On conjugate systems with respect to completely positive maps

TL;DR

This paper investigates the structure of conjugate systems related to completely positive maps on von Neumann algebras, providing a cumulant-based characterization and exploring implications for algebra centers.

Contribution

It introduces a cumulant characterization for conjugate variables and analyzes the structural implications for von Neumann algebras generated by these systems.

Findings

Center of the algebra equals the center of B when conjugate variables exist

Provides a cumulant-based criterion for the existence of conjugate variables

Studies the structural implications of conjugate systems on von Neumann algebras

Abstract

We study the operator-valued partial derivative associated with covariance matrices on a von Neumann algebra B. We provide a cumulant characterization for the existence of conjugate variables and study some structure implications of their existence. Namely, we show that the center of the von Neumann algebra generated by B and its relative commutant is the center of B.