Haagerup tensor products and Schur multipliers

TL;DR

This paper explores different classes of Schur multipliers and establishes that those with respect to measures and spectral measures are isometrically equivalent to Haagerup tensor products of $L^e$ spaces, unifying their structure.

Contribution

It proves that Schur multipliers with respect to measures and spectral measures coincide isometrically with Haagerup tensor products of $L^e$ spaces, extending known results for discrete Schur multipliers.

Findings

Schur multipliers with respect to measures and spectral measures are isometrically equivalent to Haagerup tensor products.

The main result generalizes the known discrete case to measure and spectral measure contexts.

The paper establishes a unifying framework for various classes of Schur multipliers.

Abstract

In this paper we compare various classes of Schur multipliers: classical matrix Schur multipliers, discrete Schur multipliers, Schur multipliers with respect to measures and Schur multipliers with respect to spectral measures. The main result says that in the case of Schur multipliers with respect to measures and spectral measures such Schur multipliers coincide isometrically with the Haagerup tensor products of the corresponding $L^\infty$ spaces. We deduce this result from a well known analogue of it for discrete Schur multipliers.