On convolution dominated operators (Q346467)

Let \(G\) be a locally compact group. An operator \(A:L^{2}(G)\rightarrow L^{2}(G) \) is called convolution dominated if there exists \(a\in L^{1}(G)\) such that \(|Af(x)|\leq a\ast |f(x)|\) for almost all \(x\in G\). The set of all convolution dominated operators is denoted by \(CD(G)\). For non-discrete groups, the subalgebra of regular dominated operators \((CD_{\mathrm{reg}}(G))\) studied. The authors show that, for an amenable group \(G\) (which is rigidly symmetric as a discrete group), the elements of \(CD_{\mathrm{reg}}(G)\) are invertible with respect to \(CD_{\mathrm{reg}}(G)\) if and only if these elements are bounded in \(L^{2}(G).\)