Nuclear dimension and $$\mathcal Z$$ Z -stability

Sato, Yasuhiko; White, Stuart; Winter, Wilhelm

doi:10.1007/s00222-015-0580-1

Cited by 71 publications

(84 citation statements)

References 57 publications

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“…It was shown that (iii) implies (ii) in the case that T (A) is a Bauer simplex whose extreme boundary is finite-dimensional (see [KR12], [Sat12], and [TWW12], and the precursor [MS12]). It was proved that (ii) implies (i) in the case that A has a unique tracial case ( [SWW14]), which was very recently generalized to the case that T (A) is any Bauer simplex; see [BBS + 14]. It follows that the Toms-Winter conjecture is verified for all C * -algebras A such that T (A) is a Bauer simplex with finite-dimensional extreme boundary.…”

Section: Strongly Self-absorbing Cmentioning

confidence: 98%

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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Section: Strongly Self-absorbing Cmentioning

confidence: 98%

Tensor products and regularity properties of Cuntz semigroups

2018

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“…(2): We have α ≃ cc α ⊗ id Z , and there exist two c.p.c. order zero maps ψ 0 , ψ 1 : Q → Z ∞ ∩ Z ′ with ψ 0 (1) + ψ 1 (1) = 1; see [66,Section 5] or [76,Section 6]. Consider two sequences ψ 0,n , ψ 1,n : Q → Z of c.p.c.…”

Section: Corollary 53 Let γ Be a Countable Discrete Residually Fimentioning

confidence: 99%

Rokhlin dimension: absorption of model actions

Szabó

2019

Analysis & PDE

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In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly selfabsorbing C * -algebras. Namely, as to be made precise in the paper, let G be a well-behaved locally compact group. If D is a strongly selfabsorbing C * -algebra, and α : G A is an action on a separable, Dabsorbing C * -algebra that has finite Rokhlin dimension with commuting towers, then α tensorially absorbs every semi-strongly self-absorbing Gactions on D. In particular, this is the case when α satisfies any version of what is called the Rokhlin property, such as for G = R or G = Z k . This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some nontrivial applications. Most notably it is shown that for any k ≥ 1 and on any strongly self-absorbing Kirchberg algebra, there exists a unique R k -action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.

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“…Given a simple, separable, nuclear, unital, infinite-dimensional C * -algebra A whose trace simplex T (A) is a Bauer simplex, Ozawa showed that a certain tracial completion A u of A is a W * -bundle over the space of extreme traces ∂ e T (A) with fibers all isomorphic to R. When A has finite nuclear dimension, this bundle is trivial by combining results of [30] and [20]. In the reverse direction, the results of [16,17,20] (see also [22,28]) and [2] (which builds on [18,23]) show that triviality of the bundle A u combines with strict comparison, a mild condition on positive elements analogous to the order on projections in a II 1 factor being determined by their trace, to give finite nuclear dimension. This equivalence of regularity properties for C * -algebras forms part of the Toms-Winter conjecture; see [27,Sec.…”

Section: Introductionmentioning

confidence: 99%