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Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems

1993; American Mathematical Society; Volume: 337; Issue: 1 Linguagem: Inglês

10.1090/s0002-9947-1993-1147402-7

1088-6850

Anthony Lau, A. Ülger,

Advanced Banach Space Theory

Let $G$ be a locally compact topological group and $A(G)\;[B(G)]$ be, respectively, the Fourier and Fourier-Stieltjes algebras of $G$. It is one of the purposes of this paper to investigate the ${\text {RNP}}$ (= Radon-Nikodym property) and some other geometric properties such as weak $RNP$, the Dunford-Pettis property and the Schur property on the algebras $A(G)$ and $B(G)$, and to relate these properties to the properties of the multiplication operator on the group ${C^\ast }$-algebra ${C^\ast }(G)$. We also investigate the problem of Arens regularity of the projective tensor products ${C^\ast }(G)\hat \otimes A$, when $B(G) = {C^\ast }{(G)^\ast }$ has the ${\text {RNP}}$ and $A$ is any ${C^\ast }$-algebra. Some related problems on the measure algebra, the group algebra and the algebras ${A_p}(G)$, $P{F_p}(G)$, $P{M_p}(G)\;(1 < p < \infty )$ are also discussed.