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.

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