The real and stable rank of tracially complete C*-algebras

TL;DR

This paper proves that certain factorial tracially complete C*-algebras have real rank zero and stable rank one, providing a comprehensive description of their Cuntz semigroup.

Contribution

It establishes the real and stable rank properties for factorial tracially complete C*-algebras with CPoU, extending to uniform tracial completions of Z-stable C*-algebras.

Findings

These algebras have real rank zero.

They have stable rank one.

The Cuntz semigroup is fully described for these algebras.

Abstract

We prove that a factorial tracially complete C*-algebra with CPoU has real rank zero and stable rank one. This leads to an essentially complete description of the Cuntz semigroup of these algebras. In particular, the results of this paper hold for the uniform tracial completions of $\mathcal{Z}$-stable C*-algebras.