Operator Figa-Talamanca-Herz algebras

Abstract

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 $O{A}_p(G)$ of the classical Figa-Talamanca-Herz algebras $A_p(G)$. If $p\in (1,\infty )$ is arbitrary, then $A_p(G)\subset O{A}_p(G)$ such that the inclusion is a contraction; if p = 2, then $O{A}_2(G)\cong A(G)$ as Banachspaces spaces, but not necessarily as operator spaces. We show that $O{A}_p(G)$ is a completely contractive Banach algebra for each $p\in (1,\infty )$, and that $O{A}_q(G)\subset O{A}_p(G)$ completely contractively for amenable $G$ if $1<p\le q\le 2$ or $2\le q\le p<\infty$. Finally, we characterize the amenability of G through the existence of a bounded approximate identity in $O{A}_p(G)$ for one (or equivalently for all) $p\in (1,\infty )$.

20 pages; some typos eliminated