MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

Strung, Karen R.; Winter, Wilhelm

doi:10.1142/s0129167x10006665

Cited by 12 publications

(12 citation statements)

References 34 publications

(66 reference statements)

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“…By now there exist classification results for large classes of crossed products by Z-actions: early results of Putnam about characterizing crossed products of minimal homeomorphisms on the Cantor set as AT algebras (see [17]) or of Elliott and Evans about irrational rotation algebras (see [2]) have set the stage for this project. In one of the more recent breakthroughs, Toms, Strung and Winter proved that crossed products of uniquely ergodic minimal homeomorphisms on infinite compact metrizable spaces with finite covering dimension are classified by ordered K-theory (see [21,19]).…”

Section: Introductionmentioning

confidence: 99%

The Rokhlin dimension of topological ℤ^m -actions

Szabó

2015

Proceedings of the London Mathematical Society

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Abstract. We study the topological variant of Rokhlin dimension for topological dynamical systems (X, α, Z m ) in the case where X is assumed to have finite covering dimension. Finite Rokhlin dimension in this sense is a property that implies finite Rokhlin dimension of the induced action on C * -algebraic level, as was discussed in a recent paper by Hirshberg, Winter and Zacharias. In particular, it implies under these conditions that the transformation group C * -algebra has finite nuclear dimension. Generalizing partial results of Lindenstrauss and Gutman, we show that free Z m -actions on finite dimensional spaces satisfy a strengthened version of the so-called marker property, which yields finite Rokhlin dimension for such actions.

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Section: Introductionmentioning

confidence: 99%

The Rokhlin dimension of topological ℤ^m -actions

Szabó

2015

Proceedings of the London Mathematical Society

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“…Proof: By [17, Theorem 3.2] A has property (SP) and by assumption, strict comparison. After noting this, the proof is essentially the same as that of [23,Lemma 4.4] and [30,Lemma 4.5], replacing the C * -subalgebra of tracial rank zero (respecively TAS) with the TAI C * -subalgebra B, and replacing the finite-dimensional (respectively in the class S) C * -subalgebra with a C *subalgebra from the class I.…”

Section: Lemmamentioning

confidence: 94%

“…Nevertheless, substantial progress has been made in this more general setting, for example [23,7,21,22,32]. A particularly wide-reaching result follows from the work of Andrew Toms and Wilhelm Winter in [34,33] (which uses a special case of a more general theorem found in [30]), where they give classification for the class of C * -algebras of minimal dynamical systems (X, α) of infinite finitedimensional metric spaces under the additional assumption that projections in the C * -algebras separate their tracial states.…”

Section: Introductionmentioning

confidence: 99%

C⁎-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Strung

2015

Journal of Functional Analysis

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Let β : S n → S n , for n = 2k + 1, k ≥ 1, be one of the known examples of a nonuniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (S n , β) there is a Cantor minimal system (X, α) such that the corresponding product system (X × S n , α × β) is minimal and the resulting crossed product C * -algebra C(X ×S n )⋊ α×β Z is tracially approximately an interval algebra (TAI). This entails classification for such C * -algebras. Moreover, the minimal Cantor system (X, α) is such that each tracial state on C(X × S n ) ⋊ α×β Z induces the same state on the K 0 -group and such that the embedding of C(S n ) ⋊ β Z into C(X × S n ) ⋊ α×β Z preserves the tracial state space. This implies C(S n ) ⋊ β Z is TAI after tensoring with the universal UHF algebra, which in turn shows that the C * -algebras of these examples of minimal diffeomorphisms of odd dimensional spheres are classified by their tracial state spaces.

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“…is an ordered group isomorphism [31,Lemma 4.3], (see also [22,Theorem 4.1 (5)]). Since K 0 (C(Z ϕ ) ⋊ ζ Z)) ∼ = Z, we have that S(K 0 (C(Z ϕ ) ⋊ ζ Z))) is a point.…”

Section: Theoremmentioning

confidence: 99%

Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

Deeley

Putnam

Strung

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. The principal aim of this paper is to give a dynamical presentation of the Jiang-Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang-Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott's classification programme for separable, nuclear C * -algebras. Here, we exhibit anétale equivalence relation whose groupoid C * -algebra is isomorphic to the Jiang-Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the LefschetzHopf Theorem would imply that it does not admit a minimal homeomorphism. IntroductionThe fields of operator algebras and dynamical systems have a long history of mutual influence. On the one hand, dynamical systems provide interesting examples of operator algebras and have often provided techniques which are successfully imported into the operator algebra framework. On the other hand, results in operator algebras are often of interest to those in dynamical systems. In ideal situations, significant information is retained when passing from dynamics to operator algebras, and vice versa. This relationship has been particularly interesting for the classification of C * -algebras. An extraordinary result in this setting is the classification, up to strong orbit equivalence, of the minimal dynamical systems on a Cantor set and the corresponding K-theoretical classification of the associated crossed product C * -algebras [9,24]. Classification for separable, simple, nuclear C * -algebras remains an interesting open problem. To every simple separable nuclear C * -algebra one assigns a computable set of invariants involving K-theory, tracial state spaces, and the pairing between these objects. George Elliott conjectured that for all such C * -algebras,

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MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

Cited by 12 publications

References 34 publications

The Rokhlin dimension of topological ℤ^m -actions

The Rokhlin dimension of topological ℤ^m -actions

C⁎-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

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MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

Cited by 12 publications

References 34 publications

The Rokhlin dimension of topological ℤm -actions

The Rokhlin dimension of topological ℤm -actions

C⁎-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

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The Rokhlin dimension of topological ℤ^m -actions

The Rokhlin dimension of topological ℤ^m -actions