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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

  1. 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

  2. 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 ω 1 but not bigger Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 4 / 13

  3. 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 ω 1 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

  4. 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 ω 1 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

  5. 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 ω 1 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

  6. 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 ω 1 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

  7. C*-algebras without big commutative subalgebras Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

  8. C*-algebras without big commutative subalgebras Results Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 5 / 13

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. Akemann-Donner algebras: Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

  17. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

  18. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

  19. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) If B ⊆ A is nonseparable and commutative, there are ( q ( n ) : n ∈ N ) ⊆ P 1 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

  20. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) If B ⊆ A is nonseparable and commutative, there are ( q ( n ) : n ∈ N ) ⊆ P 1 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 ′ n ∈ F q ( n ) = p F . lim Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

  21. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) If B ⊆ A is nonseparable and commutative, there are ( q ( n ) : n ∈ N ) ⊆ P 1 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 ′ n ∈ F q ( n ) = p F . lim For two uncountable G , H ⊆ F ′ there are s , r ∈ P 1 such that � p F − s � < � r − s � / 2 for all F ∈ G and � p F − r � < � r − s � / 2 for all F ∈ H and Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 6 / 13

  22. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) If B ⊆ A is nonseparable and commutative, there are ( q ( n ) : n ∈ N ) ⊆ P 1 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 ′ n ∈ F q ( n ) = p F . lim For two uncountable G , H ⊆ F ′ there are s , r ∈ P 1 such that � p F − s � < � r − s � / 2 for all F ∈ G and � p F − 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

  23. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) If B ⊆ A is nonseparable and commutative, there are ( q ( n ) : n ∈ N ) ⊆ P 1 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 ′ n ∈ F q ( n ) = p F . lim For two uncountable G , H ⊆ F ′ there are s , r ∈ P 1 such that � p F − s � < � r − s � / 2 for all F ∈ G and � p F − 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

  24. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) If B ⊆ A is nonseparable and commutative, there are ( q ( n ) : n ∈ N ) ⊆ P 1 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 ′ n ∈ F q ( n ) = p F . lim For two uncountable G , H ⊆ F ′ there are s , r ∈ P 1 such that � p F − s � < � r − s � / 2 for all F ∈ G and � p F − 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

  25. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) If B ⊆ A is nonseparable and commutative, there are ( q ( n ) : n ∈ N ) ⊆ P 1 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 ′ n ∈ F q ( n ) = p F . lim For two uncountable G , H ⊆ F ′ there are s , r ∈ P 1 such that � p F − s � < � r − s � / 2 for all F ∈ G and � p F − 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

  26. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) If B ⊆ A is nonseparable and commutative, there are ( q ( n ) : n ∈ N ) ⊆ P 1 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 ′ n ∈ F q ( n ) = p F . lim For two uncountable G , H ⊆ F ′ there are s , r ∈ P 1 such that � p F − s � < � r − s � / 2 for all F ∈ G and � p F − 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

  27. Akemann-Donner algebras: Fix an uncountable almost disjoint family F of subsets of N and distinct rank 1 projections P = ( P F : F ∈ F ) ⊆ P 1 ⊆ M 2 such that � p F − p 0 � < 1 / 4 for some p 0 ∈ P 1 A = A F , P ⊆ ℓ ∞ ( M 2 ) generated by ( p F χ F : F ∈ F ) and c 0 ( M 2 ) If B ⊆ A is nonseparable and commutative, there are ( q ( n ) : n ∈ N ) ⊆ P 1 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 ′ n ∈ F q ( n ) = p F . lim For two uncountable G , H ⊆ F ′ there are s , r ∈ P 1 such that � p F − s � < � r − s � / 2 for all F ∈ G and � p F − 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

  28. Counterexamples to permanence of stability: Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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 ( ℓ 2 ) satisfying the following short exact sequence 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 , − where ι [ K ( ℓ 2 )] 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

  35. 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 ( ℓ 2 ) satisfying the following short exact sequence 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 , − where ι [ K ( ℓ 2 )] 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<ω 1 I α where the sequence is increasing and I α + 1 / I α ≡ K ( ℓ 2 ) and so each I α is stable. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

  36. 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 ( ℓ 2 ) satisfying the following short exact sequence 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 , − where ι [ K ( ℓ 2 )] 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<ω 1 I α where the sequence is increasing and I α + 1 / I α ≡ K ( ℓ 2 ) and so each I α is stable. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

  37. 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 ( ℓ 2 ) satisfying the following short exact sequence 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 , − where ι [ K ( ℓ 2 )] 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<ω 1 I α where the sequence is increasing and I α + 1 / I α ≡ K ( ℓ 2 ) and so each I α is stable. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 7 / 13

