Free analysis, convexity and LMI domains

This paper concerns the geometry of noncommutative domains and analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of noncommuting variables amongst which there are no relations they are free variables. Analytic free maps include vector-valued polynomials in free (noncommuting) variables and form a canonical class of mappings from one noncommutative domain D in say g variables to another noncommutative domain D̃ in g̃ variables. This article contains rigidity results paralleling those in the commutative world of several complex variables – particularly, in the case that the domains are circular and bounded. For instance, we show that proper free maps are one-to-one. Furthermore, between two freely biholomorphic bounded noncommutative domains there exists a linear biholomorphism. Because of its role in systems engineering, convexity is a major topic. Hence of particular interest is the case of domains defined by a linear matrix inequality, or LMI domains. Our main theorem yields the following nonconvexification result: If a bounded circular noncommutative domain D is freely biholomorphic to a bounded circular LMI domain, then D is itself an LMI domain.