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