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

For a locally compact group $G$ and $p\in (1,\mathrm{\infty })$, we define $B_p(G)$ 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 $B_p(G)$ is a commutative Banach algebra that contractively (isometrically, if $G$ is amenable) contains the Fig`a-Talamanca–Herz algebra $A_p(G)$. If $2\le q\le p$ or $p \leq q \leq 2$,wehaveacontractiveinclusion$B_q(G) \subset B_p(G)$\mathrm{.}Wealsoshowthat$B_p(G)$embedscontractivelyintothemultiplieralgebraof$A_p(G)$ and is a dual space. For amenable $G$, this multiplier algebra and $B_p(G)$ are isometrically isomorphic.