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: [email protected] (K.R. Davidson), [email protected] (C. Ramsey), [email protected] (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 -