Idempotent multipliers of Figà-Talamanca-Herz algebras.

For a locally compact group G and p ∈ (1,8), let Bp(G) is the multiplier algebra of the Figà-Talamanca-Herz algebra Ap(G). For p = 2 and G amenable, the algebra B(G) := B2(G) is the usual Fourier-Stieltjes algebra. In this paper, we show that Ap(G) is a Bochner-Schoenberg-Eberlin (BSE) algebra and every clopen subset of G is a synthetic set for Ap(G). Furthermore, we characterize idempotent elements of the Banach algebra Bp(G). This result generalizes the Cohen-Host idempotent theorems for the case of Figà-Talamanca-Herz algebras. Characterization of idempotent elements of Bp(G) is of paramount importance to study homomorphisms in Figà-Talamanca-Herz algebras. [ABSTRACT FROM AUTHOR]