Proper Analytic Free Maps
arXiv:1004.1381 · doi:10.1016/j.jfa.2010.11.007
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 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'.
17 pages, final version. To appear in the Journal of Functional Analysis
References in corpus (4)
Cited by in corpus (27)
- The inverse function theorem and the resolution of the Jacobian conjecture in free analysis
- Free function theory through matrix invariants
- Progress in noncommutative function theory
- Optimization over trace polynomials
- Free Convex Algebraic Geometry
- Bianalytic Maps Between Free Spectrahedra
- Global holomorphic functions in several non-commuting variables II
- Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation
- Null- and Positivstellensätze for rationally resolvable ideals
- Semidefinite programming in matrix unknowns which are dimension free
- Bianalytic free maps between spectrahedra and spectraballs
- The free Grothendieck theorem
- Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras
- On the classification of function algebras on subvarieties of noncommutative operator balls
- Projective Spectrum and Cyclic Cohomolgy
- NC Automorphisms of nc-bounded domains
- A Model Theoretic Perspective on Matrix Rings
- Noncommutative hyperbolic metrics
- Free potential functions
- Symmetric functions of two noncommuting variables
- The implicit function theorem and free algebraic sets
- Fixed point theorems for noncommutative functions
- Noncommutative functions are weakly algebraic on operatorial polynomial polyhedra
- Hyper-Reinhardt Free Spectrahedra
- On Automorphism Groups of Hardy Algebras
- Pick Interpolation for free holomorphic functions
- Integrability of Free Noncommutative Functions