Regularity for stably projectionless, simple C*-algebras

TL;DR

This paper investigates regularity properties such as Jiang-Su stability, unperforation, and slow dimension growth in simple, stably projectionless C*-algebras, providing examples and establishing conditions linking these properties.

Contribution

It introduces an example of a simple, nuclear, stably projectionless C*-algebra with a non-almost unperforated Cuntz semigroup and clarifies the relationship between slow dimension growth and unperforation.

Findings

Example of a stably projectionless C*-algebra with non-almost unperforated Cuntz semigroup

Slow dimension growth implies almost unperforation in approximately subhomogeneous algebras

Contrast between properties in different classes of C*-algebras

Abstract

This paper explores the following regularity properties and their relationships for simple, not-necessarily-unital C*-algebras: (i) Jiang-Su stability, (ii) Unperforation in the Cuntz semigroup, and (iii) slow dimension growth (applying only in the case that the C*-algebra is approximately subhomogeneous). An example is given of a simple, separable, nuclear, stably projectionless C*-algebra whose Cuntz semigroup is not almost unperforated. This example is in fact approximately subhomogeneous. It is also shown that, in contrast to this example, when an approximately subhomogeneous simple C*-algebra has slow dimension growth, its Cuntz semigroup is necessarily almost unperforated.