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

TL;DR

This paper investigates when the inclusion of the group algebra into the reduced group $C^*$-algebra allows for the recovery of closed ideals via intersection, revealing important properties for harmonic analysis and noncommutative geometry.

Contribution

It establishes permanence properties of groups where the algebra inclusion has the ideal separation property, advancing understanding of the structure of reduced group $C^*$-algebras.

Findings

Identifies conditions under which the ideal separation property holds

Shows stability of this property under certain group operations

Connects algebraic properties with harmonic analysis applications

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.