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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups

Author: Ching Chou
Journal: Trans. Amer. Math. Soc. 274 (1982), 141-157
MSC: Primary 43A60
MathSciNet review: 670924
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Abstract: A noncompact locally compact group $ G$ is called an Eberlein group if $ W(G) = B{(G)^ - }$ where $ W(G)$ is the algebra of continuous weakly almost periodic functions on $ G$ and $ B{(G)^ - }$ is the uniform closure of the Fourier-Stieltjes algebra of $ G$. We show that if $ G$ is a noncompact $ [IN]$-group or a noncompact nilpotent group then $ W(G)/B{(G)^ - }$ contains a linear isometric copy of $ {l^\infty }$. In particular, $ G$ is not an Eberlein group. On the other hand, finite direct products of Euclidean motion groups and, by a result of W. Veech, noncompact semisimple analytic groups with finite centers are Eberlein groups.

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Keywords: Locally compact groups, weakly almost periodic functions, Fourier-Stieltjes algebras, unitary representations, weak Sidon sets, relatively dense sets, $ {C^ \ast }$-algebras, $ [IN]$-groups, nilpotent groups, motion groups, semisimple groups, weakly almost periodic compactification
Article copyright: © Copyright 1982 American Mathematical Society