p-Operator spaces and Figà-Talamanca-Herz algebras

We study a generalisation of operator spaces modelled on L spaces, instead of Hilbert spaces, using the notion of p-complete boundedness, as studied by Pisier and Le Merdy. We show that the Figà-Talamanca-Herz algebras A(G) become quantised Banach algebras in this framework, and that amenability of these algebras corresponds to amenability of the locally compact group G, extending the result of Ruan about A(G). We also show that various notions of multipliers of A(G) (including Herz's generalisation of the Fourier-Stieltjes algebra) naturally fit into this framework.