Single commutators in $C^{\ast }$-algebras

Abstract

Let $\mathcal{A}$ be a simple unital $C^{\ast }$-algebra with real rank zero, nonempty tracial state space $T(\mathcal{A})$, and strict comparison of projections. We prove that every element $T$ in $\mathcal{A}$ with $\tau (T)=0$ for all $\tau \in T(\mathcal{A})$ is a single commutator if one of the following conditions holds: $(1)$ $\mathcal{A}$ has unique trace; $(2)$ $\mathcal{A}$ is separable and has stable rank one. We also prove that every nonscalar element in a simple unital purely infinite $C^{\ast }$-algebra is a single commutator. Applications include UHF algebras, irrational rotation algebras, and Cuntz algebras.