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Weakly almost periodic elements in $ L\sb \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
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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