ, η < κ, Let A = A(T ) be the C∗-subalgebra of B(ℓ2) generated by K and T = (Tη,ξ : ξ, η ∈ 2ω) Every multiplier R ∈ M(A) ⊆ B(ℓ2) defines a 2ω × 2ω-matrix Λ(R) satisfying Tξ,ξRTη,η =K ΛT (R)ξ,ηTξ,η. An R as above is called a potential multiplier for T

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 70

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

2

Tβ,α Tη,ξ =K δα,ηTβ,ξ.

holds for every α, β, ξ, η < κ, Let A = A(T ) be the C∗-subalgebra of B(ℓ2) generated by K and T = (Tη,ξ : ξ, η ∈ 2ω) Every multiplier R ∈ M(A) ⊆ B(ℓ2) defines a 2ω × 2ω-matrix Λ(R) satisfying Tξ,ξRTη,η =K ΛT (R)ξ,ηTξ,η. An R as above is called a potential multiplier for T ΛT (R) is a matrix of a bounded operator in B(ℓ2(2ω))

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

slide-71
SLIDE 71

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

2

Tβ,α Tη,ξ =K δα,ηTβ,ξ.

holds for every α, β, ξ, η < κ, Let A = A(T ) be the C∗-subalgebra of B(ℓ2) generated by K and T = (Tη,ξ : ξ, η ∈ 2ω) Every multiplier R ∈ M(A) ⊆ B(ℓ2) defines a 2ω × 2ω-matrix Λ(R) satisfying Tξ,ξRTη,η =K ΛT (R)ξ,ηTξ,η. An R as above is called a potential multiplier for T ΛT (R) is a matrix of a bounded operator in B(ℓ2(2ω)) If T is maximal, then Λ(R) is a matrix of a compact plus identity

  • perator, iff R ∈

A(T ).

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 72

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

2

Tβ,α Tη,ξ =K δα,ηTβ,ξ.

holds for every α, β, ξ, η < κ, Let A = A(T ) be the C∗-subalgebra of B(ℓ2) generated by K and T = (Tη,ξ : ξ, η ∈ 2ω) Every multiplier R ∈ M(A) ⊆ B(ℓ2) defines a 2ω × 2ω-matrix Λ(R) satisfying Tξ,ξRTη,η =K ΛT (R)ξ,ηTξ,η. An R as above is called a potential multiplier for T ΛT (R) is a matrix of a bounded operator in B(ℓ2(2ω)) If T is maximal, then Λ(R) is a matrix of a compact plus identity

  • perator, iff R ∈

A(T ).

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 73

An extension of compact operators by compact operators: An extension 0 → K(ℓ2) ι − → A → K(ℓ2(2ω)) → 0 corresponds to choice of almost matrix units, i.e., noncompact

  • perators T = (Tη,ξ : ξ, η ∈ 2ω) such that

1

T ∗

η,ξ =K Tξ,η,

2

Tβ,α Tη,ξ =K δα,ηTβ,ξ.

holds for every α, β, ξ, η < κ, Let A = A(T ) be the C∗-subalgebra of B(ℓ2) generated by K and T = (Tη,ξ : ξ, η ∈ 2ω) Every multiplier R ∈ M(A) ⊆ B(ℓ2) defines a 2ω × 2ω-matrix Λ(R) satisfying Tξ,ξRTη,η =K ΛT (R)ξ,ηTξ,η. An R as above is called a potential multiplier for T ΛT (R) is a matrix of a bounded operator in B(ℓ2(2ω)) If T is maximal, then Λ(R) is a matrix of a compact plus identity

  • perator, iff R ∈

A(T ).

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

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SLIDE 74

An extension of compact operators by compact operators:

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

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SLIDE 75

An extension of compact operators by compact operators: If T is maximal, R is a potential multiplier and Λ(R) is a matrix of a ‘countably many entries plus identity’ operator, then Λ(R) is a matrix of a compact plus identity operator

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

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SLIDE 76

An extension of compact operators by compact operators: If T is maximal, R is a potential multiplier and Λ(R) is a matrix of a ‘countably many entries plus identity’ operator, then Λ(R) is a matrix of a compact plus identity operator We work in H = ℓ2({0, 1}<N)

