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Transactions of the American Mathematical Society

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



Uniformly continuous functionals on the Fourier algebra of any locally compact group

Author: Anthony To Ming Lau
Journal: Trans. Amer. Math. Soc. 251 (1979), 39-59
MSC: Primary 43A60; Secondary 22D25
MathSciNet review: 531968
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Abstract: Let G be any locally compact group. Let $ VN\,(G)$ be the von Neumann algebra generated by the left regular representation of G. We study in this paper the closed subspace $ UBC\mathop {(G)}\limits^ \wedge $ of $ VN\, (G)$ consisting of the uniformly continuous functionals as defined by E. Granirer. When G is abelian, $ UBC\mathop {(G)}\limits^ \wedge $ is precisely the bounded uniformly continuous functions on the dual group Ĝ. We prove among other things that if G is amenable, then the Banach algebra $ UBC\mathop {(G)}\limits^ \wedge {\ast}$ (with the Arens product) contains a copy of the Fourier-Stieltjes algebra in its centre. Furthermore, $ UBC\mathop {(G)}\limits^ \wedge {\ast}$ is commutative if and only if G is discrete. We characterize $ W\mathop {(G)}\limits^ \wedge $, the weakly almost periodic functionals, as the largest subspace X of $ VN\, (G)$ for which the Arens product makes sense on $ {X^ {\ast} }$ and $ {X^ {\ast} }$ is commutative. We also show that if G is amenable, then for certain subspaces Y of $ VN\, (G)$ which are invariant under the action of the Fourier algebra $ A\, (G)$, the algebra of bounded linear operators on Y commuting with the action of $ A\, (G)$ is isometric and algebra isomorphic to $ {X^ {\ast} }$ for some $ X \subseteq UBC(\mathop {G)}\limits^ \wedge $.

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Keywords: Locally compact group, amenable group, regular representation, Fourier algebra, Fourier-Stieltjes algebra, positive definite function, $ {C^ {\ast} }$-group algebra, almost periodic functionals, uniformly continuous functionals, multipliers, invariant mean, second conjugate algebra, Arens product
Article copyright: © Copyright 1979 American Mathematical Society