The isomorphism problem for some universal operator algebras

By K. Davidson, Christopher Ramsey, Orr Shalit. 2010

90 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7416d56671373181dd87aa3515da0ff6f939ae7b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Michael Hartz
2. Raphael Clouatre
3. E. Shamovich
4. Guy Salomon
5. Orr Shalit

In Applied Mathematics

1. Michael Hartz
2. Raphael Clouatre
3. E. Shamovich
4. Guy Salomon
5. Orr Shalit
6. Joerg Eschmeier
7. Miroslav Englis

In Mathematical Physics

1. Michael Hartz
2. E. Shamovich
3. Guy Salomon
4. Raphael Clouatre
5. Orr Shalit

Citations in the corpus

• Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, methodology): “In fact Davidson, Ramsey and Shalit in [23] and Hartz in [35] proved that if d < ∞, then for homogeneous varieties V and W we have that MV ∼= MW algebraically if and only if there exists a linear map φ ∈ GLd(C), such that φ(V ) = W .”
• Multiplier algebras of complete Nevanlinna–Pick spaces: Dilations, boundary representations and hyperrigidity (2016, background): “Moreover, they fit in well with the recent interest in operator algebras associated to geometric varieties [23, 24, 35].”
• On the Isomorphism Problem for Multiplier Algebras of Nevanlinna-Pick Spaces (2015, methodology): “4 in [16], where it was established by adjusting the proof of Cartan’s uniqueness theorem from [30, Theorem 2.”
• Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, unknown): “The case of homogeneous varieties is completely settled by results of [50] and [78].”