  For a locally compact group $G$ we consider the algebra $CD(G)$ of
convolution dominated operators on $L^{2}(G)$: An operator $A:L^2(G)\to L^2(G)$
is called convolution dominated if there exists $a\in L^1(G)$ such that for all
$f \in L^2(G)$ $ |Af(x)| \leq a * |f| (x)$, for almost all $x \in G$.
  In the case of discrete groups those operators can be dealt with quite
sufficiently if the group in question is rigidly symmetric. For non-discrete
groups we investigate the subalgebra of regular convolution dominated operators
$CD_{reg}(G)$.
  For amenable $G$ which is rigidly symmetric as a discrete group, we show that
any element of $CD_{reg}(G)$ is invertible in $CD_{reg}(G)$ if it is invertible
as a bounded operator on $L^2(G)$.
  We give an example of a symmetric group $E$ for which the convolution
dominated operators are not inverse-closed in the bounded operators on
$L^2(E)$.
