The isomorphism problem for some universal operator algebras

Kenneth R. Davidson, Christopher Ramsey, Orr Moshe Shalit

Research output: Contribution to journalArticlepeer-review

52 Scopus citations

Abstract

This paper addresses the isomorphism problem for the universal (non-self-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 a radical ideal of 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 wot-closures of these algebras as well.

Original languageEnglish
Pages (from-to)167-218
Number of pages52
JournalAdvances in Mathematics
Volume228
Issue number1
DOIs
StatePublished - 10 Sep 2011
Externally publishedYes

Keywords

  • Non-self-adjoint operator algebras
  • Reproducing kernel hilbert spaces
  • Subproduct systems

ASJC Scopus subject areas

  • General Mathematics

Fingerprint

Dive into the research topics of 'The isomorphism problem for some universal operator algebras'. Together they form a unique fingerprint.

Cite this