Analytic functions relative to a covariance map $\eta$: I. Generalized Haagerup products and analytic relations

TL;DR

This paper develops a framework for analytic functions related to covariance maps in von Neumann algebras, introducing new operator space constructions and analyzing relations among non-commutative variables with finite Fisher information.

Contribution

It generalizes Haagerup tensor products to construct dual operator spaces and defines a matrix normed algebra of analytic functions capturing free semicircular relations.

Findings

Constructed dual operator spaces from generalized Haagerup tensor products.

Defined a matrix normed algebra of analytic functions related to covariance maps.

Proved variables with finite Fisher information have no analytic relations in this framework.

Abstract

We generalize module weak-* Haagerup tensor products to obtain complete quotients of normal Haagerup tensor product included in canonical Hilbert spaces associated to completely positive normal (covariance) maps $\eta$ on a finite von Neumann algebra $B$. We construct in this way dual operator spaces, providing new examples even in the case of module extended Haagerup tensor products. This is the basis for defining a matrix normed algebra of analytic functions that captures the relations of free semicircular variables with covariance $\eta$. We prove that a class of non-commutative random variables having finite Fisher information relative to $\eta$ have also no analytic relations among our class of analytic functions.