Unconditionality, Fourier multipliers and Schur multipliers

Volume 127 / 2012

Cédric Arhancet Colloquium Mathematicum 127 (2012), 17-37 MSC: Primary 43A15, 43A22, 46L07; Secondary 46L51. DOI: 10.4064/cm127-1-2

Abstract

Let $G$ be an infinite locally compact abelian group and $X$ be a Banach space. We show that if every bounded Fourier multiplier $T$ on $L^2(G)$ has the property that $T\otimes {\rm Id}_X$ is bounded on $L^2(G,X)$ then $X$ is isomorphic to a Hilbert space. Moreover, we prove that if $1< p< \infty $, $p\not =2$, then there exists a bounded Fourier multiplier on $L^p(G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions for an operator space to be completely isomorphic to an operator Hilbert space.

Authors

  • Cédric ArhancetLaboratoire de Mathématiques
    Université de Franche-Comté
    25030 Besançon Cedex, France
    e-mail

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