Operator algebras for analytic varieties

Abstract

Abstract. We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions M V of the multiplier algebra M of Drury-Arveson space to a holomorphic subvariety V of the unit ball B d .We find that M V is completely isometrically isomorphic to M W if and only if W is the image of V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthend to show that, when d < ∞, every isometric isomorphism is completely isometric.The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When V and W are each a finite union of irreducible varieties and a discrete variety in B d with d < ∞, then an isomorphism between M V and M W determines 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.We also discuss the norm closed algebras associated to a variety, and point out some of the differences.

Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients

“…The answer is negative in general [45,52]. The first positive result in this direction was obtained by D. Alpay, M. Putinar and V. Vinnikov.…”

“…(1) S is commuting, i.e., Many results on the d-shift can be obtained by "compressing theorems" about the noncommutative d-shift down to F + (E); see, e.g., [47,48,52,101], the proof of Theorem 7.2.4 or Sections 4.9 and 8.2 below. This is a powerful technique, due to the availability of strong results for the noncommutative d-shift, e.g., [49,50,97,98] or more generally [92].…”

Erratum to: Multipliers of Embedded Discs

“…It is not hard to modify Example 6.12 in [3] to see that the conditions in the preceding lemma are not always satisfied. Now recall that for s ∈ R, H s is the reproducing kernel Hilbert space on the unit disc with kernel…”

“…4 and statements (1) and (3) in Theorem 8.5. Statements (1) and (3) are not known to hold in full generality, but do hold if one assumes that one of the spaces is H s , s < −1, so Example 8.6 still exhibits an uncountable family of non-isomorphic multiplier algebras associated to compact varieties. Lemma 8.4 holds if the compact variety V arises from one of the spaces H s , s < −1, by the results in Section 9 of [4].…”

Applications of Hilbert Module Approach to Multivariable Operator Theory

“…This is one of the most active research areas in multivariable operator theory. For instance, if S is a submodule of L 2 a (B n ) and generated by a polynomial (by Douglas and Wang [DoW11]) or a submodule of H 2 n and generated by a homogeneous polynomial (by Guo and Wang [GuW08]), then S is p-essentially normal for all p > n (see also [FXi09], [Es11] and [DaRS14]). 9.2.…”