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 L²(G) has the property that $T\otimes I{d}_X$ is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on $L^p\left(G\right)$ 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...