This is an announcement for the paper "Unconditionality, Fourier multipliers and Schur multipliers" by Cedric Arhancet.

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. Indeed, we give several sufficient conditions to know if a Banach space or an operator space is isomorphic to a Hilbert space or completely isomorphic to an operator Hilbert space.

Archive classification: math.FA math.OA

Remarks: 16 pages

Submitted from: cedric.arhancet@univ-fcomte.fr

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1111.1664

or

http://arXiv.org/abs/1111.1664