A Note on Approximately Divisible C$^*$-algebras

TL;DR

This paper proves that separable, unital, approximately divisible C*-algebras are generated by two self-adjoint elements, have topological free entropy dimension at most 1, and satisfy Kadison's similarity problem with a similarity degree at most 5.

Contribution

It establishes new structural properties of approximately divisible C*-algebras, including generation, entropy dimension bounds, and similarity degree, advancing understanding of their operator algebraic features.

Findings

Generated by two self-adjoint elements

Topological free entropy dimension ≤ 1

Similarity degree ≤ 5

Abstract

Let $\mathcal A$ be a separable, unital, approximately divisible C$^*$-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$^*$-algebra has an affirmative answer to Kadison's similarity problem.