<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="normal">C</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math>-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Strung, Karen R.

doi:10.1016/j.jfa.2014.10.014

Cited by 13 publications

(11 citation statements)

References 27 publications

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“…The classification of locally ASH algebras (Theorem 5.9) in fact allow us to recover the recent classification result for the C*-algebra of a minimal homeomorphism-assumed to have mean dimension zero but not to be uniquely ergodic ( [27], [18]-the uniquely ergodic case was dealt with in [3], or in [29] on the ease the space is finite dimensional):…”

Section: Tracial Factorization and Tracial Approximationmentioning

confidence: 69%

The classification of simple separable unitalZ-stable locally ASH algebras

Elliott

Gong

Lin

et al. 2017

Journal of Functional Analysis

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Let A be a simple separable unital locally approximately subhomogeneous C*algebra (locally ASH algebra). It is shown that A ⊗ Q can be tracially approximated by unital Elliott-Thomsen algebras with trivial K 1 -group, where Q is the universal UHF algebra. In particular, it follows that A is classifiable by the Elliott invariant if A is Jiang-Su stable.Date: July 16, 2018. 1 THE CLASSIFICATION OF SIMPLE SEPARABLE UNITAL LOCALLY ASH ALGEBRAS 2behaved if in addition matrix algebras over it are also finite in the Murray-von Neumann sense (i.e., if the algebra is stably finite)-the class considered by Gong, Lin, and Niu does not on the face of it include the class of all well behaved-"Jiang-Su stable"-simple unital ASH algebras.Using the recent result of Santiago, Tikuisis, and the present authors ([5]) that any Jiang-Su stable simple unital ASH algebra has finite nuclear dimension (one of the Toms-Winter well behavedness properties-the important concept of nuclear dimension was introduced by Winter and Zacharias in [34])-and also using the notion of non-commutative cell complex introduced in [5] in the proof of this result-, the present note shows that indeed such an algebra (Jiang-Su stable simple unital ASH) belongs to the class dealt with by Gong, Lin, and Niu. (Even if the ASH algebra has slow dimension growth, so that by [28] and [31] it is indeed Jiang-Su stable, it is not known to belong to the class studied by Gong, Lin, and Niu-the class of rationally tracially approximately point-line algebras-see below.)Acknowledgements.

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Section: Tracial Factorization and Tracial Approximationmentioning

confidence: 69%

The classification of simple separable unitalZ-stable locally ASH algebras

Elliott

Gong

Lin

et al. 2017

Journal of Functional Analysis

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“…Minimal dynamical systems on a compact metric space X (systems in which the orbit of every point under the homeomorphism ϕ is dense in X) have provided a wealth of examples of simple separable unital nuclear C * -algebras via the crossed product construction. The question of their classification has seen a lot of interest, for example [8,27,21,44,45,39] to name but a few. The case of a system with mean dimension zero (see [22]) was recently fully resolved by Huaxin Lin [18].…”

Section: Introductionmentioning

confidence: 99%

Nuclear dimension and classification of 𝐶*-algebras associated to Smale spaces

Deeley¹,

Strung

2017

Trans. Amer. Math. Soc.

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We show that the homoclinic C * -algebras of mixing Smale spaces are classifiable by the Elliott invariant. To obtain this result, we prove that the stable, unstable, and homoclinic C * -algebras associated to such Smale spaces have finite nuclear dimension. Our proof of finite nuclear dimension relies on Guentner, Willett, and Yu's notion of dynamic asymptotic dimension.2000 Mathematics Subject Classification. 46L35, 37D20.

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“…In the uniquely ergodic case, classification for the resulting crossed product follows from [34,35] (which uses the main result in [31]). Without assuming unique ergodicity, we may appeal to Lin's generalisation [15] of the third author's classification result for products with Cantor systems [30], to show all our minimal dynamical systems result in classifiable crossed products. We note that as we were finishing this paper, Lin posted a classification theorem for all crossed product C * -algebras associated to minimal dynamical systems with mean dimension zero [16], but our results do not rely on his proof.…”

Section: Introductionmentioning

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

Cited by 13 publications

References 27 publications

The classification of simple separable unitalZ-stable locally ASH algebras

The classification of simple separable unitalZ-stable locally ASH algebras

Nuclear dimension and classification of 𝐶*-algebras associated to Smale spaces

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

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