On conjugate systems with respect to completely positive maps

TL;DR

The paper extends Dabrowski's factoriality results to the operator-valued setting by introducing the $\eta$-partial derivative $\partial_\eta$ and $(B,\eta)$-conjugate systems for a tuple of self-adjoint generators. It provides a cumulant-based characterization of conjugate variables and proves that the existence of such systems with diagonal covariance $\eta=(\delta_{ij}E_B)$ forces the center to satisfy $Z(B\vee(B'\cap M))=Z(B)$, yielding $Z(M)\subset Z(B)$. In the diagonal CP-map case, the authors show relative diffuseness of $B\vee(B'\cap M)$ with respect to $B$ and, under suitable conditions, irreducibility of intermediate algebras when $B$ is a factor. They also establish absence of atoms for certain $B$-linear polynomials in the presence of a conjugate system and provide a robust cumulant framework that generalizes operator-valued free probability techniques to study the center and structure of amalgamated 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.

Theorems & Definitions (44)

• Theorem A: Theorem \ref{['thm:main']}
• Theorem B: Theorem \ref{['thm:no atom']}
• Theorem C: Theorem \ref{['thm:cumulants']}
• Theorem 1.1
• Lemma 1.2
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4: Theorem \ref{['thm c']}
• proof