Regularity for stably projectionless, simple C⁎-algebras

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 ((iii) 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.