Operator Figà-Talamanca–Herz algebras

Volker Runde¹

¹ Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta Canada, T6G 2G1

Studia Mathematica, Tome 155 (2003) no. 2, pp. 153-170

Résumé

Let $G$ be a locally compact group. We use the canonical operator space structure on the spaces $L^p(G)$ for $p \in [1,\infty]$ introduced by G. Pisier to define operator space analogues $OA_p(G)$ of the classical Figà-Talamanca–Herz algebras $A_p(G)$. If $p \in (1,\infty)$ is arbitrary, then $A_p(G) \subset OA_p(G)$ and the inclusion is a contraction; if $p = 2$, then $OA_2(G) \cong A(G)$ as Banach spaces, but not necessarily as operator spaces. We show that $OA_p(G)$ is a completely contractive Banach algebra for each $p \in (1,\infty)$, and that $OA_q(G) \subset OA_p(G)$ completely contractively for amenable $G$ if $1 p \leq q \leq 2$ or $2 \leq q \leq p \infty$. Finally, we characterize the amenability of $G$ through the existence of a bounded approximate identity in $OA_p(G)$ for one (or equivalently for all) $p \in (1,\infty)$.

Détail

DOI : 10.4064/sm155-2-5

Mots-clés : locally compact group canonical operator space structure spaces infty introduced pisier define operator space analogues classical fig talamanca herz algebras infty arbitrary subset inclusion contraction cong banach spaces necessarily operator spaces completely contractive banach algebra each infty subset completely contractively amenable leq leq leq leq infty finally characterize amenability through existence bounded approximate identity equivalently infty