A Note on Approximately Divisible C${}^{\ast }$-algebras

Abstract

Let $\mathcal{A}$ be a separable, unital, approximately divisible C${}^{\ast }$-algebra. We show that $\mathcal{A}$ is generated by two self-adjoint elements and the topological free entropy dimension of any finite generating set of $\mathcal{A}$ is less than or equal to 1. In addition, we show that the similarity degree of $\mathcal{A}$ is at most 5. Thus an approximately divisible C${}^{\ast }$-algebra has an affirmative answer to Kadison's similarity problem.