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arXiv · 1111.1664

Unconditionality, Fourier multipliers and Schur multipliers

Cédric Arhancet
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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 $L^2(G)$ has the property that $T\ot Id_X$ is bounded on $L^2(G,X)$ then the Banach space $X$ is isomorphic to a Hilbert space. Moreover, if $1<p<\infty$, $p\not=2$, 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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Cédric Arhancet. 2011-11-07. Unconditionality, Fourier multipliers and Schur multipliers. https://arxiv.org/abs/1111.1664

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