$p$-Operator Spaces and Figá-Talamanca-Herz Algebras

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 Figá-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.

LaTeX, 30 pages. Updated with minor corrections and added references

Cited by in corpus (8)

• Quantitative $K$-theory for Banach algebras
• Noncommutative Figa-Talamanca-Herz algebras for Schur multipliers
• The $p$-approximation property for simple Lie groups with finite center
• Column and row operator spaces over QSL_p-spaces and their use in abstract harmonic analysis
• On the isometrisability of group actions on p-spaces
• Weak*-continuity of invariant means on spaces of matrix coefficients
• Representations of étale groupoids on $L^p$-spaces
• On the functorial properties of the p-analog of the Fourier-Stieltjes algebras and their homomorphisms