Absence of algebraic relations and of zero divisors under the assumption of finite non-microstates free Fisher information

TL;DR

This paper demonstrates that in certain non-commutative probability spaces, the existence of conjugate variables prevents algebraic relations among generators and ensures the absence of zero divisors under finite free Fisher information.

Contribution

It establishes a link between conjugate variables, algebraic independence, and zero divisors in the context of free probability with finite Fisher information.

Findings

Conjugate variables exclude algebraic relations among generators.

Finite free Fisher information implies no zero divisors in the algebra.

Results apply to tracial, finitely generated W*-probability spaces.

Abstract

We show that in a tracial and finitely generated $W^\ast$-probability space existence of conjugate variables in an appropriate sense exclude algebraic relations for the generators. Moreover, under the assumption of finite non-microstates free Fisher information, we prove that there are no zero divisors in the sense that the product of any non-commutative polynomial in the generators with any element from the von Neumann algebra is zero if and only if at least one of those factors is zero.