idempotent multipliers of figà-talamanca-herz algebras

For a locally compact group g and p ∈ (1, ∞), 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.