Representations of locally compact groups on QSLp-spaces and a p-analog of the Fourier–Stieltjes algebra

Abstract

For a locally compact group G and p ∈ (1, oo), we define Bp(G) to be the space of all coefficient functions of isometric representations of G on quotients of subspaces of L p spaces. For p = 2, this is the usual Fourier-Stieltjes algebra. We show that Bp(G) is a commutative Banach algebra that contractively (isometrically, if G is amenable) contains the Figa-Talamanca-Herz algebra Ap(G). If 2 ≤ q ≤ p or p ≤ q ≤ 2, we have a contractive inclusion B q (G) ⊂ Bp(G). We also show that Bp(G) embeds contractively into the multiplier algebra of Ap(G) and is a dual space. For amenable G, this multiplier algebra and Bp(G) are isometrically isomorphic.