Weak Centrality: AF-algebras, C(X)-algebras, and group C*-algebras

TL;DR

This paper characterizes weak centrality in AF-algebras and unital C*-algebras, providing new insights into their structure and relations with group C*-algebras, and resolves an open question in the field.

Contribution

It proves that all AF-algebras are weakly central and offers a new characterization of weak centrality for unital C*-algebras using C(X)-algebras.

Findings

All AF-algebras are weakly central.

New characterization of weak centrality for unital C*-algebras.

Analysis of weak centrality in full group C*-algebras.

Abstract

We first prove that every AF-algebra is weakly central, thereby resolving a question left open by Archbold--Gogi\'c. We then establish a new characterization of weak centrality for unital $C^*$-algebras in terms of $C(X)$-algebras. The paper concludes with an appendix that examines weak centrality in full group $C^*$-algebras and places these examples within the hierarchy of group classes.