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Journal of Operator Theory

Volume 63, Issue 1, Winter 2010  pp. 47-83.

$p$-Operator spaces and Figa-Talamanca--Herz algebras

Authors: Matthew Daws
Author institution:St John's College, Oxford, OX1 3JP, United Kingdom

Summary:  We study a generalisation of operator spaces modelled on\break $L_p$ spaces, instead of Hilbert spaces, using the notion of $p$-complete boundedness, as studied by Pisier and Le Merdy. We show that the Fig\-Talamanca--Herz algebras $A_p(G)$ 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 $A_p(G)$ (including Herz's generalisation of the Fourier--Stieltjes algebra) naturally fit into this framework.

Keywords: Operator space, locally compact group, $SQ_p$-space, Figa-Talamanca--Herz algebra, multiplier algebra, amenability.

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