On classification of non-unital amenable simple C*-algebras, II

Gong, Guihua; Lin, Huaxin

doi:10.1016/j.geomphys.2020.103865

Cited by 34 publications

(56 citation statements)

References 142 publications

(585 reference statements)

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“…This second condition has a geometric flavour and generalises the notation of finite covering dimension for topological spaces. Recent results ( [29,23,24,30]) are now converging on a similar classification result in the stably projectionless case; the state of the art will be discussed below.…”

Section: Introductionmentioning

confidence: 73%

“…We end this introduction with a discussion of the state of the art for the classification of simple, stably projectionless C * -algebras. As mentioned above, there has been impressive progress in recent years ( [29,30,23]). As in the unital case, the classification is via a functor constructed from the K-theory and the tracial data of the C * -algebra; this functor is called the Elliott invariant and is typically denoted Ell(•) (see [30,Definition 2.9] for a precise definition).…”

Section: Theorem Bmentioning

confidence: 99%

“…By combining Theorem A with [30,Theorem 1.2], one obtains a classification of simple, separable, nuclear C * -algebras in the UCT class that tensorially absorb the C * -algebra Z 0 -a stably projectionless analogue of the Jiang-Su algebra introduced in [30, Definition 8.1].…”

Section: Theorem Bmentioning

confidence: 99%

“…Corollary D reduces to the celebrated Kirchberg-Phillips classification ( [36,48]) in the traceless case and is otherwise a result about stably projectionless C * -algebras. For these C * -algebras, the difference between Z 0 -stability and Z-stability, roughly speaking, comes down to how complex the interaction between the K-theory and traces is allowed to be; see [30] for more details.…”

Section: Theorem Bmentioning

confidence: 99%

“…(This dichotomy follows from Brown's Theorem ( [10]) and is discussed further below.) Recall that a C *algebra A is said to be stably projectionless if there are no non-zero projections in the matrix amplification M n (A) for any n ∈ N. Stably projectionless, simple, separable, nuclear C * -algebras arise naturally as crossed products ( [40]), and can also be constructed using inductive limits with a wide variety of K-theoretic and tracial invariants occurring ( [3,49,64,33,29,30,24,25]).…”

Section: Introductionmentioning

confidence: 99%

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Nuclear dimension of simple stably projectionless C∗-algebras

Castillejos

Evington

2020

Analysis & PDE

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We prove that Z-stable, simple, separable, nuclear, non-unital C * -algebras have nuclear dimension at most 1. This completes the equivalence between finite nuclear dimension and Zstability for simple, separable, nuclear, non-elementary C * -algebras.1 i.e. (a) A is a simple, separable, nuclear and Z-stable C * -algebra if and only if A ⊗ K is likewise, and (b) dim nuc A ≤ 1 if and only if dim nuc (A ⊗ K) ≤ 1; see Proposition 2.3 for details and references.

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Section: Introductionmentioning

confidence: 73%

Section: Theorem Bmentioning

confidence: 99%

Section: Theorem Bmentioning

confidence: 99%

Section: Theorem Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nuclear dimension of simple stably projectionless C∗-algebras

Castillejos

Evington

2020

Analysis & PDE

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Equivariant $${\mathcal {Z}}$$-stability for single automorphisms on simple $$C^*$$-algebras with tractable trace simplices

Wouters

2023

Math. Z.

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The Classification of Rokhlin Flows on $$\mathrm {C}^{*}$$-algebras

Szabó

2020

Commun. Math. Phys.

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We study flows on C * -algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C * -algebras satisfying certain technical properties, which hold for many C * -algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto's conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, O∞-absorbing C * -algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable KK-contractible C * -algebras: Two Rokhlin flows on such a C * -algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate. Contents 19 4. O ∞ -absorbing C * -algebras 30 5. Simple KK-contractible algebras 42 References 54

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On classification of non-unital amenable simple C*-algebras, II

Cited by 34 publications

References 142 publications

Nuclear dimension of simple stably projectionless C∗-algebras

Nuclear dimension of simple stably projectionless C∗-algebras

Equivariant $${\mathcal {Z}}$$-stability for single automorphisms on simple $$C^*$$-algebras with tractable trace simplices

The Classification of Rokhlin Flows on $$\mathrm {C}^{*}$$-algebras

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