Rings of non-commutative functions and their fields of fractions
Méric L. Augat, Robert T. W. Martin, Eli Shamovich
TL;DR
The work develops a comprehensive framework for algebras of non-commutative analytic functions, proving that rings of NC entire or meromorphic functions $\mathscr{O}_d(R)$ are semifirs and possess universal skew fields of fractions $\mathscr{M}_d(R)$. It extends Bergman’s nullstellensatz to these semifirs and establishes evaluation of meromorphic NC expressions in stably finite algebras, including finite $C^*$-algebras, using an operator-algebraic approach via the free Hardy space. A key contribution is the strong convergence result that extends Haagerup–Thorbjørnsen-type limits from polynomials to NC meromorphic expressions, enabling robust connections to free probability and random matrix theory. The paper outlines a rich program of open problems about NC function theory, semirings, and the nature of meromorphic NC expressions, with potential implications for NC Poincaré-type questions and operator-algebraic topology.
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 $C^*$-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.