Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension

TL;DR

This paper demonstrates that in certain non-commutative probability spaces, maximal free entropy dimension ensures the absence of algebraic relations, zero divisors, and atomic distributions for polynomials in generators.

Contribution

It establishes that maximal non-microstates free entropy dimension implies no algebraic relations, zero divisors, or atoms in the distribution of polynomials in generators.

Findings

Existence of conjugate variables excludes algebraic relations.

Maximal free entropy dimension implies no zero divisors.

Distributions of non-constant polynomials have no atoms.

Abstract

We show that in a tracial and finitely generated $W^\ast$-probability space existence of conjugate variables excludes algebraic relations for the generators. Moreover, under the assumption of maximal non-microstates free entropy dimension, 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. In particular, this shows that in this case the distribution of any non-constant self-adjoint non-commutative polynomial in the generators does not have atoms. Questions on the absence of atoms for polynomials in non-commuting random variables (or for polynomials in random matrices) have been an open problem for quite a while. We solve this general problem by showing that maximality of free entropy dimension excludes atoms.