The ideal separation property for reduced group C⁎-algebras

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 $L^1\left(G\right)\subseteq C_{\mathrm{red}}^{⁎}\left(G\right)$ has the ideal separation property.