$p$-Operator Spaces and Fig\'a-Talamanca-Herz Algebras

TL;DR

This paper extends the theory of operator spaces to $L_p$-based spaces, demonstrating that Figà-Talamanca-Herz algebras become quantised Banach algebras with properties reflecting the underlying group's amenability.

Contribution

It introduces a $p$-operator space framework for $A_p(G)$, linking algebraic properties to group amenability and unifying multiplier concepts within this setting.

Findings

$A_p(G)$ becomes a quantised Banach algebra in the $p$-operator space framework.

Amenability of $A_p(G)$ corresponds to the amenability of $G$.

Various multipliers of $A_p(G)$ fit naturally into the new framework.

Abstract

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.