Isomorphisms of noncommutative domain algebras II

TL;DR

This paper classifies aspherical noncommutative domain algebras up to isomorphism using Sunada's classification, showing they are equivalent if their defining symbols differ by permutation and scaling, and explores their automorphism groups.

Contribution

It extends classification results of noncommutative domain algebras, linking algebraic isomorphisms to geometric symbol equivalences and analyzing automorphism groups.

Findings

Aspherical noncommutative domain algebras are isomorphic iff their symbols are equivalent via permutation and scaling.

Automorphism groups of these algebras form subgroups of finite dimensional unitary groups.

Methods can extend classical analysis results, like Cartan's lemma, to noncommutative settings.

Abstract

This paper extends the results of the previous work of the authors on the classification on noncommutative domain algebras up to completely isometric isomorphism. Using Sunada's classification of Reinhardt domains in $C^n$, we show that aspherical noncommutative domain algebras are isomorphic if and only if their defining symbols are equivalent, in the sense that one can be obtained from the other via permutation and scaling of the free variables. Our result also shows that the automorphism groups of aspherical noncommutative domain algebras consists of a subgroup of some finite dimensional unitary group. We conclude by illustrating how our methods can be used to extend to noncommutative domain algebras some results from analysis in $C^n$ with the example of Cartan's lemma.