Operator algebras for analytic varieties

Abstract

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions ${\mathcal{M}}_V$ of the multiplier algebra $\mathcal{M}$ of Drury-Arveson space to a holomorphic subvariety $V$ of the unit ball ${\mathbb{B}}_d$. We find that ${\mathcal{M}}_V$ is completely isometrically isomorphic to ${\mathcal{M}}_W$ 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 $d<\infty$, 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 ${\mathbb{B}}_d$ with $d<\infty$, then an isomorphism between ${\mathcal{M}}_V$ and ${\mathcal{M}}_W$ determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-$\ast$ 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.

39 pages. It has come to light that a result of Davidson and Pitts, that the fibre of the maximal ideal space of the multiplier algebra over a point in the open ball consists only of point evaluation, is not true for $d=\infty$. Version 5 addresses this with some minor changes to the $d=\infty$ case. To appear in Trans. Amer. Math. Soc