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For a locally compact group G and 1 < p < ∞, let Ap (G) be the Herz-Figà-Talamanca algebra and let PMp (G) be its dual Banach space. For a Banach Ap (G)-module X of PMp (G), we prove that the multiplier space ℳ(Ap (G); X*) is the dual Banach space of QX , where QX is the norm closure of the linear span Ap (G)X of u f for u 2 Ap (G) and f ∈ X in the dual of ℳ(Ap (G); X*). If p = 2 and PFp (G) ⊆ X, then Ap (G)X is closed in X if and only if G is amenable. In particular, we prove that the multiplier algebra MAp (G) of Ap (G) is the dual of Q, where Q is the completion of L 1(G) in the ‖ · ‖ M -norm. Q is characterized by the following: f ∈ Q if an only if there are ui ∈ Ap (G) and fi ∈ PFp (G) (i = 1; 2, … ) with such that on MAp (G). It is also proved that if Ap (G) is dense in MAp (G) in the associated w*-topology, then the multiplier norm and ‖ · ‖ Ap (G)-norm are equivalent on Ap (G) if and only if G is amenable.
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