Idempotent multipliers of Figa-Talamanca-Herz algebras

For a locally compact group G and p\in(۱,\infty), let B_p(G) is the multiplier algebra of the Fig\`{a}-Talamanca-Herz algebra A_p(G). For p=۲ and G amenable, the algebra B(G):= B_۲(G) is the usual Fourier-Stieltjes algebra. In this paper, we show that A_p(G) is a Bochner-Schoenberg-Eberlin (BSE) algebra and every clopen subset of G is a synthetic set for A_p(G). Furthermore, we characterize idempotent elements of the Banach algebra B_p(G). This result generalizes the Cohen-Host idempotent theorems for the case of Fig\`{a}-Talamanca-Herz algebras. Characterization of idempotent elements of B_p(G) is of paramount importance to study homomorphisms in Fig\`{a}-Talamanca-Herz algebras.

Keywords:

Figa-Talamanca-Herz algebra , Multiplier algebra , Idempotent element , Fourier algebra , Fourier-Stieltjes algebra