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

TL;DR

This paper explores how finite nuclear dimension relates to Z-stability and algebraic regularity in non-simple C*-algebras, providing new cases where nuclear dimension implies Z-stability, advancing the classification program.

Contribution

It proves the conjecture that finite nuclear dimension implies Z-stability for certain classes of non-simple C*-algebras, extending previous results.

Findings

Finite nuclear dimension implies algebraic regularity in the Cuntz semigroup.

Finite nuclear dimension can be used to analyze the structure of the central sequence algebra.

The conjecture that finite nuclear dimension implies Z-stability is proved for specific classes of non-simple C*-algebras.

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 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).