Unconditionality, Fourier multipliers and Schur multipliers (Q2893050)

Let \(G\) be an infinite locally compact abelian group and \(X\) be a Banach space. Suppose \(1\leq p\leq \infty.\) Denote by \(S^p=S^p(l^2)\) the Schatten space. Let \(\Omega\) be a measure space. A linear map \(T: L^p(\Omega)\to L^p(\Omega)\) is completely bounded if \(T\otimes \mathrm{Id}_{S^p}\) extends to a bounded operator \(T\otimes \mathrm{Id}_{S^p}: L^p(\Omega,S^p)\to L^p(\Omega,S^p).\)NEWLINENEWLINEThe author shows that if every bounded Fourier multiplier \(T\) on \(L^2(G)\) has the property that \(T\otimes \mathrm{Id}_{X}\) is bounded on \(L^2(G,X)\), then \(X\) is isomorphic to a Hilbert space. He generalizes a result obtained by M. Defant and M. Junge to infinite compact abelian groups. Moreover, he proves that there exists a bounded Fourier multiplier on \(L^p(G)\) which is not completely bounded if \(1<p<\infty\), \(p\neq 2.\)NEWLINENEWLINELet \(E\) be an operator space and \(S^p(E)\) denote the vector-valued noncommutative \(L^p\)-space. The author examines unconditionality from the point of view of Schur multipliers. More precisely, he establishes the following two equivalent conditions:NEWLINENEWLINE1. There exists a positive constant \(C\) such that NEWLINE\[NEWLINE\left\| \sum^{n}_{i,j=1}t_{ij}e_{ij}\otimes x_{ij}\right\|_{S^2(E)}\leq C \sup_{1\leq i,j\leq n}|t_{ij}|\left\|\sum^{n}_{i,j=1} e_{ij}\otimes x_{ij}\right\|_{S^2(E)}NEWLINE\]NEWLINE for any \(n\in \mathbb{N}\), \(t_{ij}\in \mathbb{C}\) and \(x_{ij}\in E\).NEWLINENEWLINE2. The operator space \(E\) is completely isomorphic to an operator Hilbert space \(\mathrm{OH}(I)\) for some index set \(I\).