Weakly almost periodic elements in 𝐿_{∞}(𝐺) of a locally compact group

Abstract

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