Unconditionality, Fourier multipliers and Schur multipliers

2011/11/07 by Cédric Arhancet, Arhancet, Cédric
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA

paper · pdf · doi:10.48550/arxiv.1111.1664

minor corrections; 17 pages ; to appear in Colloquium Mathematicum

arxiv created 2012/04/02 · arxiv updated 2012/04/03

Abstract

Let G be an infinite locally compact abelian group. If X is Banach space, we show that if every bounded Fourier multiplier T on L²(G) has the property that T\ot Id_X is bounded on L²(G,X) then the Banach space X is isomorphic to a Hilbert space. Moreover, if 1<p<∞, p\not=2, we prove that 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 to determine if an operator space is completely isomorphic to an operator Hilbert space.

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