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

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SLIDE 77

An extension of compact operators by compact operators: If T is maximal, R is a potential multiplier and Λ(R) is a matrix of a ‘countably many entries plus identity’ operator, then Λ(R) is a matrix of a compact plus identity operator We work in H = ℓ2({0, 1}<N) Define a system of almost matrix units T2N = (Tη,ξ : ξ, η ∈ {0, 1}N) by Tη,ξ(eρ|k) =

  • eη|k

if ρ = ξ,

  • therwise,

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

slide-78
SLIDE 78

An extension of compact operators by compact operators: If T is maximal, R is a potential multiplier and Λ(R) is a matrix of a ‘countably many entries plus identity’ operator, then Λ(R) is a matrix of a compact plus identity operator We work in H = ℓ2({0, 1}<N) Define a system of almost matrix units T2N = (Tη,ξ : ξ, η ∈ {0, 1}N) by Tη,ξ(eρ|k) =

  • eη|k

if ρ = ξ,

  • therwise,

Assume that R is a potential multiplier for T2N and U is a Borel subset of C, then the set BR

U = {(η, ξ) ∈ {0, 1}N × {0, 1}N : λ T2N η,ξ (R) ∈ U}

is Borel in {0, 1}N × {0, 1}N. In particular, BR

U is either countable or

  • f size of the continuum.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

slide-79
SLIDE 79

An extension of compact operators by compact operators: If T is maximal, R is a potential multiplier and Λ(R) is a matrix of a ‘countably many entries plus identity’ operator, then Λ(R) is a matrix of a compact plus identity operator We work in H = ℓ2({0, 1}<N) Define a system of almost matrix units T2N = (Tη,ξ : ξ, η ∈ {0, 1}N) by Tη,ξ(eρ|k) =

  • eη|k

if ρ = ξ,

  • therwise,

Assume that R is a potential multiplier for T2N and U is a Borel subset of C, then the set BR

U = {(η, ξ) ∈ {0, 1}N × {0, 1}N : λ T2N η,ξ (R) ∈ U}

is Borel in {0, 1}N × {0, 1}N. In particular, BR

U is either countable or

  • f size of the continuum.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

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SLIDE 80

An extension of compact operators by compact operators: If T is maximal, R is a potential multiplier and Λ(R) is a matrix of a ‘countably many entries plus identity’ operator, then Λ(R) is a matrix of a compact plus identity operator We work in H = ℓ2({0, 1}<N) Define a system of almost matrix units T2N = (Tη,ξ : ξ, η ∈ {0, 1}N) by Tη,ξ(eρ|k) =

  • eη|k

if ρ = ξ,

  • therwise,

Assume that R is a potential multiplier for T2N and U is a Borel subset of C, then the set BR

U = {(η, ξ) ∈ {0, 1}N × {0, 1}N : λ T2N η,ξ (R) ∈ U}

is Borel in {0, 1}N × {0, 1}N. In particular, BR

U is either countable or

  • f size of the continuum.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

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SLIDE 81

≪-increasing approximate units a ≪ b iff a = ab,

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

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SLIDE 82

≪-increasing approximate units a ≪ b iff a = ab, an approximate unit (uλ)λ∈Λ ⊆ A1

+ is ≪-increasing iff λ < λ′

implies uλ ≪ uλ′,

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

slide-83
SLIDE 83

≪-increasing approximate units a ≪ b iff a = ab, an approximate unit (uλ)λ∈Λ ⊆ A1

+ is ≪-increasing iff λ < λ′

implies uλ ≪ uλ′, Commutative C*-algebras have such units: elements of A1

+ with

compact support

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

slide-84
SLIDE 84

≪-increasing approximate units a ≪ b iff a = ab, an approximate unit (uλ)λ∈Λ ⊆ A1

+ is ≪-increasing iff λ < λ′

implies uλ ≪ uλ′, Commutative C*-algebras have such units: elements of A1

+ with

compact support (Blackadar’s book) σ-unital C*-algebras have ≪-increasing approximate units

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

slide-85
SLIDE 85

≪-increasing approximate units a ≪ b iff a = ab, an approximate unit (uλ)λ∈Λ ⊆ A1

+ is ≪-increasing iff λ < λ′

implies uλ ≪ uλ′, Commutative C*-algebras have such units: elements of A1

+ with

compact support (Blackadar’s book) σ-unital C*-algebras have ≪-increasing approximate units (Bice +K) ω1-unital C*-algebras have ≪-increasing approximate units

