Invariant Subsets and Homological Properties of Orlicz Modules over Group Algebras

Abstract

Let $G$ be a locally compact group with left Haar measure. We study the closed convex left invariant subsets of $L^{\Phi}(G)$ and characterize affine mappings from the space of nonnegative functions in $L^{1}(G)$ of norm $1$ into $L^{\Phi}(G)$ spaces. We apply the results to the study of the multipliers of $L^{\Phi}(G)$. We also investigate the homological properties of $L^{\Phi}(G)$ as a Banach left $L^{1}(G)$-module such as projectivity, injectivity and flatness.