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


Authors: Anthony To Ming Lau and Ali Ülger
Journal: Trans. Amer. Math. Soc. 337 (1993), 321-359
MSC: Primary 22D15; Secondary 22D25, 46H99, 46M05
DOI: https://doi.org/10.1090/S0002-9947-1993-1147402-7
MathSciNet review: 1147402
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Abstract: 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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1147402-7
Keywords: Locally compact groups, amenability, Fourier and Fourier-Stieltjes algebras, group algebra, measure algebra, group $ {C^\ast}$-algebra, multiplier algebra, regular representation, Arens regularity, Radon-Nikodym property, Dunford-Pettis property, Schur property
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