Nuclear dimension, Z-stability, and algebraic simplicity for stably projectionless C*-algebras

TL;DR

This paper proves that simple separable C*-algebras with finite nuclear dimension or slow dimension growth are Z-stable, extending previous results to nonunital algebras and introducing algebraic simplicity as a key concept.

Contribution

It generalizes Z-stability results to nonunital C*-algebras with finite nuclear dimension or slow dimension growth, using algebraic simplicity.

Findings

Finite nuclear dimension implies Z-stability in separable C*-algebras.

Approximate subhomogeneity with slow dimension growth implies Z-stability.

Algebraic simplicity is a useful weakening of simplicity for these results.

Abstract

The main result here is that a simple separable C*-algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [Toms, "K-theoretic rigidity and slow dimension growth"; Winter, "Nuclear dimension and Z-stability of pure C*-algebras"] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C*-algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept.