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The Radon-Nikodym property for some Banach algebras related to the Fourier algebra. (English) Zbl 1230.43003

Let \(G\) be a locally compact group and \(A_p(G)\) the Figa-Talamanca-Herz Banach algebra (\(1<p<\infty\)), i.e. the algebra generated by \(L^{p'}(G)\ast L^{p\vee}(G)\) under a suitable norm. The author studies the Radon-Nikodym Property (RNP) for the Banach algebra \(A^r_p(G)=A_p(G)\cap L^p(G)\). His main result is Theorem 2.2:
Let \(G\) be unimodular and either
(1) \(1<p<\infty\) and \(G\) is amenable, or
(2) \(p=2\) and \(A_p(G)\) has a multiplier bounded approximate identity.
Then for each \(r, 1\leq r\leq \max\{p,p'\}\), \(A_p^r(G)\) has the RNP.
The author provides examples that for larger \(r\) the above fails to be true. The results are new even for \(G={\mathbb R}^n\).

MSC:

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
46J10 Banach algebras of continuous functions, function algebras
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46J20 Ideals, maximal ideals, boundaries
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A80 Analysis on other specific Lie groups
22E30 Analysis on real and complex Lie groups
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