The Jiang-Su algebra does not always embed

Dǎdǎrlat, Marius; Hirshberg, Ilan; Toms, Andrew S.; Winter, Wilhelm

doi:10.4310/mrl.2009.v16.n1.a3

Cited by 13 publications

(13 citation statements)

References 7 publications

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“…It was shown in [6] that there is a simple unital infinite-dimensional nuclear C * -algebra A such that the dimension drop C * -algebra Z 3,4 , and hence the Jiang-Su algebra Z, do not embed unitally into A. The divisibility properties of A were not explicitly mentioned in [6], but it is easily seen (using Lemma 6.1, that is paraphrased from [27,Lemma 4.3]) that Div 3 (A) > 4. We shall, in Section 7, give further examples of simple unital infinite-dimensional C * -algebras where the divisibility numbers attain non-trivial values.…”

Section: Examples and Remarksmentioning

confidence: 92%

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Divisibility properties for C *-algebras

Robert

Rørdam

2013

Proceedings of the London Mathematical Society

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We consider three notions of divisibility in the Cuntz semigroup of a C * -algebra, and show how they reflect properties of the C * -algebra. We develop methods to construct (simple and nonsimple) C * -algebras with specific divisibility behaviour. As a byproduct of our investigations, we show that there exists a sequence (An) of simple unital infinite dimensional C * -algebras such that the product ∞ n=1 An has a character.

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Section: Examples and Remarksmentioning

confidence: 92%

“…It was shown in [DHTW09] that there exists a simple unital infinite dimensional C * -algebra which does not admit a unital embedding of the Jiang-Su algebra Z. This answered in the negative a question posed by the second named author.…”

Section: Introductionmentioning

confidence: 98%

Divisibility properties for C *-algebras

Robert

Rørdam

2013

Proceedings of the London Mathematical Society

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“…However, in Theorem 7.3.11(3) we not only require the Cu-semigroup to be almost unperforated but also almost divisible. We remark that not every Cuntz semigroup of a simple C * -algebra is almost divisible; see [DHTW09]. On the other hand, it seems possible that the Cuntz semigroup of a simple C * -algebra is automatically almost divisible whenever it is almost unperforated.…”

Section: Almost Unperforated and Almost Divisible Cu-semigroupsmentioning

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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“…recently discovered counterexamples to Elliott's conjecture; cf. the articles [21], [20], [2], [32], [42], [43], [14], [9] and [6], to mention but a few.…”

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confidence: 99%

Decomposition rank and $\mathcal{Z}$ -stability

Winter

2009

Invent. math.

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We show that separable, simple, nonelementary, unital C * -algebras with finite decomposition rank absorb the Jiang-Su algebra Z tensorially. This has a number of consequences for Elliott's program to classify nuclear C *algebras by their K-theory data. In particular, it completes the classification of C * -algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K-groups.

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The Jiang-Su algebra does not always embed

Abstract: ABSTRACT. We exhibit a unital simple nuclear non-type-I C * -algebra into which the Jiang-Su algebra does not embed unitally. This answers a question of M. Rørdam.

Cited by 13 publications

References 7 publications

Divisibility properties for C *-algebras

Divisibility properties for C *-algebras

Tensor products and regularity properties of Cuntz semigroups

Decomposition rank and $\mathcal{Z}$ -stability

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