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 (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.