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

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SLIDE 86

≪-increasing approximate units a ≪ b iff a = ab, an approximate unit (uλ)λ∈Λ ⊆ A1

+ is ≪-increasing iff λ < λ′

implies uλ ≪ uλ′, Commutative C*-algebras have such units: elements of A1

+ with

compact support (Blackadar’s book) σ-unital C*-algebras have ≪-increasing approximate units (Bice +K) ω1-unital C*-algebras have ≪-increasing approximate units (Bice+K) There are (scattered) C*-algebras without ≪-increasing approximate units, in particular there are scattered not AF algebras

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

slide-87
SLIDE 87

≪-increasing approximate units a ≪ b iff a = ab, an approximate unit (uλ)λ∈Λ ⊆ A1

+ is ≪-increasing iff λ < λ′

implies uλ ≪ uλ′, Commutative C*-algebras have such units: elements of A1

+ with

compact support (Blackadar’s book) σ-unital C*-algebras have ≪-increasing approximate units (Bice +K) ω1-unital C*-algebras have ≪-increasing approximate units (Bice+K) There are (scattered) C*-algebras without ≪-increasing approximate units, in particular there are scattered not AF algebras (Bice+K) Whether there are (scattered) C*-subalgebras of B(ℓ2) without ≪-increasing approximate units, in particular LF not AF subalgebras of B(ℓ2) is independent.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

slide-88
SLIDE 88

≪-increasing approximate units a ≪ b iff a = ab, an approximate unit (uλ)λ∈Λ ⊆ A1

+ is ≪-increasing iff λ < λ′

implies uλ ≪ uλ′, Commutative C*-algebras have such units: elements of A1

+ with

compact support (Blackadar’s book) σ-unital C*-algebras have ≪-increasing approximate units (Bice +K) ω1-unital C*-algebras have ≪-increasing approximate units (Bice+K) There are (scattered) C*-algebras without ≪-increasing approximate units, in particular there are scattered not AF algebras (Bice+K) Whether there are (scattered) C*-subalgebras of B(ℓ2) without ≪-increasing approximate units, in particular LF not AF subalgebras of B(ℓ2) is independent.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

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SLIDE 89

≪-increasing approximate units a ≪ b iff a = ab, an approximate unit (uλ)λ∈Λ ⊆ A1

+ is ≪-increasing iff λ < λ′

implies uλ ≪ uλ′, Commutative C*-algebras have such units: elements of A1

+ with

compact support (Blackadar’s book) σ-unital C*-algebras have ≪-increasing approximate units (Bice +K) ω1-unital C*-algebras have ≪-increasing approximate units (Bice+K) There are (scattered) C*-algebras without ≪-increasing approximate units, in particular there are scattered not AF algebras (Bice+K) Whether there are (scattered) C*-subalgebras of B(ℓ2) without ≪-increasing approximate units, in particular LF not AF subalgebras of B(ℓ2) is independent.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

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SLIDE 90

Non-≪-unital C*-algebras Define D = {(an)n∈N ∈ ℓ∞(M2) : limn→∞ an ∈ CP0}.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

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SLIDE 91

Non-≪-unital C*-algebras Define D = {(an)n∈N ∈ ℓ∞(M2) : limn→∞ an ∈ CP0}. p = (pn)n∈N and q = (qn)n∈N in D are defined by pn = P0 and qn = P1/n for each n ∈ N.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

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SLIDE 92

Non-≪-unital C*-algebras Define D = {(an)n∈N ∈ ℓ∞(M2) : limn→∞ an ∈ CP0}. p = (pn)n∈N and q = (qn)n∈N in D are defined by pn = P0 and qn = P1/n for each n ∈ N. There is no a ∈ D1

+ with p, q ≪ a.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

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SLIDE 93

Non-≪-unital C*-algebras Define D = {(an)n∈N ∈ ℓ∞(M2) : limn→∞ an ∈ CP0}. p = (pn)n∈N and q = (qn)n∈N in D are defined by pn = P0 and qn = P1/n for each n ∈ N. There is no a ∈ D1

+ with p, q ≪ a.

