Column and row operator spaces - which we denote by COL and ROW, respectively - over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group $G$ and $p,p' \in (1,\infty)$ with $\frac{1}{p} + \frac{1}{p'} = 1$, we use the operator space structure on $CB(COL(L^{p'}(G)))$ to equip the Figa-Talamanca-Herz algebra $A_p(G)$ with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for $p \leq q \leq 2$ or $2 \leq q \leq p$ and amenable $G$, the canonical inclusion $A_q(G) \subset A_p(G)$ is completely bounded (with cb-norm at most $K_G^2$, where $K_G$ is Grothendieck's constant). As an application, we show that $G$ is amenable if and only if $A_p(G)$ is operator amenable for all - and equivalently for one - $p \in (1,\infty)$; this extends a theorem by Z.-J. Ruan.

25 pages; some minor, hopefully clarifying revisions