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Summary: ISOMORPHISMS OF NON-COMMUTATIVE DOMAIN
ALGEBRAS
ALVARO ARIAS AND FRÉDÉRIC LATRÉMOLIÈRE
Abstract. Noncommutative domain algebras were introduced by Popescu as
the non-selfadjoint operator algebras generated by weighted shifts on the Full
Fock space. This paper uses results from several complex variables to clas-
sify many noncommutative domain algebras, and it uses results from operator
theory to obtain new bounded domains in Cn with non-compact automorphic
group.
1. Introduction
Noncommutative domain algebras, introduced in [10], generalize the non-
commutative disk algebras and are de...ned as norm closures of the algebras gener-
ated by a family of weighted shifts on the Full Fock space. This paper investigates
the isomorphism problem for this class of algebras. The fundamental tool we use
is the theory of functions in several complex variables in Cn
. Formally, we apply
Cartan's Lemma [3] and Sunada's Theorem [15] to domains in Cn
naturally as-
sociated with our noncommutative domain algebras to derive a ...rst classi...cation
result: if there exists an isometric isomorphism between two noncommutative do-
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