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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.