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

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

2006-11-27

arxiv.org/abs/math/0611833v3

J. Operator Theory 63 (2010), no. 1, 47-83

Mathematics

Operator Algebras

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

Scientific paper

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.

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