Convolution-dominated integral operators
Volume 89 / 2010
Abstract
For a locally compact group $G$ we consider the algebra $CD(G)$ of convolution-dominated operators on $L^{2}(G)$, where 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 $f \in L^2(G)$ $$ |Af(x)| \leq a \star |f|(x), \quad\ \hbox{for almost all } x\in G.\tag{1} $$ The case of discrete groups was treated in previous publications \cite{fgl08a, fgl08}. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the products are restricted to those given by multiplication with left uniformly continuous functions. This algebra, $CD_{reg}(G)$, is canonically isomorphic to a twisted $L^{1}$-algebra. For amenable $G$ that is rigidly symmetric as a discrete group we show the following result: An element of $CD_{reg}(G)$ is invertible in $CD_{reg}(G)$ if and only if it is invertible as a bounded operator on $L^2(G)$. This report is about work in progress. Complete details and further results will be given in a paper still in preparation.
