A characterization of absolutely dilatable Schur multipliers

Abstract

Let $M$ be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let $T:M\to M$ be a contraction. We say that $T$ is absolutely dilatable if there exist another von Neumann algebra $M^{\prime }$ equipped with a nsf trace, a $w^{\ast }$-continuous trace preserving unital $\ast$-homomorphim $J:M\to M^{\prime }$ and a trace preserving $\ast$-automomorphim $U:M^{\prime }\to M^{\prime }$ such that $T^k=E{U}^{k}J$ for all integer $k\ge 0$, where $E:M^{\prime }\to M$ is the conditional expectation associated with $J$. Given a $\ddot{I}ƒ$-finite measure space $(\hat{I}\text{c◯},\hat{I}¼)$, we characterize bounded Schur multipliers $\ddot{I}•\in L^{\infty }(\hat{I}{\text{c◯}}^2)$ such that the Schur multiplication operator $T_{\ddot{I}}•:B(L^2(\hat{I}\text{c◯}))\to B(L^2(\hat{I}\text{c◯}))$ is absolutely dilatable. In the separable case, they are characterized by the existence of a von Neumann algebra $N$ with a separable predual, equipped with a normalized normal faithful trace $\ddot{I}{„}_N$, and of a $w^{\ast }$-continuous essentially bounded function $d:\hat{I}\text{c◯}\to N$ such that $\ddot{I}•(s,t)=\ddot{I}{„}_N(d(s{)}^{\ast }d(t))$ for almost every $(s,t)\in \hat{I}{\text{c◯}}^2$.

Revised version, published in Avances in Mathematics