Group algebra modules. III

Abstract:

Let $\Gamma$ be a locally compact group and $K$ a Banach space. The left ${L^1}(\Gamma )$ module $K$ is by definition absolutely continuous under the composition $\ast$ if for $k \in K$ there exist $f \in {L^1}(\Gamma ),k’ \in K$ with $k = f \ast k’$. If the locally compact Hausdorff space $X$ is a transformation group over $\Gamma$ and has a measure quasi-invariant with respect to $\Gamma$, then ${L^1}(X)$ is an absolutely continuous ${L^1}(\Gamma )$ module—the main object we study. If $Y \subseteq X$ is measurable, let ${L_Y}$ consist of all functions in ${L^1}(X)$ vanishing outside $Y$. For $\Omega \subseteq \Gamma$ not locally null and $B$ a closed linear subspace of $K$, we observe the connection between the closed linear span (denoted ${L_\Omega } \ast B$) of the elements $f \ast k$, with $f \in {L_\Omega }$ and $k \in B$, and the collection of functions of $B$ shifted by elements in $\Omega$. As a result, a closed linear subspace of ${L^1}(X)$ is an ${L_Z}$ for some measurable $Z \subseteq X$ if and only if it is closed under pointwise multiplication by elements of ${L^\infty }(X)$. This allows the theorem stating that if $\Omega \subseteq \Gamma$ and $Y \subseteq X$ are both measurable, then there is a measurable subset $Z$ of $X$ such that ${L_\Omega } \ast {L_Y} = {L_Z}$. Under certain restrictions on $\Gamma$, we show that this $Z$ is essentially open in the (usually stronger) orbit topology on $X$. Finally we prove that if $\Omega$ and $Y$ are both relatively sigma-compact, and if also ${L_\Omega } \ast {L_Y} \subseteq {L_Y}$, then there exist ${\Omega _1}$ and ${Y_1}$ locally almost everywhere equal to $\Omega$ and $Y$ respectively, such that ${\Omega _1}{Y_1} \subseteq {Y_1}$; in addition we characterize those $\Omega$ and $Y$ for which ${L_\Omega } \ast {L_\Omega } = {L_\Omega }$ and ${L_\Omega } \ast {L_Y} = {L_Y}$.