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. 
Aaron Tikuisis
math.OA