Schur multipliers of Cartan pairs

TL;DR

This paper generalizes classical Schur multipliers to the setting of separable von Neumann algebras with Cartan masas, characterizing them as normal A-bimodule maps and exploring their structure in relation to the hyperfinite II_1 factor.

Contribution

It introduces a new framework for Schur multipliers in von Neumann algebras with Cartan masas and characterizes their properties and limitations.

Findings

Schur multipliers are characterized as normal A-bimodule maps.

In certain cases, Schur multipliers from the extended Haagerup tensor product are strictly contained in all Schur multipliers.

The structure of Schur multipliers depends on the presence of hyperfinite II_1 summands.

Abstract

We define the Schur multipliers of a separable von Neumann algebra M with Cartan masa A, generalising the classical Schur multipliers of $B(\ell^2)$. We characterise these as the normal A-bimodule maps on M. If M contains a direct summand isomorphic to the hyperfinite II_1 factor, then we show that the Schur multipliers arising from the extended Haagerup tensor product $A \otimes_{eh} A$ are strictly contained in the algebra of all Schur multipliers.