Generalized Bunce-Deddens algebras

TL;DR

This paper introduces a broad class of crossed product C*-algebras associated with residually finite groups and their profinite completions, establishing their structural properties and classification features.

Contribution

It defines a new class of crossed product C*-algebras for residually finite groups and analyzes their structural and classification properties.

Findings

They are unital, separable, simple, nuclear, quasidiagonal C*-algebras.

They have real rank zero and stable rank one.

They possess a unique trace.

Abstract

We define a broad class of crossed product C*-algebras of the form C(G)xG, where G is a discrete countable amenable residually finite group, and G is a profinite completion of G. We show that they are unital separable simple nuclear quasidiagonal C*-algebras, or real rank zero, stable rank one, with comparability of projections and with a unique trace.