C*-algebras with and without ≪-increasing approximate units

Bice, Tristan; Koszmider, Piotr

doi:10.1016/j.jmaa.2019.05.020

Cited by 3 publications

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“…• There are 2-subhomogenous non-AF extensions of nonseparable AF-algebras by AF-algebras (Theorem 1.12 of [2]). • There are nonstable C * -algebras A satisfying the following short exact sequence:…”

Section: Introductionmentioning

confidence: 99%

A nonstable 𝐶*-algebra with an elementary essential composition series

Ghasemi

Koszmider

2019

Proc. Amer. Math. Soc.

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A C * -algebra A is said to be stable if it is isomorphic to A ⊗ K(ℓ 2 ). Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C * -algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra A of B(ℓ 2 ), which is the inductive limit of length ω 1 of its separable stable ideals Iα (α < ω 1 ) satisfying I α+1 /Iα ∼ = K(ℓ 2 ) for each α < ω 1 , while A is not stable. The sequence (Iα) α≤ω 1 is the GCR composition series of A which in this case coincides with the Cantor-Bendixson composition series as a scattered C *algebra. A has the property that all of its proper two-sided ideals are listed as Iαs for some α < ω 1 and therefore the family of stable ideals of A has no maximal element.By taking A ′ = A ⊗ K(ℓ 2 ) we obtain a stable C * -algebra with analogous composition series (Jα)α<ω 1 whose ideals Jαs are isomorphic to Iαs for each α < ω 1 . In particular, there are nonisomorphic scattered C * -algebras whose GCR composition series (Iα) α≤ω 1 satisfy I α+1 /Iα ∼ = K(ℓ 2 ) for all α < ω 1 , for which the composition series differ first at α = ω 1 .Proof. Combine Theorems 2.5, 3.10 and 4.2.

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Section: Introductionmentioning

confidence: 99%

A nonstable 𝐶*-algebra with an elementary essential composition series

Ghasemi

Koszmider

2019

Proc. Amer. Math. Soc.

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“…We adopt the terminology of [23] that is, the above notion is called locally finite dimensional (LF). Morover it was recently proved that the properties LF and AF are not equivalent even for for scattered C * -algebras ( [6]).…”

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confidence: 99%

Noncommutative Cantor–Bendixson derivatives and scattered C⁎-algebras

Ghasemi

Koszmider

2018

Topology and its Applications

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We analyze the sequence obtained by consecutive applications of the Cantor-Bendixson derivative for a noncommutative scattered C * -algebra A, using the ideal I At (A) generated by the minimal projections of A. With its help, we present some fundamental results concerning scattered C * -algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the "cardinal sequences" programme in the classical topology, in the noncommutative context. This leads to some new constructions of noncommutative scattered C * -algebras and new open problems. In particular, we construct a type I C * -algebra which is the inductive limit of stable ideals Aα, along an uncountable limit ordinal λ, such that A α+1 /Aα is * -isomorphic to the algebra of all compact operators on a separable Hilbert space and A α+1 is σ-unital and stable for each α < λ, but A is not stable and where all ideals of A are of the form Aα. In particular, A is a nonseparable C * -algebra with no ideal which is maximal among the stable ideals. This answers a question of M. Rørdam in the nonseparable case. All the above C * -algebras Aαs and A satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers [25,26].Proof. We prove J α = I α ⊗ K(ℓ 2 ) by induction on α ≤ ht(A). For α = 1 this follows from Lemma 5.2. At a successor ordinal by Lemma 5.2 and the inductive assumption we have J α+1 /J α =

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Soft operators in C*-algebras

Thiel,

Vilalta

2024

Journal of Mathematical Analysis and Applications

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C*-algebras with and without ≪-increasing approximate units

Cited by 3 publications

References 17 publications

A nonstable 𝐶*-algebra with an elementary essential composition series

A nonstable 𝐶*-algebra with an elementary essential composition series

Noncommutative Cantor–Bendixson derivatives and scattered C⁎-algebras

Soft operators in C*-algebras

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