Mathematics – Functional Analysis
Scientific paper
2010-04-08
J. Funct. Anal. 260, No. 5, 1476-1490 (2011)
Mathematics
Functional Analysis
17 pages, final version. To appear in the Journal of Functional Analysis
Scientific paper
10.1016/j.jfa.2010.11.007
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'.
Helton John William
Klep Igor
McCullough Scott
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