The isomorphism problem for some universal operator algebras

Abstract

This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by radical relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C*-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the weak-operator closures of these algebras as well.Comment: 46 pages. Final version, to appear in Advances in Mathematic

Operator algebras for analytic varieties

“…As an operator algebra concretely represented on a reproducing kernel Hilbert space, M V also has the weak-operator topology (wot). In [16,Lemma 11.9] we observed that, as is the case for the free semigroup algebras L d [13], the weak-operator and weak- * topologies on M V coincide. The setting there was slightly different, but the proof remains the same.…”

“…In [16], now improved by [20], we saw that in the case of homogeneous varieties V and W , the algebras A V and A W are isomorphic if and only if the algebras M V and M W are isomorphic. The above discussion shows that this is not true in general.…”

“…With these results in hand, we may repeat the arguments in [16,Section 11.3] word for word to obtain the following automatic continuity result. Recall that the weak-operator and the weak- * topologies on M V coincide by Lemma 3.1.…”

“…It is not clear when this happens in general. A large class of varieties for which this condition holds is the class of homogeneous varieties [16,Section 6]. Another class is described below.…”

“…In an earlier paper [16], the authors considered the case of homogeneous varieties. We showed, under some extra assumptions on the varieties, that the algebras are isomorphic if and only if there is a biholomorphic map of one variety onto the other.…”