Given an ω1-tree T for each uncountable branch b through T, define a projection pb ∈ ℓT

∞(D) by

pb(t) =      p if t ∈ b, q if t ∈ b#,

  • therwise.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

slide-94
SLIDE 94

Non-≪-unital C*-algebras Define D = {(an)n∈N ∈ ℓ∞(M2) : limn→∞ an ∈ CP0}. p = (pn)n∈N and q = (qn)n∈N in D are defined by pn = P0 and qn = P1/n for each n ∈ N. There is no a ∈ D1

+ with p, q ≪ a.

Given an ω1-tree T for each uncountable branch b through T, define a projection pb ∈ ℓT

∞(D) by

pb(t) =      p if t ∈ b, q if t ∈ b#,

  • therwise.

Let AT be the C*-algebra generated by (pb)b∈Br(T) in ℓT

∞(D).

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

slide-95
SLIDE 95

Non-≪-unital C*-algebras Define D = {(an)n∈N ∈ ℓ∞(M2) : limn→∞ an ∈ CP0}. p = (pn)n∈N and q = (qn)n∈N in D are defined by pn = P0 and qn = P1/n for each n ∈ N. There is no a ∈ D1

+ with p, q ≪ a.

Given an ω1-tree T for each uncountable branch b through T, define a projection pb ∈ ℓT

∞(D) by

pb(t) =      p if t ∈ b, q if t ∈ b#,

  • therwise.

Let AT be the C*-algebra generated by (pb)b∈Br(T) in ℓT

∞(D).

AT is a scattered C*-algebra without a ≪-increasing approximate unit.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

slide-96
SLIDE 96

Non-≪-unital C*-algebras Define D = {(an)n∈N ∈ ℓ∞(M2) : limn→∞ an ∈ CP0}. p = (pn)n∈N and q = (qn)n∈N in D are defined by pn = P0 and qn = P1/n for each n ∈ N. There is no a ∈ D1

+ with p, q ≪ a.

Given an ω1-tree T for each uncountable branch b through T, define a projection pb ∈ ℓT

∞(D) by

pb(t) =      p if t ∈ b, q if t ∈ b#,

  • therwise.

Let AT be the C*-algebra generated by (pb)b∈Br(T) in ℓT

∞(D).

AT is a scattered C*-algebra without a ≪-increasing approximate unit. Using Q-sets or an embedding of ℘(ω1) into ℘(N)/Fin we can find B ⊆ B(ℓ2) such that AT is a quotient of B, then B has no ≪-increasing approximate unit and B can be LF or even scattered.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

slide-97
SLIDE 97

Non-≪-unital C*-algebras Define D = {(an)n∈N ∈ ℓ∞(M2) : limn→∞ an ∈ CP0}. p = (pn)n∈N and q = (qn)n∈N in D are defined by pn = P0 and qn = P1/n for each n ∈ N. There is no a ∈ D1

+ with p, q ≪ a.

Given an ω1-tree T for each uncountable branch b through T, define a projection pb ∈ ℓT

∞(D) by

pb(t) =      p if t ∈ b, q if t ∈ b#,

  • therwise.

Let AT be the C*-algebra generated by (pb)b∈Br(T) in ℓT

∞(D).

AT is a scattered C*-algebra without a ≪-increasing approximate unit. Using Q-sets or an embedding of ℘(ω1) into ℘(N)/Fin we can find B ⊆ B(ℓ2) such that AT is a quotient of B, then B has no ≪-increasing approximate unit and B can be LF or even scattered.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

slide-98
SLIDE 98

Non-≪-unital C*-algebras Define D = {(an)n∈N ∈ ℓ∞(M2) : limn→∞ an ∈ CP0}. p = (pn)n∈N and q = (qn)n∈N in D are defined by pn = P0 and qn = P1/n for each n ∈ N. There is no a ∈ D1

+ with p, q ≪ a.

Given an ω1-tree T for each uncountable branch b through T, define a projection pb ∈ ℓT

∞(D) by

pb(t) =      p if t ∈ b, q if t ∈ b#,

  • therwise.

Let AT be the C*-algebra generated by (pb)b∈Br(T) in ℓT

∞(D).

AT is a scattered C*-algebra without a ≪-increasing approximate unit. Using Q-sets or an embedding of ℘(ω1) into ℘(N)/Fin we can find B ⊆ B(ℓ2) such that AT is a quotient of B, then B has no ≪-increasing approximate unit and B can be LF or even scattered.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

slide-99
SLIDE 99

Open problems Are there in ZFC scattered C*-subalgebras of B(ℓ2) of density 2ω without a nonseparable commutative subalgebra?

