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

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.

not intended for publication in this form; an extended version will appear under the title "Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension"