Free biholomorphic classification of noncommutative domains

TL;DR

This paper develops a classification theory for noncommutative domains using free holomorphic functions, linking geometric domain equivalence to algebraic isomorphisms in operator algebras.

Contribution

It establishes a correspondence between free biholomorphic equivalence of noncommutative domains and isometric isomorphisms of their associated algebras.

Findings

Classification of noncommutative domains via algebraic isomorphisms

Equivalence of geometric and algebraic classifications in noncommutative setting

Framework for analyzing free holomorphic functions on Reinhardt domains

Abstract

In this paper we develop a theory of free holomorphic functions on noncommutative Reinhardt domains generated by positive regular free holomorphic functions in n noncommuting variables. We show that the free biholomorphic classification of these domains is the same as the classification, up to unital completely isometric isomorphisms, of the corresponding noncommutative domain algebras.