50 research outputs found
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 āRegularity for Stably Projectionless, Simple C ā -Algebrasā
Peer reviewedPostprin
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
Peer reviewedPostprin