Non-selfadjoint operator algebras, weighted shifts, biholomorphisms
Noncommutative domain algebras are noncommutative analogues of the algebras of holomorphic functions on domains of Cn defined by holomorphic polynomials, and they generalize the noncommutative Hardy algebras. We present here a complete classification of these algebras based upon techniques inspired by multivariate complex analysis, and more specifically the classification of domains in hermitian spaces up to biholomorphic equivalence.
Arias, Alvaro and Latrémolièr, Frédéric, "Classification of Noncommutative Domain Algebras" (2012). Mathematics Preprint Series. 32.