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

Abstract

For a locally compact group $G$ and $p\in (1,\infty )$, 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 Figà-Talamanca--Herz algebra $A_p(G)$. If $2\le q\le p$ or $p\le q\le 2$, we have a contractive inclusion $B_q(G)\subset B_p(G)$. We also show that $B_p(G)$ embeds contractively into the multiplier algebra of $A_p(G)$ and is a dual space. For amenable $G$, this multiplier algebra and $B_p(G)$ are isometrically isomorphic.

19 pages; LaTeX2e; two references added