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

slide-100
SLIDE 100

Open problems Are there in ZFC scattered C*-subalgebras of B(ℓ2) of density 2ω without a nonseparable commutative subalgebra? Are there in ZFC scattered algebras which are not AF of all densities ≥ ω2?

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

slide-101
SLIDE 101

Open problems Are there in ZFC scattered C*-subalgebras of B(ℓ2) of density 2ω without a nonseparable commutative subalgebra? Are there in ZFC scattered algebras which are not AF of all densities ≥ ω2? Are there is ZFC C*-subalgebras of B(ℓ2(ω1)) without ≪-increasing approximate units?

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

slide-102
SLIDE 102

Open problems Are there in ZFC scattered C*-subalgebras of B(ℓ2) of density 2ω without a nonseparable commutative subalgebra? Are there in ZFC scattered algebras which are not AF of all densities ≥ ω2? Are there is ZFC C*-subalgebras of B(ℓ2(ω1)) without ≪-increasing approximate units? Does the existence of a C*-subalgebra of B(ℓ2) without ≪-increasing approximate unit follows from the negation of CH alone?

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

slide-103
SLIDE 103

Open problems Are there in ZFC scattered C*-subalgebras of B(ℓ2) of density 2ω without a nonseparable commutative subalgebra? Are there in ZFC scattered algebras which are not AF of all densities ≥ ω2? Are there is ZFC C*-subalgebras of B(ℓ2(ω1)) without ≪-increasing approximate units? Does the existence of a C*-subalgebra of B(ℓ2) without ≪-increasing approximate unit follows from the negation of CH alone? as above for algebras which are LF not AF.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

slide-104
SLIDE 104

Open problems Are there in ZFC scattered C*-subalgebras of B(ℓ2) of density 2ω without a nonseparable commutative subalgebra? Are there in ZFC scattered algebras which are not AF of all densities ≥ ω2? Are there is ZFC C*-subalgebras of B(ℓ2(ω1)) without ≪-increasing approximate units? Does the existence of a C*-subalgebra of B(ℓ2) without ≪-increasing approximate unit follows from the negation of CH alone? as above for algebras which are LF not AF. as above for algebras which are scattered not AF.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

slide-105
SLIDE 105

Open problems Are there in ZFC scattered C*-subalgebras of B(ℓ2) of density 2ω without a nonseparable commutative subalgebra? Are there in ZFC scattered algebras which are not AF of all densities ≥ ω2? Are there is ZFC C*-subalgebras of B(ℓ2(ω1)) without ≪-increasing approximate units? Does the existence of a C*-subalgebra of B(ℓ2) without ≪-increasing approximate unit follows from the negation of CH alone? as above for algebras which are LF not AF. as above for algebras which are scattered not AF.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

slide-106
SLIDE 106

Open problems Are there in ZFC scattered C*-subalgebras of B(ℓ2) of density 2ω without a nonseparable commutative subalgebra? Are there in ZFC scattered algebras which are not AF of all densities ≥ ω2? Are there is ZFC C*-subalgebras of B(ℓ2(ω1)) without ≪-increasing approximate units? Does the existence of a C*-subalgebra of B(ℓ2) without ≪-increasing approximate unit follows from the negation of CH alone? as above for algebras which are LF not AF. as above for algebras which are scattered not AF.

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

slide-107
SLIDE 107

References:

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 13 / 13

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SLIDE 108

References:

  • T. Bice, P

. Koszmider, A note on the Akemann-Doner and Farah-Wofsey constructions, Proc. Amer. Math. Soc. 145 (2017),

  • no. 2, 681–687.
  • T. Bice, P

. Koszmider, C*-algebras with and without ≪-increasing approximate units. Matharxiv.

  • S. Ghasemi, P

. Koszmider; An extension of compact operators by compact operators with no nontrivial multipliers. Matharxiv.

  • S. Ghasemi, P

. Koszmider, Noncommutative Cantor-Bendixson derivatives and scattered C*-algebras. Matharxiv.

  • S. Ghasemi, P

. Koszmider, On the stability of thin-tall scattered C*-algebras (final stages of preparation).

Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 13 / 13