Orr Moshe Shalit

The isomorphism problem for some universal operator algebras

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.

Operator algebras for analytic varieties

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictionsMV of the multiplier algebraM of Drury-Arveson space to a holomorphic subvariety V of the unit ball Bd. We nd that MV is completely isometrically isomorphic toMW if and only if W is the image of V under a biholomorphic auto- morphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthend to show that, when d <1, every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (al- gebraically) isomorphic is also studied. When V and W are each a nite union of irreducible varieties and a discrete variety in Bd with d <1, then an isomorphism betweenMV andMW deter- mines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak- continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold|particularly, smooth curves and Blaschke sequences.

Stable polynomial division and essential normality of graded Hilbert modules

The purpose of this paper is to initiate a new attack on Arvesons resistant conjecture, that all graded submodules of the

Multipliers of Embedded Discs

d

Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations

-shift Hilbert module

Unitary N-dilations for tuples of commuting matrices

H^2

The Isomorphism Problem for Complete Pick Algebras: A Survey

are essentially normal. We introduce the stable division property for modules (and ideals): a normed module

Dilation theory in finite dimensions: The possible, the impossible and the unknown

M

Spaces of Dirichlet series with the complete Pick property

over the ring of polynomials in

Minimal and maximal matrix convex sets

d