  38. An extension of compact operators by compact operators: Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  39. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that holds for every α, β, ξ, η < κ , Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  40. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 holds for every α, β, ξ, η < κ , Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  41. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 T β,α T η,ξ = K δ α,η T β,ξ . 2 holds for every α, β, ξ, η < κ , Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  42. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 T β,α T η,ξ = K δ α,η T β,ξ . 2 holds for every α, β, ξ, η < κ , Let A = A ( T ) be the C ∗ -subalgebra of B ( ℓ 2 ) generated by K and T = ( T η,ξ : ξ, η ∈ 2 ω ) Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  43. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 T β,α T η,ξ = K δ α,η T β,ξ . 2 holds for every α, β, ξ, η < κ , Let A = A ( T ) be the C ∗ -subalgebra of B ( ℓ 2 ) generated by K and T = ( T η,ξ : ξ, η ∈ 2 ω ) Every multiplier R ∈ M ( A ) ⊆ B ( ℓ 2 ) defines a 2 ω × 2 ω -matrix Λ( R ) satisfying T ξ,ξ RT η,η = K Λ T ( R ) ξ,η T ξ,η . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  44. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 T β,α T η,ξ = K δ α,η T β,ξ . 2 holds for every α, β, ξ, η < κ , Let A = A ( T ) be the C ∗ -subalgebra of B ( ℓ 2 ) generated by K and T = ( T η,ξ : ξ, η ∈ 2 ω ) Every multiplier R ∈ M ( A ) ⊆ B ( ℓ 2 ) defines a 2 ω × 2 ω -matrix Λ( R ) satisfying T ξ,ξ RT η,η = K Λ T ( R ) ξ,η T ξ,η . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  45. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 T β,α T η,ξ = K δ α,η T β,ξ . 2 holds for every α, β, ξ, η < κ , Let A = A ( T ) be the C ∗ -subalgebra of B ( ℓ 2 ) generated by K and T = ( T η,ξ : ξ, η ∈ 2 ω ) Every multiplier R ∈ M ( A ) ⊆ B ( ℓ 2 ) defines a 2 ω × 2 ω -matrix Λ( R ) satisfying T ξ,ξ RT η,η = K Λ T ( R ) ξ,η T ξ,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

  46. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 T β,α T η,ξ = K δ α,η T β,ξ . 2 holds for every α, β, ξ, η < κ , Let A = A ( T ) be the C ∗ -subalgebra of B ( ℓ 2 ) generated by K and T = ( T η,ξ : ξ, η ∈ 2 ω ) Every multiplier R ∈ M ( A ) ⊆ B ( ℓ 2 ) defines a 2 ω × 2 ω -matrix Λ( R ) satisfying T ξ,ξ RT η,η = K Λ T ( R ) ξ,η T ξ,η . An R as above is called a potential multiplier for T Λ T ( R ) is a matrix of a bounded operator in B ( ℓ 2 ( 2 ω )) Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  47. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 T β,α T η,ξ = K δ α,η T β,ξ . 2 holds for every α, β, ξ, η < κ , Let A = A ( T ) be the C ∗ -subalgebra of B ( ℓ 2 ) generated by K and T = ( T η,ξ : ξ, η ∈ 2 ω ) Every multiplier R ∈ M ( A ) ⊆ B ( ℓ 2 ) defines a 2 ω × 2 ω -matrix Λ( R ) satisfying T ξ,ξ RT η,η = K Λ T ( R ) ξ,η T ξ,η . An R as above is called a potential multiplier for T Λ T ( R ) is a matrix of a bounded operator in B ( ℓ 2 ( 2 ω )) If T is maximal, then Λ( R ) is a matrix of a compact plus identity operator, iff R ∈ � A ( T ) . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  48. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 T β,α T η,ξ = K δ α,η T β,ξ . 2 holds for every α, β, ξ, η < κ , Let A = A ( T ) be the C ∗ -subalgebra of B ( ℓ 2 ) generated by K and T = ( T η,ξ : ξ, η ∈ 2 ω ) Every multiplier R ∈ M ( A ) ⊆ B ( ℓ 2 ) defines a 2 ω × 2 ω -matrix Λ( R ) satisfying T ξ,ξ RT η,η = K Λ T ( R ) ξ,η T ξ,η . An R as above is called a potential multiplier for T Λ T ( R ) is a matrix of a bounded operator in B ( ℓ 2 ( 2 ω )) If T is maximal, then Λ( R ) is a matrix of a compact plus identity operator, iff R ∈ � A ( T ) . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  49. An extension of compact operators by compact operators: An extension 0 → K ( ℓ 2 ) ι → A → K ( ℓ 2 ( 2 ω )) → 0 − corresponds to choice of almost matrix units, i.e., noncompact operators T = ( T η,ξ : ξ, η ∈ 2 ω ) such that η,ξ = K T ξ,η , T ∗ 1 T β,α T η,ξ = K δ α,η T β,ξ . 2 holds for every α, β, ξ, η < κ , Let A = A ( T ) be the C ∗ -subalgebra of B ( ℓ 2 ) generated by K and T = ( T η,ξ : ξ, η ∈ 2 ω ) Every multiplier R ∈ M ( A ) ⊆ B ( ℓ 2 ) defines a 2 ω × 2 ω -matrix Λ( R ) satisfying T ξ,ξ RT η,η = K Λ T ( R ) ξ,η T ξ,η . An R as above is called a potential multiplier for T Λ T ( R ) is a matrix of a bounded operator in B ( ℓ 2 ( 2 ω )) If T is maximal, then Λ( R ) is a matrix of a compact plus identity operator, iff R ∈ � A ( T ) . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 8 / 13

