Dilation properties of measurable Schur multipliers and Fourier multipliers

TL;DR

This paper establishes new dilation results for measurable Schur and Fourier multipliers on non-commutative Lp spaces, demonstrating their isometric dilations and extending to multivariable cases, advancing operator algebra theory.

Contribution

It introduces novel dilation theorems for measurable Schur and Fourier multipliers on non-commutative Lp spaces, including multivariable extensions.

Findings

Existence of invertible isometric dilations for Schur multipliers.

Similar dilation results for Fourier multipliers on unimodular groups.

Extension of dilation results to multivariable settings.

Abstract

In the article, we find new dilatation results on non-commutative $L_p$ spaces. We prove that any selfadjoint, unital, positive measurable Schur multiplier on some $B(L^2(\Sigma))$ admits, for all $1\leq p<\infty$, an invertible isometric dilation on some non-commutative $L^p$-space. We obtain a similar result for selfadjoint, unital, completely positive Fourier multiplier on $VN(G)$, when $G$ is a unimodular locally compact group. Furthermore, we establish multivariable versions of these results.