UCT-Kirchberg algebras have nuclear dimension one

Ruiz, Efren; Sims, Aidan; Sørensen, Adam P. W.

doi:10.1016/j.aim.2014.12.042

Cited by 36 publications

(35 citation statements)

References 24 publications

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“…We end the paper by showing how our techniques can also be used to calculate the nuclear dimension of Kirchberg algebras without a UCT assumption. Aiming for the exact value of 1 (and hence lowering the bound for Kirchberg algebras from [60]) was inspired by recent work of Ruiz, Sims and Sørensen ( [77]) who obtain the optimal estimate for UCT Kirchberg algebras.…”

Section: Introductionmentioning

confidence: 99%

Covering Dimension of C*-Algebras and 2-Coloured Classification

et al. 2019

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We introduce the concept of finitely coloured equivalence for unital * -homomorphisms between C * -algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify *homomorphisms from separable, unital, nuclear C * -algebras into ultrapowers of simple, unital, nuclear, Z-stable C * -algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data.As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z-stable C * -algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a "homotopy equivalence implies isomorphism" result for large classes of C * -algebras with finite nuclear dimension.

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Section: Introductionmentioning

confidence: 99%

Covering Dimension of C*-Algebras and 2-Coloured Classification

et al. 2019

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“…In addition to their relevance for C * -algebraic classification [78,86], the C * -algebras of higher-rank graphs are closely linked with orbit equivalence for shift spaces [17] and with symbolic dynamics more generally [79,88,80], with fractals and self-similar structures [35,36], and with renormalization problems in physics [40]. More links between higher-rank graphs and symbolic dynamics can be seen via [8,9,7] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Separable representations, KMS states, and wavelets for higher-rank graphs

Farsi

Gillaspy

Kang

et al. 2016

Journal of Mathematical Analysis and Applications

124

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Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C * (Λ) on certain separable Hilbert spaces of the form L 2 (X, µ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation ofwhere M is the Perron-Frobenius probability measure on the infinite path space Λ ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C * (Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of, where X is a fractal subspace of [0, 1] by embedding Λ ∞ into [0, 1] as a fractal subset X of [0,1]. In this latter case we additionally show that there exists a KMS state for C * (Λ) whose inverse temperature is equal to the Hausdorff dimension of X. Finally, we construct a wavelet system for L 2 (Λ ∞ , M ) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.2010 Mathematics Subject Classification: 46L05.

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“…Concrete models of Kirchberg algebras have proved extremely useful in classification theorysee for example [9,30,31]. In this paper, we develop a technique for realising many Kirchberg algebras as the C * -algebras of amenable principal groupoids.…”

Section: Introductionmentioning

confidence: 99%

“…Algebraically, a principal groupoid is just an equivalence relation; so in spirit at least, the C * -algebras of amenable principal groupoids are akin to matrix algebras. Also, most of the existing groupoid models for Kirchberg algebras are based on graphs and their analogues [13,30,31,33], and are not principal. But recent work of the third-named author with Rørdam shows that there are indeed examples of amenable principal groupoids whose C * -algebras are Kirchberg algebras: [29,Theorem 6.11] shows that every nonamenable exact group admits a free and amenable action on the Cantor set for which the associated crossed-product is a Kirchberg algebra.…”

Section: Introductionmentioning

confidence: 99%