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

Leonel Robert, Aaron Tikuisis

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

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 simple purely infinite ideals of quotients 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).
Original languageEnglish
Pages (from-to)4631-4670
Number of pages40
JournalTransactions of the American Mathematical Society
Volume369
Issue number7
Early online date27 Dec 2016
DOIs
Publication statusPublished - Jul 2017

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