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Transactions of the American Mathematical Society

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



The second conjugate algebra of the Fourier algebra of a locally compact group

Author: Anthony To Ming Lau
Journal: Trans. Amer. Math. Soc. 267 (1981), 53-63
MSC: Primary 43A30; Secondary 22D25
MathSciNet review: 621972
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Abstract: Let $ G$ be a locally compact group and let $ VN(G)$ denote the von Neumann algebra generated by the left translations of $ G$ on $ {L_2}(G)$. Then $ VN{(G)^{\ast}}$, when regarded as the second conjugate space of the Fourier algebra of $ G$, is a Banach algebra with the Arens product. We prove among other things that when $ G$ is amenable, $ VN{(G)^{\ast}}$ is neither commutative nor semisimple unless $ G$ is finite. We study in detail the class of maximal regular left ideals in $ VN{(G)^{\ast}}$. We also show that if $ {G_1}$ and $ {G_2}$ are discrete groups, then $ {G_1}$ and $ {G_2}$ are isomorphic if and only if $ VN{({G_1})^{\ast}}$ and $ VN{({G_2})^{\ast}}$ are isometric order isomorphic.

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Keywords: Arens product, second conjugate algebra, Fourier algebra, Fourier-Stieltjes algebra, left regular representation, amenable locally compact group, regular ideal, group $ {C^{\ast}}$-algebra, Tauberian property, radical, uniformly continuous functionals, topological invariant mean
Article copyright: © Copyright 1981 American Mathematical Society