Free Biholomorphic Classification of Noncommutative Domains

Abstract

In this paper, we develop a theory of free holomorphic functions on noncommutative Reinhardt domains , generated by positive regular free holomorphic functions f in n noncommuting variables and by positive integers m ≥ 1, where is the algebra of all bounded linear operators on a Hilbert space . Noncommutative Berezin transforms are used to study Hardy algebras H ∞ (D f, rad m ) and domain algebras A(D f, rad m ) associated with and compositions of free holomorphic functions. We obtain noncommutative Cartan type results for formal power series, in several noncommuting indeterminates, which leave invariant the nilpotent parts of the corresponding domains. As a consequence, we characterize the set of all free biholomorphic functions with F(0) = 0. We show that the free biholomorphic classification of the domains is the same as the classification, up to unital completely isometric isomorphisms having completely contractive hereditary extension, of the corresponding noncommutative domain algebras A(D f, rad m ). In particular, we prove that Ψ : A(D f, rad 1 ) → A(D g, rad 1 ) is a unital completely isometric isomorphism if and only if there is a free biholomorphic map φ ∈ Bih(D g 1 , D f 1 ) such that This implies that the noncommutative domains and are free biholomorphic equivalent if and only if the domain algebras A(D f, rad 1 ) and A(D g, rad 1 ) are completely isometrically isomorphic. Using the interaction between the theory of functions in several complex variables and our noncommutative theory, we provide several results concerning the free biholomorphic classification of the noncommutative domains and the classification, up to completely isometric isomorphisms, of the associated noncommutative domain (resp. Hardy) algebras. In particular, we characterize the unit ball of among the noncommutative domains , up to free biholomorphisms. We also obtain characterizations for the unitarily implemented isomorphisms of noncommutative Hardy (resp. domain) algebras in terms of free biholomorphic functions between the corresponding noncommutative domains.