Proper Analytic Free Maps

TL;DR

This paper studies proper analytic free maps between non-commutative domains, proving that such maps are injective and invertible under certain conditions, extending classical complex analysis results to free variables.

Contribution

It establishes that proper analytic free maps fixing zero are injective and invertible when domains contain zero, extending classical properties to free non-commutative settings.

Findings

Proper analytic free maps are injective if they fix zero.

When domains are equal, such maps are invertible with free analytic inverses.

The results are optimal without additional domain assumptions.

Abstract

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain D' in g' variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of D'. Assuming that both domains contain 0, we show that if f:D->D' is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also g=g', then f is invertible and f^(-1) is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and D'.