High-dimensional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">Z</mml:mi></mml:math>-stable AH algebras

Tikuisis, Aaron

doi:10.1016/j.jfa.2015.05.013

Cited by 3 publications

(2 citation statements)

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“…With regard to how it is weakened, the (non-commutative) result is more like saying that n-dimensional compact metrizable spaces are inverse limits of n-dimensional cell complexes. As with the commutative case, the result provides building blocks for (approximately) subhomogeneous algebras that are much more tractable and amenable to further analysis (e.g., Theorem A, [8,34,39]), compared to general subhomogeneous algebras-or even to Phillips's recursive subhomogeneous algebras.…”

Section: Non-commutative Cell Complexesmentioning

confidence: 99%

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Decomposition rank of approximately subhomogeneous C^*-algebras

et al. 2020

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AbstractIt is shown that every Jiang–Su stable approximately subhomogeneous {{\mathrm{C}^{*}}}-algebra has finite decomposition rank. This settles a key direction of the Toms–Winter conjecture for simple approximately subhomogeneous {{\mathrm{C}^{*}}}-algebras. A key step in the proof is that subhomogeneous {{\mathrm{C}^{*}}}-algebras are locally approximated by a certain class of more tractable subhomogeneous algebras, namely a non-commutative generalization of the class of cell complexes. The result is applied, in combination with other recent results, to show classifiability of crossed product {{\mathrm{C}^{*}}}-algebras associated to minimal homeomorphisms with mean dimension zero.

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Section: Non-commutative Cell Complexesmentioning

confidence: 99%

“…Their tractable structure allows them to be used to prove Theorem A. The author A.T. has also made use of Theorem B, in an argument that shows that C(X, Q) is not locally approximated by subhomogeneous C * -algebras of topological dimension less than the dimension of X, where Q is the universal UHF algebra [39].…”

Section: Theorem Bmentioning

confidence: 99%

Decomposition rank of approximately subhomogeneous C^*-algebras

et al. 2020

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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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The Weak Ideal Property and Topological Dimension Zero

Pasnicu¹,

Phillips

2017

Can. j. math.

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Abstract. Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or topological dimension zero, and some results for C*-algebras with related properties. Some of the more important results include:• e weak ideal property implies topological dimension zero.• For a separable C*-algebra A, topological dimension zero is equivalent to RR(O ⊗ A) = , to D ⊗ A having the ideal property for some (or any) Kirchberg algebra D, and to A being residually hereditarily in the class of all C*-algebras B such that O ∞ ⊗ B contains a nonzero projection.• Extending the known result for Z , the classes of C*-algebras with residual (SP), which are residually hereditarily (properly) in nite, or which are purely in nite and have the ideal property, are closed under crossed products by arbitrary actions of abelian -groups.• If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A ⊗ min B has the weak ideal property.• If X is a totally disconnected locally compact Hausdor space and A is a C (X)-algebra all of whose bers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure in niteness and the ideal property, then A also has the corresponding property (for topological dimension zero, provided A is separable).• Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH algebras.• e weak ideal property does not imply the ideal property for separable Z-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for.

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High-dimensionalZ-stable AH algebras

Cited by 3 publications

References 11 publications

Decomposition rank of approximately subhomogeneous C^*-algebras

Decomposition rank of approximately subhomogeneous C^*-algebras

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

The Weak Ideal Property and Topological Dimension Zero

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High-dimensionalZ-stable AH algebras

Cited by 3 publications

References 11 publications

Decomposition rank of approximately subhomogeneous C*-algebras

Decomposition rank of approximately subhomogeneous C*-algebras

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

The Weak Ideal Property and Topological Dimension Zero

Contact Info

Product

Resources

About

Decomposition rank of approximately subhomogeneous C^*-algebras

Decomposition rank of approximately subhomogeneous C^*-algebras