Weakly almost periodic elements in $L_ \infty (G)$ of a locally compact group

Abstract:

Let $G$ be a locally compact abelian group with dual group $G$ acting on ${L_\infty }(G)$ by pointwise multiplication. We show that if ${L_\infty }(G)$ contains a nonzero element $f$ such that $O(f) = \left \{ {x \cdot f:\chi \in \hat G} \right \}$ is relatively compact in the weak (or norm) topology of ${L_\infty }(G)$, then $G$ is discrete. In this case $O(f)$ is relatively compact in the weak or norm topology of ${L_\infty }(G)$ if and only if $f$ vanishes at infinity. A related result when $G$ acts on the von Neumann algebra $VN(G)$ is also determined.