Strict comparison and $ \mathcal{Z} $-absorption of nuclear C∗-algebras

Matui, Hiroki; Sato, Yasuhiko

doi:10.1007/s11511-012-0084-4

Cited by 133 publications

(185 citation statements)

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“…If A has no tracial states then it is purely infinite and the conjecture has been confirmed in that case using [KP00], [Rør04], and [MS14]. 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].…”

Section: Strongly Self-absorbing Cmentioning

confidence: 92%

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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: 92%

“…Similarly, if A is nuclear, then as Z ′ has only one normalized functional, we would get that A is monotracial. In that situation, the solution of the Toms-Winter conjecture (see [MS12]) would imply that A is Z-stable, a contradiction.…”

Section: Cu (C(t) ⊗ A)mentioning

confidence: 99%

Tensor products and regularity properties of Cuntz semigroups

2018

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“…Last year, in a remarkable paper by Matui and Sato,[13], it was shown that the strict comparison property is equivalent to Z-stability for all unital, separable, simple, nuclear C -algebras whose trace simplex has finite dimension, i.e., its extreme boundary is a finite set. Matui and Sato employed new techniques involving the central sequence C -algebra, A !…”

Section: Introductionmentioning

confidence: 99%

“…Let us describe some of the main ingredients in Matui and Sato's paper [13] that we shall build on and further develop. It is well known that if A is a separable unital C -algebra, then A is Z-absorbing, i.e., A Š A˝Z if and only if there is a unital -homomorphism from Z into the central sequence algebra A !…”

Section: Introductionmentioning

confidence: 99%

Central sequence C^* -algebras and tensorial absorption of the Jiang–Su algebra

Kirchberg

Rørdam

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

100

139

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We study properties of the central sequence algebra A ! \ A 0 of a C -algebra A, and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C -algebra, whose trace simplex is finitedimensional, tensorially absorbs the Jiang-Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C -algebra of having the strict comparison property to a weaker property that we call local weak comparison. Namely, we prove that a unital separable simple nuclear C -algebra, whose trace simplex has finite-dimensional closed extreme boundary, tensorially absorbs the Jiang-Su algebra if and only if it has the local weak comparison property.We can also eliminate the nuclearity assumption if instead we assume the (SI) property of Matui and Sato, and, moreover, that each II 1 -factor representation of the C -algebra is a McDuff factor.

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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%