  50. An extension of compact operators by compact operators: Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

  51. 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

  52. 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 = ℓ 2 ( { 0 , 1 } < N ) Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

  53. 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 = ℓ 2 ( { 0 , 1 } < N ) Define a system of almost matrix units T 2 N = ( T η,ξ : ξ, η ∈ { 0 , 1 } N ) by � e η | k if ρ = ξ, T η,ξ ( e ρ | k ) = 0 otherwise, Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

  54. 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 = ℓ 2 ( { 0 , 1 } < N ) Define a system of almost matrix units T 2 N = ( T η,ξ : ξ, η ∈ { 0 , 1 } N ) by � e η | k if ρ = ξ, T η,ξ ( e ρ | k ) = 0 otherwise, Assume that R is a potential multiplier for T 2 N and U is a Borel subset of C , then the set U = { ( η, ξ ) ∈ { 0 , 1 } N × { 0 , 1 } N : λ T 2 N B R η,ξ ( R ) ∈ U } is Borel in { 0 , 1 } N × { 0 , 1 } N . In particular, B R U is either countable or of size of the continuum. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

  55. 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 = ℓ 2 ( { 0 , 1 } < N ) Define a system of almost matrix units T 2 N = ( T η,ξ : ξ, η ∈ { 0 , 1 } N ) by � e η | k if ρ = ξ, T η,ξ ( e ρ | k ) = 0 otherwise, Assume that R is a potential multiplier for T 2 N and U is a Borel subset of C , then the set U = { ( η, ξ ) ∈ { 0 , 1 } N × { 0 , 1 } N : λ T 2 N B R η,ξ ( R ) ∈ U } is Borel in { 0 , 1 } N × { 0 , 1 } N . In particular, B R U is either countable or of size of the continuum. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

  56. 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 = ℓ 2 ( { 0 , 1 } < N ) Define a system of almost matrix units T 2 N = ( T η,ξ : ξ, η ∈ { 0 , 1 } N ) by � e η | k if ρ = ξ, T η,ξ ( e ρ | k ) = 0 otherwise, Assume that R is a potential multiplier for T 2 N and U is a Borel subset of C , then the set U = { ( η, ξ ) ∈ { 0 , 1 } N × { 0 , 1 } N : λ T 2 N B R η,ξ ( R ) ∈ U } is Borel in { 0 , 1 } N × { 0 , 1 } N . In particular, B R U is either countable or of size of the continuum. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 9 / 13

  57. ≪ -increasing approximate units a ≪ b iff a = ab , Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

  58. ≪ -increasing approximate units a ≪ b iff a = ab , an approximate unit ( u λ ) λ ∈ Λ ⊆ A 1 + is ≪ -increasing iff λ < λ ′ implies u λ ≪ u λ ′ , Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

  59. ≪ -increasing approximate units a ≪ b iff a = ab , an approximate unit ( u λ ) λ ∈ Λ ⊆ A 1 + is ≪ -increasing iff λ < λ ′ implies u λ ≪ u λ ′ , Commutative C*-algebras have such units: elements of A 1 + with compact support Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

  60. ≪ -increasing approximate units a ≪ b iff a = ab , an approximate unit ( u λ ) λ ∈ Λ ⊆ A 1 + is ≪ -increasing iff λ < λ ′ implies u λ ≪ u λ ′ , Commutative C*-algebras have such units: elements of A 1 + 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

