Group algebra modules. IV

Abstract:

Let $\Gamma$ be a locally compact group, $\Omega$ a measurable subset of $\Gamma$, and let ${L_\Omega }$ denote the subspace of ${L^1}(\Gamma )$ consisting of all functions vanishing off $\Omega$. Assume that ${L_\Omega }$ is a subalgebra of ${L^1}(\Gamma )$. We discuss the collection ${\Re _\Omega }(K)$ of all module homomorphisms from ${L_\Omega }$ into an arbitrary Banach space $K$ which is simultaneously a left ${L^1}(\Gamma )$ module. We prove that ${\Re _\Omega }(K) = {\Re _\Omega }({K_0}) \oplus {\Re _\Omega }({K_{\text {abs} }})$, where ${K_0}$ is the collection of all $k \in K$ such that $fk = 0$, for all $f \in {L^1}(\Gamma )$, and where ${K_{\text {abs} }}$ consists of all elements of $K$ which can be factored with respect to the module composition. We prove that ${\Re _\Omega }({K_0})$ is the collection of linear continuous maps from ${L_\Omega }$ to ${K_0}$ which are zero on a certain measurable subset of $X$. We reduce the determination of ${\Re _\Omega }({K_{\text {abs} }})$ to the determination of ${\Re _\Gamma }({K_{\text {abs} }})$. Denoting the topological conjugate space of $K$ by ${K^ \ast }$, we prove that ${({K_{\text {abs} }})^ \ast }$ is isometrically isomorphic to ${\Re _\Omega }({K^ \ast })$. Finally, we discuss module homomorphisms $R$ from ${L_\Omega }$ into ${L^1}(X)$ such that for each $f \in {L_\Omega },Rf$ vanishes off $Y$.