Remarks on some simple $C^*$-algebras admitting a unique lower semicontinuous $2$-quasitrace

TL;DR

This paper characterizes certain simple $ ext{C}^*$-algebras with unique lower semicontinuous 2-quasitraces, showing they are either stable or algebraically simple under specific conditions involving the Cuntz semigroup and Pedersen ideal.

Contribution

It provides new criteria for classifying simple $ ext{C}^*$-algebras with unique 2-quasitraces using Cuntz semigroup and Pedersen ideal descriptions.

Findings

Algebras are either stable or algebraically simple.

Conditions involve almost unperforated Cuntz semigroup and almost stable rank 1.

Unique lower semicontinuous 2-quasitrace plays a key role.

Abstract

Using different descriptions of the Cuntz semigroup and of the Pedersen ideal, it is shown that $\sigma$-unital simple $C^*$-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.