  61. ≪ -increasing approximate units a ≪ b iff a = ab , an approximate unit ( u λ ) λ ∈ Λ ⊆ A 1 + is ≪ -increasing iff λ < λ ′ implies u λ ≪ u λ ′ , Commutative C*-algebras have such units: elements of A 1 + with compact support (Blackadar’s book) σ -unital C*-algebras have ≪ -increasing approximate units (Bice +K) ω 1 -unital C*-algebras have ≪ -increasing approximate units Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

  62. ≪ -increasing approximate units a ≪ b iff a = ab , an approximate unit ( u λ ) λ ∈ Λ ⊆ A 1 + is ≪ -increasing iff λ < λ ′ implies u λ ≪ u λ ′ , Commutative C*-algebras have such units: elements of A 1 + with compact support (Blackadar’s book) σ -unital C*-algebras have ≪ -increasing approximate units (Bice +K) ω 1 -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

  63. ≪ -increasing approximate units a ≪ b iff a = ab , an approximate unit ( u λ ) λ ∈ Λ ⊆ A 1 + is ≪ -increasing iff λ < λ ′ implies u λ ≪ u λ ′ , Commutative C*-algebras have such units: elements of A 1 + with compact support (Blackadar’s book) σ -unital C*-algebras have ≪ -increasing approximate units (Bice +K) ω 1 -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 ( ℓ 2 ) without ≪ -increasing approximate units, in particular LF not AF subalgebras of B ( ℓ 2 ) is independent. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

  64. ≪ -increasing approximate units a ≪ b iff a = ab , an approximate unit ( u λ ) λ ∈ Λ ⊆ A 1 + is ≪ -increasing iff λ < λ ′ implies u λ ≪ u λ ′ , Commutative C*-algebras have such units: elements of A 1 + with compact support (Blackadar’s book) σ -unital C*-algebras have ≪ -increasing approximate units (Bice +K) ω 1 -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 ( ℓ 2 ) without ≪ -increasing approximate units, in particular LF not AF subalgebras of B ( ℓ 2 ) is independent. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

  65. ≪ -increasing approximate units a ≪ b iff a = ab , an approximate unit ( u λ ) λ ∈ Λ ⊆ A 1 + is ≪ -increasing iff λ < λ ′ implies u λ ≪ u λ ′ , Commutative C*-algebras have such units: elements of A 1 + with compact support (Blackadar’s book) σ -unital C*-algebras have ≪ -increasing approximate units (Bice +K) ω 1 -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 ( ℓ 2 ) without ≪ -increasing approximate units, in particular LF not AF subalgebras of B ( ℓ 2 ) is independent. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 10 / 13

  66. Non- ≪ -unital C*-algebras Define D = { ( a n ) n ∈ N ∈ ℓ ∞ ( M 2 ) : lim n →∞ a n ∈ C P 0 } . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

  67. Non- ≪ -unital C*-algebras Define D = { ( a n ) n ∈ N ∈ ℓ ∞ ( M 2 ) : lim n →∞ a n ∈ C P 0 } . p = ( p n ) n ∈ N and q = ( q n ) n ∈ N in D are defined by p n = P 0 and q n = P 1 / n for each n ∈ N . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

  68. Non- ≪ -unital C*-algebras Define D = { ( a n ) n ∈ N ∈ ℓ ∞ ( M 2 ) : lim n →∞ a n ∈ C P 0 } . p = ( p n ) n ∈ N and q = ( q n ) n ∈ N in D are defined by p n = P 0 and q n = P 1 / n for each n ∈ N . There is no a ∈ D 1 + with p , q ≪ a . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

  69. Non- ≪ -unital C*-algebras Define D = { ( a n ) n ∈ N ∈ ℓ ∞ ( M 2 ) : lim n →∞ a n ∈ C P 0 } . p = ( p n ) n ∈ N and q = ( q n ) n ∈ N in D are defined by p n = P 0 and q n = P 1 / n for each n ∈ N . There is no a ∈ D 1 + with p , q ≪ a . Given an ω 1 -tree T for each uncountable branch b through T , define a projection p b ∈ ℓ T ∞ ( D ) by   p if t ∈ b ,  if t ∈ b # , p b ( t ) = q   0 otherwise . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

  70. Non- ≪ -unital C*-algebras Define D = { ( a n ) n ∈ N ∈ ℓ ∞ ( M 2 ) : lim n →∞ a n ∈ C P 0 } . p = ( p n ) n ∈ N and q = ( q n ) n ∈ N in D are defined by p n = P 0 and q n = P 1 / n for each n ∈ N . There is no a ∈ D 1 + with p , q ≪ a . Given an ω 1 -tree T for each uncountable branch b through T , define a projection p b ∈ ℓ T ∞ ( D ) by   p if t ∈ b ,  if t ∈ b # , p b ( t ) = q   0 otherwise . Let A T be the C*-algebra generated by ( p b ) b ∈ Br ( T ) in ℓ T ∞ ( D ) . Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

