  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.
