  We study a generalisation of operator spaces modelled on $L_p$ spaces,
instead of Hilbert spaces, using the notion of $p$-complete boundedness, as
studied by Pisier and Le Merdy. We show that the Fig\'a-Talamanca-Herz Algebras
$A_p(G)$ becomes quantised Banach algebras in this framework, and that the
cohomological notion of amenability of these algebras corresponds to
amenability of the locally compact group $G$. We thus argue that we have
presented a generalised of the use of operator spaces in studying the Fourier
algebra $A(G)$, in the spirit of Ruan. Finally, we show that various notions of
multipliers of $A_p(G)$ (including Herz's generalisation of the
Fourier-Stieltjes algebra) naturally fit into this framework.