  71. Non- ≪ -unital C*-algebras Define D = { ( a n ) n ∈ N ∈ ℓ ∞ ( M 2 ) : lim n →∞ a n ∈ C P 0 } . p = ( p n ) n ∈ N and q = ( q n ) n ∈ N in D are defined by p n = P 0 and q n = P 1 / n for each n ∈ N . There is no a ∈ D 1 + with p , q ≪ a . Given an ω 1 -tree T for each uncountable branch b through T , define a projection p b ∈ ℓ T ∞ ( D ) by   p if t ∈ b ,  if t ∈ b # , p b ( t ) = q   0 otherwise . Let A T be the C*-algebra generated by ( p b ) b ∈ Br ( T ) in ℓ T ∞ ( D ) . A T is a scattered C*-algebra without a ≪ -increasing approximate unit. Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 11 / 13

  72. Non- ≪ -unital C*-algebras Define D = { ( a n ) n ∈ N ∈ ℓ ∞ ( M 2 ) : lim n →∞ a n ∈ C P 0 } . p = ( p n ) n ∈ N and q = ( q n ) n ∈ N in D are defined by p n = P 0 and q n = P 1 / n for each n ∈ N . There is no a ∈ D 1 + with p , q ≪ a . Given an ω 1 -tree T for each uncountable branch b through T , define a projection p b ∈ ℓ T ∞ ( D ) by   p if t ∈ b ,  if t ∈ b # , p b ( t ) = q   0 otherwise . Let A T be the C*-algebra generated by ( p b ) b ∈ Br ( T ) in ℓ T ∞ ( D ) . A T is a scattered C*-algebra without a ≪ -increasing approximate unit. Using Q -sets or an embedding of ℘ ( ω 1 ) into ℘ ( N ) / Fin we can find B ⊆ B ( ℓ 2 ) such that A T 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

  73. Non- ≪ -unital C*-algebras Define D = { ( a n ) n ∈ N ∈ ℓ ∞ ( M 2 ) : lim n →∞ a n ∈ C P 0 } . p = ( p n ) n ∈ N and q = ( q n ) n ∈ N in D are defined by p n = P 0 and q n = P 1 / n for each n ∈ N . There is no a ∈ D 1 + with p , q ≪ a . Given an ω 1 -tree T for each uncountable branch b through T , define a projection p b ∈ ℓ T ∞ ( D ) by   p if t ∈ b ,  if t ∈ b # , p b ( t ) = q   0 otherwise . Let A T be the C*-algebra generated by ( p b ) b ∈ Br ( T ) in ℓ T ∞ ( D ) . A T is a scattered C*-algebra without a ≪ -increasing approximate unit. Using Q -sets or an embedding of ℘ ( ω 1 ) into ℘ ( N ) / Fin we can find B ⊆ B ( ℓ 2 ) such that A T 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

  74. Non- ≪ -unital C*-algebras Define D = { ( a n ) n ∈ N ∈ ℓ ∞ ( M 2 ) : lim n →∞ a n ∈ C P 0 } . p = ( p n ) n ∈ N and q = ( q n ) n ∈ N in D are defined by p n = P 0 and q n = P 1 / n for each n ∈ N . There is no a ∈ D 1 + with p , q ≪ a . Given an ω 1 -tree T for each uncountable branch b through T , define a projection p b ∈ ℓ T ∞ ( D ) by   p if t ∈ b ,  if t ∈ b # , p b ( t ) = q   0 otherwise . Let A T be the C*-algebra generated by ( p b ) b ∈ Br ( T ) in ℓ T ∞ ( D ) . A T is a scattered C*-algebra without a ≪ -increasing approximate unit. Using Q -sets or an embedding of ℘ ( ω 1 ) into ℘ ( N ) / Fin we can find B ⊆ B ( ℓ 2 ) such that A T 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

  75. Open problems Are there in ZFC scattered C*-subalgebras of B ( ℓ 2 ) of density 2 ω without a nonseparable commutative subalgebra? Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

  76. Open problems Are there in ZFC scattered C*-subalgebras of B ( ℓ 2 ) of density 2 ω without a nonseparable commutative subalgebra? Are there in ZFC scattered algebras which are not AF of all densities ≥ ω 2 ? Piotr Koszmider (IMPAN) LF algebras and transfinite structures Houston, 01-08-2017 12 / 13

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