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

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

2004-02-03

arxiv.org/abs/math/0402018v4

Pacific J. Math. 221 (2005), 379-397

Mathematics

Functional Analysis

19 pages; LaTeX2e; two references added

Scientific paper

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 Fig\`a-Talamanca--Herz algebra $A_p(G)$. If $2 \leq q \leq p$ or $p \leq q \leq 2$, 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.

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