On the classification of simple amenable
                    C*-algebras with finite decomposition rank

Elliott, George A.; Niu, Zhuang

doi:10.1090/conm/671/13506

Cited by 104 publications

(96 citation statements)

References 44 publications

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“…By Theorem 7.6, C 0 (Y ) R is stable. By the main result of [TWW15] every trace on C 0 (Y ) R is quasidiagonal, so if the crossed product is the stabilization of a unital C * -algebra the main result of [EGLN15] applies and yields classification. The remaining statements follow from Connes' Thom isomorphism and Corollary 2 of [Con81] (and its proof); see also [KP06] for a more detailed account of the Ruelle-Sullivan map associated to an invariant measure.…”

Section: Corollary 102 Let Y Be a Locally Compact And Metrizable Spmentioning

confidence: 98%

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Rokhlin Dimension for Flows

Hirshberg

Szabó

Winter

et al. 2016

Commun. Math. Phys.

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We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C * -algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C * -algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C * -algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product C * -algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological K -theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map).

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Section: Corollary 102 Let Y Be a Locally Compact And Metrizable Spmentioning

confidence: 98%

“…[GLN15,EGLN15,Win16] and [TWW15]-it is particularly relevant how Rokhlin dimension behaves in connection with nuclear dimension in the sense of [WZ10], and with D-absorption, where D is a strongly self-absorbing C * -algebra; cf. [TW07].…”

Section: Introductionmentioning

confidence: 99%

Rokhlin Dimension for Flows

Hirshberg

Szabó

Winter

et al. 2016

Commun. Math. Phys.

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“…In this way, Z-stability postulates itself as a regularity property for C * -algebras. For the said algebras that furthermore fall in the UCT class and have finite nuclear dimension, classification is now complete by the work of many authors (see [GLN15], [EGLN15], [TWW15] and the references therein).…”

Section: Strongly Self-absorbing Cmentioning

confidence: 99%

Tensor products and regularity properties of Cuntz semigroups

2018

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Abstract. The Cuntz semigroup of a C * -algebra is an important invariant in the structure and classification theory of C * -algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups.Given a C * -algebra A, its (concrete) Cuntz semigroup Cu(A) is an object in the category Cu of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu in [CEI08]. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter Cu-semigroups.We establish the existence of tensor products in the category Cu and study the basic properties of this construction. We show that Cu is a symmetric, monoidal category and relate Cu(A ⊗ B) with Cu(A) ⊗ Cu Cu(B) for certain classes of C * -algebras.As a main tool for our approach we introduce the category W of precompleted Cuntz semigroups. We show that Cu is a full, reflective subcategory of W. One can then easily deduce properties of Cu from respective properties of W, for example the existence of tensor products and inductive limits. The advantage is that constructions in W are much easier since the objects are purely algebraic.For every (local) C * -algebra A, the classical Cuntz semigroup W (A) together with a natural auxiliary relation is an object of W. This defines a functor from C * -algebras to W which preserves inductive limits. We deduce that the assignment A → Cu(A) defines a functor from C * -algebras to Cu which preserves inductive limits. This generalizes a result from [CEI08].We also develop a theory of Cu-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C * -algebra has a natural product giving it the structure of a Cu-semiring. For C * -algebras, it is an important regularity property to tensorially absorb a strongly self-absorbing C * -algebra. Accordingly, it is of particular interest to analyse the tensor products of Cu-semigroups with the Cu-semiring of a strongly self-absorbing C * -algebra. This leads us to define 'solid' Cu-semirings (adopting the terminology from solid rings), as those Cu-semirings S for which the product induces an isomorphism between S ⊗ Cu S and S. This can be considered as an analog of being strongly self-absorbing for Cu-semirings. As it turns out, if a strongly self-absorbing C * -algebra satisfies the UCT, then its Cu-semiring is solid. We prove a classification theorem for solid Cu-semirings. This raises the question of whether the Cuntz semiring of every strongly self-absorbing C * -algebra is solid.If R is a solid Cu-semiring, then a Cu-semigroup S is a semimodule over R if and only if R ⊗ Cu S is isomorphic to S. Thus, analogous to the case for C * -algebras, we can think of semimodules over R as Cu-semigroups that tensorially absorb R. We give explicit characterizations when a Cu-semigroup is such a semimodule for the cases that R is the Cu-semiring of a strongly self-absorbing C * -algebra satisfying the UCT. For instance, we show that ...

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“…The C * -algebra pU p is separable, simple, unital, has finite nuclear dimension and satisfies the UCT. Hence by [12,Theorem 4.3] and [32, Theorem A], pU p fits within the Elliott classification program. It follows from [11] that pU p is an approximately subhomogeneous C * -algebra.…”

Section: Projections In the Totally Disconnected Stable Sets Casementioning

confidence: 95%

Smale space $C^*$-algebras have nonzero projections

Deeley¹,

Goffeng²,

Yashinski³

2019

Proc. Amer. Math. Soc.

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The main result of the present paper is that the stable and unstable C * -algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C * -algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.

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On the classification of simple amenable C*-algebras with finite decomposition rank

Cited by 104 publications

References 44 publications

Rokhlin Dimension for Flows

Rokhlin Dimension for Flows

Tensor products and regularity properties of Cuntz semigroups

Smale space $C^*$-algebras have nonzero projections

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