  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
Figa-Talamanca-Herz algebras $A_p(G)$. If $p \in (1,\infty)$ is arbitrary, then
$A_p(G) \subset OA_p(G)$ such that the inclusion is a contraction; if p = 2,
then $OA_2(G) \cong A(G)$ as Banachspaces 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) $p \in (1,\infty)$.
