Proper analytic free maps

• Published in: Journal of functional analysis Vol. 260; no. 5; pp. 1476 - 1490
• Main Authors: Helton, J. William, Klep, Igor, McCullough, Scott
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2011
• Subjects: Analytic map; Free analysis; Free real algebraic geometry; Linear matrix inequality; Non-commutative set and function; Proper map; Rigidity; Several complex variables; Free analysis; Proper map; Free real algebraic geometry; Linear matrix inequality; Several complex variables; Non-commutative set and function; Analytic map; Rigidity
• ISSN: 0022-1236; 1096-0783
• DOI: 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 ˜ .