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The Radon-Nikodym property for the Banach algebras Ap(G) = Ap ∩ Lr(G), where A2(G) is the Fourier algebra, is investigated. A complete solution is given for amenable groups G if 1 < p < ∞ and for arbitrary G if p = 2 and A2(G) has a multiplier bounded approximate identity. The results are new even for G = Rn.

The Radon-Nikodym property for some Banach algebras related to the Fourier algebra