Proper analytic free maps

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) over tilde in (g) over tilde 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) over tilde. Assuming that both domains contain 0, we show that if f : D -> (D) over tilde is a proper analytic free map, and f(0) = 0, then f is one-to-one. Moreover, if also g = (g) over tilde, 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) over tilde. (C) 2010 Elsevier Inc. All rights reserved.