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

TL;DR

This work resolves the question of weak centrality for AF-algebras by proving every AF-algebra is weakly central, via the center quotient property in the Bratteli inductive-limit framework. It then develops a bundle-theoretic, $C(X)$-algebra characterization of weak centrality, linking it to fiberwise maximal-ideal structure $A(x)=A/I_x$ and to the base space $X$ through the center $\mathcal{Z}(A)$. The authors apply this framework to full group $C^*$-algebras, showing both positive results (preservation under certain Rokhlin-type actions) and substantial obstructions (e.g., the discrete Heisenberg group and many nilpotent or crystallographic groups yield non-weakly central algebras). The paper thereby clarifies how weak centrality sits relative to centrality and the Dixmier property, and provides concrete criteria and examples to analyze group $C^*$-algebras via bundle-theoretic methods, including open questions about the existence of weakly central but non-central algebras and the behavior of various group classes.

Abstract

We first prove that every AF-algebra is weakly central, thereby resolving a question left open by Archbold--Gogić. 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.

Figures (1)

• Figure 1: Relation between different classes of locally compact groups.

Theorems & Definitions (8)

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