Smale space $C^*$-algebras have nonzero projections

Deeley, Robin J.; Goffeng, Magnus; Yashinski, Allan

doi:10.1090/proc/14837

Cited by 7 publications

(3 citation statements)

References 31 publications

(89 reference statements)

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“…(We include a small remark: the results of [9] need the hypothesis that the C * -algebras contain a projection. This was later shown to hold in general [10], but, in our case, we will explicitly provide clopen subsets of the unit spaces of both G s (X ξ , σ ξ , P ξ ) and G s (X ξ , σ ξ , P ξ ) which show this holds. )…”

Section: Binary Factors Of Shifts Of Finite Type 915mentioning

confidence: 59%

Binary factors of shifts of finite type

Putnam

2023

Ergod. Th. Dynam. Sys.

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We construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$ -algebras associated to them, in terms of the pair of original graphs.

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Section: Binary Factors Of Shifts Of Finite Type 915mentioning

confidence: 59%

Binary factors of shifts of finite type

Putnam

2023

Ergod. Th. Dynam. Sys.

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“…Noticing this situation, the authors of the present paper were hopeful that the class of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant would be sufficiently well-behaved to completely understand the stable algebra and Ruelle algebras in the case of Wieler solenoids, see for example the introduction of [4]. By sufficiently wellbehaved, we mean that results from the continuous trace case would generalize without much issue or change to our setting.…”

Section: Introductionmentioning

confidence: 93%

Fell algebras, groupoids, and projections

Deeley¹,

Goffeng²,

Yashinski³

2021

Preprint

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Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous trace C * -algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arise naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper.

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“…It is worth mentioning that these C * -algebras fit into Elliott's classification program of simple, separable, nuclear C * -algebras, and satisfy the UCT [44]. In addition, the stable and unstable algebras have finite nuclear dimension [14, Corollary 3.8] and are quasidiagonal [12]. If the Smale space is mixing, the stable and unstable algebras are also simple and have unique traces [43,Theorem 3.3].…”

Section: Smale Space C*-algebrasmentioning

confidence: 99%

A geometric representative for the fundamental class in KK-duality of Smale spaces

Gerontogiannis¹,

Whittaker²,

Zacharias³

2022

Preprint

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A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ-summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions.Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module.

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Smale space $C^*$-algebras have nonzero projections

Cited by 7 publications

References 31 publications

Binary factors of shifts of finite type

Binary factors of shifts of finite type

Fell algebras, groupoids, and projections

A geometric representative for the fundamental class in KK-duality of Smale spaces

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