Continuity of translation in the dual of $L^ \infty (G)$ and related spaces

Abstract:

Let $X$ be a Banach space and $G$ a locally compact Hausdorff group that acts as a group of isometric linear operators on $X$. The operation of $x \in G$ on $X$ will be denoted by ${L_x}$. We study the set ${X_c}$ of elements $\mu \in X$ such that $x \mapsto {L_x}\mu$ is continuous with respect to the topology on $G$ and the norm-topology on $X$. The spaces $X$ studied include $M{(G)^{\ast } },{\text {LUC}}{(G)^{\ast } },{L^\infty }{(G)^{\ast } },{\text {VN}}(G)$, and ${\text {VN}}{(G)^{\ast } }$. In most cases, characterizations of ${X_c}$ do not appear to be possible, and we give constructions that illustrate this. We relate properties of ${X_c}$ to properties of $G$. For example, if ${X_c}$ is sufficiently small, then $G$ is compact, or even finite, depending on the case. We give related results and open problems.