TY - JOUR

T1 - The isomorphism problem for some universal operator algebras

AU - Davidson, Kenneth R.

AU - Ramsey, Christopher

AU - Shalit, Orr Moshe

N1 - Funding Information:
* Corresponding author. E-mail addresses: krdavids@uwaterloo.ca (K.R. Davidson), ciramsey@uwaterloo.ca (C. Ramsey), oshalit@uwaterloo.ca (O.M. Shalit). 1 Partially supported by an NSERC grant.

PY - 2011/9/10

Y1 - 2011/9/10

N2 - 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.

AB - 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.

KW - Non-self-adjoint operator algebras

KW - Reproducing kernel hilbert spaces

KW - Subproduct systems

UR - http://www.scopus.com/inward/record.url?scp=80955178386&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2011.05.015

DO - 10.1016/j.aim.2011.05.015

M3 - Article

AN - SCOPUS:80955178386

SN - 0001-8708

VL - 228

SP - 167

EP - 218

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 1

ER -