Rings of non-commutative functions and their fields of fractions

Abstract

Semi-free ideal rings, or semifirs, were introduced by Paul M. Cohn to study universal localizations in the non-commutative setting. We provide new examples of semifirs consisting of analytic functions in several non-commuting variables. These examples arise canonically in free analysis by completing the free algebra in the topology of ``uniform convergence on operator-space balls'' in the non-commutative universe of tuples of square matrices of any finite size. We show, in particular, that the ring of (uniformly) entire non-commutative (NC) functions in $d\in \mathbb{N}$ non-commuting variables, $\scr{O}_d$, is a semifir. Every finitely--generated right (or left) ideal in $\scr{O}_d$ is closed, which yields an analytic extension of G. Bergman's nullstellensatz for the free algebra. Any semifir admits a universal skew field of fractions; applying this to $\scr{O}_d$ yields the universal skew field of ``NC meromorphic expressions", $\scr{M} _d$. We show that any $f \in \scr{M} _d$ has a well-defined domain and evaluations in a large class of stably-finite topological algebras, including finite $C^{\ast }$-algebras, extending a result of Cohn for NC rational functions. As an application, we extend the almost sure convergence result of Haagerup and Thorbjörnsen for free polynomials evaluated on tuples of random matrices to the setting of NC meromorphic expressions.

Section 5 has been changed substantially. We have added a general semi-inertia theorem and a version of Amitsur's theorem that states that there are no meromorphic identities for all matrix algebras.Other parts of the paper have been changed accordingly