Proper analytic free maps |
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Authors: | J William Helton Igor Klep Scott McCullough |
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Institution: | a Department of Mathematics, University of California, San Diego, United States b Univerza v Ljubljani, Fakulteta za Matematiko in Fiziko, Slovenia c Univerza v Mariboru, Fakulteta za Naravoslovje in Matematiko, Slovenia d Department of Mathematics, University of Florida, Gainesville, United States |
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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 in 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 . Assuming that both domains contain 0, we show that if is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also , 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 . |
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Keywords: | Non-commutative set and function Analytic map Proper map Rigidity Linear matrix inequality Several complex variables Free analysis Free real algebraic geometry |
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