  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
$\mathbb{C}1+CD(G)$ (respectively $CD(G)$ if $G$ is discrete) inverse-closed in
the 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
fulfilled (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.
