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'.Comment: 17 pages, final version. To appear in the Journal of Functional
Analysi
