Convolution dominated operators on compact extensions of abelian groups

Abstract

If $G$ is a locally compact group, $CD(G)$ the algebra of convolution dominated operators on $L^2(G)$, then an important question is: Is $\mathbb {C}1+CD(G)$ (or $CD(G)$ if $G$ is discrete) inverse-closed in the algebra of bounded operators on $L^2(G)$?

In this note we answer this question in the affirmative, provided $G$ is such that one of the following properties is satisfied.

1. There is a discrete, rigidly symmetric, and amenable subgroup $H\subset G$ and a (measurable) relatively compact neighbourhood of the identity $U$, invariant under conjugation by elements of $H$, such that $\{hU\;:\;h\in H\}$ is a partition of $G$.

2. The commutator subgroup of $G$ is relatively compact. (If $G$ is connected, this just means that $G$ is an IN group.)

All known examples where $CD(G)$ is inverse-closed in $B(L^2(G))$ are covered by this.