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

Authors:
Anthony To Ming Lau and James C. S. Wong

Journal:
Proc. Amer. Math. Soc. **107** (1989), 1031-1036

MSC:
Primary 43A15; Secondary 22D25, 43A07

DOI:
https://doi.org/10.1090/S0002-9939-1989-0991701-0

MathSciNet review:
991701

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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.

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© Copyright 1989
American Mathematical Society