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Open subgroups of $G$ and almost periodic functionals on $A(G)$


Author: Zhiguo Hu
Journal: Proc. Amer. Math. Soc. 128 (2000), 2473-2478
MSC (1991): Primary 22D25, 43A30
DOI: https://doi.org/10.1090/S0002-9939-00-05299-0
Published electronically: February 25, 2000
MathSciNet review: 1662249
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Abstract: Let $G$ be a locally compact group and let $C_{\delta }^{*}(G)$ denote the $C^{*}$-algebra generated by left translation operators on $L^{2}(G)$. Let $AP(\hat {G})$ and $WAP(\hat {G})$ be the spaces of almost periodic and weakly almost periodic functionals on the Fourier algebra $A(G)$, respectively. It is shown that if $G$ contains an open abelian subgroup, then (1) $AP(\hat {G}) = C_{\delta }^{*}(G)$ if and only if $AP(\hat {G})_{c}$ is norm dense in $AP(\hat {G})$; (2) $WAP(\hat {G})$ is a $C^{*}$-algebra if $WAP(\hat {G})_{c}$ is norm dense in $WAP(\hat {G})$, where $X_{c}$ denotes the set of elements in $X$ with compact support. In particular, for any amenable locally compact group $G$ which contains an open abelian subgroup, $G$ has the dual Bohr approximation property and $WAP(\hat {G})$ is a $C^{*}$-algebra.


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Additional Information

Zhiguo Hu
Affiliation: Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada N9B 3P4
Email: zhiguohu@uwindsor.ca

DOI: https://doi.org/10.1090/S0002-9939-00-05299-0
Keywords: Locally compact groups, left regular representation, Fourier and Fourier-Stieltjes algebras, almost periodic functionals, weakly almost periodic functionals, amenable groups
Received by editor(s): December 3, 1997
Received by editor(s) in revised form: September 8, 1998
Published electronically: February 25, 2000
Additional Notes: This research was supported by an NSERC grant
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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