A characterization of absolutely dilatable Schur multipliers

TL;DR

This paper characterizes when Schur multipliers on $L^2$ spaces are absolutely dilatable, linking this property to the existence of a von Neumann algebra and a specific trace-related function.

Contribution

It provides a complete characterization of absolutely dilatable Schur multipliers in terms of von Neumann algebra structures and trace conditions.

Findings

Characterization of absolutely dilatable Schur multipliers via von Neumann algebra representations.

Identification of conditions involving a normalized trace and a bounded function into a von Neumann algebra.

Extension of the theory to $\sigma$-finite measure spaces and separable cases.

Abstract

Let $M$ be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let $T\colon M\to M$ be a contraction. We say that $T$ is absolutely dilatable if there exist another von Neumann algebra $M'$ equipped with a nsf trace, a $w^*$-continuous trace preserving unital $*$-homomorphim $J\colon M\to M'$ and a trace preserving $*$-automomorphim $U\colon M'\to M'$ such that $T^k=E U^k J$ for all integer $k\geq 0$, where $E\colon M'\to M$ is the conditional expectation associated with $J$. Given a $\sigma$-finite measure space $(\Omega,\mu)$, we characterize bounded Schur multipliers $\phi\in L^\infty(\Omega^2)$ such that the Schur multiplication operator $T_\phi\colon B(L^2(\Omega))\to B(L^2(\Omega))$ 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 $\tau_N$, and of a $w^*$-continuous essentially bounded function $d\colon\Omega\to N$ such that $\phi(s,t)=\tau_N(d(s)^*d(t))$ for almost every $(s,t)\in\Omega^2$.