# Journal of Operator Theory

Volume 70, Issue 1, Summer 2013 pp. 273-290.

Isomorphisms of noncommutative domain algebras. II**Authors**: Alvaro Arias (1) and Frederic Latremoliere (2)

**Author institution:**(1) Department of Mathematics, University of Denver, Denver, CO 80208, U.S.A.

(2) Department of Mathematics, University of Denver, Denver, CO 80208, U.S.A.

**Summary:**We classify aspherical Popescu's noncommutative domain algebras in terms of their defining symbols. An aspherical noncommutative domain algebra is defined by its one-dimensional spectrum not being the unit ball of a hermitian space. We first use the geometry of the spectra of noncommutative domain algebras, together with Sunada's classification of Reinhardt domains in $\mathbb{C}^n$, to show that isomorphisms between aspherical domains must be linear in the generators. We then employ a new combinatorial argument to show that the existence of such an isomorphism implies that the defining symbols must be equivalent, in the sense that they can be obtained from each others by permutation and rescaling of their indeterminates. This paper uses and greatly expand on previous work of the authors.

**DOI:**http://dx.doi.org/0.7900/jot.2011aug10.1917

**Keywords:**non-self-adjoint operator algebras, disk algebra, weighted shifts, biholomorphisms, Reinhardt domains

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