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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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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Article copyright: © Copyright 1989 American Mathematical Society