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

Using different descriptions of the Cuntz semigroup and of the Pedersen ideal, it is shown that $σ$-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.

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