The isomorphism problem for some universal operator algebras

The isomorphism problem for some universal operator algebras

##### Author

Davidson, Kenneth R.

Ramsey, Christopher

Shalit, Orr Moshe

##### Faculty Advisor

##### Date

2011

##### Keywords

non-self-adjoint operator algebras , subproduct systems , reproducing kernel Hilbert spaces

##### Abstract (summary)

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 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.

##### Publication Information

Davidson, K., C. Ramsey and O. Shalit. "The isomorphism problem for some universal operator algebras", Advances in Mathematics 228, 1 (2011) 167-218. https://doi.org/10.1016/j.aim.2011.05.015

##### DOI

##### Notes

##### Item Type

##### Language

English