Remarks on some simple $C^{\ast }$-algebras admitting a unique lower semicontinuous $2$-quasitrace

Abstract

Using different descriptions of the Cuntz semigroup and of the Pedersen ideal, it is shown that $\sigma$-unital simple $C^{\ast }$-algebras with almost unperforated Cuntz semigroup, a unique lower semicontinuous $2$-quasitrace and whose stabilization has almost stable rank $1$ are either stable or algebraically simple.

Replaced with accepted version, to appear in Colloquium Mathematicum