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arXiv · 1004.1381

Proper Analytic Free Maps

J. William Helton·Igor Klep·Scott McCullough
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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'.

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BibTeXRIS

J. William Helton, Igor Klep, Scott McCullough. 2010-12-01. Proper Analytic Free Maps. https://doi.org/10.1016/j.jfa.2010.11.007

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