The Cuntz semigroup of continuous functions into certain simple C*-algebras

This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C_0(X,A), where A is a unital, simple, Z-stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(T,A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.Comment: 37 pages. To appear in International Journal of Mathematic

Nuclear dimension and Z-stability of non-simple C*-algebras

We investigate the interplay of the following regularity properties for non-simple C*-algebras: finite nuclear dimension, Z-stability, and algebraic regularity in the Cuntz semigroup. We show that finite nuclear dimension implies algebraic regularity in the Cuntz semigroup, provided that known type I obstructions are avoided. We demonstrate how finite nuclear dimension can be used to study the structure of the central sequence algebra, by factorizing the identity map on the central sequence algebra, in a manner resembling the factorization arising in the definition of nuclear dimension. Results about the central sequence algebra are used to attack the conjecture that finite nuclear dimension implies Z-stability, for sufficiently non-type I, separable C*-algebras. We prove this conjecture in the following cases: (i) the C*-algebra has no purely infinite subquotients and its primitive ideal space has a basis of compact open sets, (ii) the C*-algebra has no purely infinite quotients and its primitive ideal space is Hausdorff. In particular, this covers C*-algebras with finite decomposition rank and real rank zero. Our results hold more generally for C*-algebras with locally finite nuclear dimension which are (M,N)-pure (a regularity condition of the Cuntz semigroup).Comment: Rewrote abstract and introduction. Added a couple of results. Main results unchange

Relative commutant pictures of Roe algebras

Let X be a proper metric space, which has finite asymptotic dimension in the sense of Gromov (or more generally, straight finite decomposition complexity of Dranishnikov and Zarichnyi). New descriptions are provided of the Roe algebra of X: (i) it consists exactly of operators which essentially commute with diagonal operators coming from Higson functions (that is, functions on X whose oscillation tends to 0 at infinity) and (ii) it consists exactly of quasi-local operators, that is, ones which have finite epsilon propogation (in the sense of Roe) for every epsilon>0. These descriptions hold both for the usual Roe algebra and for the uniform Roe algebra.Comment: 35 pages. Minor changes. To appear in Comm. Math. Phy

Corrigendum to “Regularity for Stably Projectionless, Simple C ∗ -Algebras”

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K-Theoretic Characterization of C*-Algebras with Approximately Inner Flip

The author is supported by an NSERC PDF.Peer reviewedPostprin

Finite dimensional ordered vector spaces with Riesz interpolation and Effros-Shen's unimodularity conjecture

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