The ideal separation property for reduced group $C^*$-algebras

TL;DR

The paper studies the ideal separation property (ISP) and the ideal intersection property (IIP) for the inclusion $L^1(G) \subseteq C^*_{\mathrm{red}}(G)$ of locally compact groups, linking them to $*$-regularity and $C^*$-uniqueness. It develops a framework of permanence results showing ISP/IIP are preserved under quotients by amenable (resp. compact) normal subgroups, under approximation by closed subgroups, and under inverse limits, and it establishes locality and Lie-quotient reduction for almost connected groups. In the tensor-product setting, the authors relate ISP/IIP to Tomiyama's property (F) and characterize ISP/IIP for Cartesian products $G\times H$, including exactness-driven simplifications when one factor is exact. Collectively, these results provide a robust toolkit for deducing ISP/IIP for broad classes of groups and connect these properties to fundamental C*-algebraic structures and notions of exactness.

Abstract

We say that an inclusion of an algebra $A$ into a $C^*$-algebra $B$ has the ideal separation property if closed ideals in $B$ can be recovered by their intersection with $A$. Such inclusions have attractive properties from the point of view of harmonic analysis and noncommutative geometry. We establish several permanence properties of locally compact groups for which $L^1(G) \subseteq C^*_{\mathrm{red}}(G)$ has the ideal separation property.

Theorems & Definitions (61)

• Definition 1
• Proposition 1
• Theorem 1
• Definition 1
• Proposition 1
• proof
• Proposition 2
• Lemma 2.1
• proof
• Proposition 3