Operator algebras for analytic varieties

TL;DR

This paper characterizes when multiplier algebras of irreducible complete Pick kernels, restricted to varieties in the unit ball, are isomorphic or isometric, linking algebraic isomorphisms to biholomorphic automorphisms of the ball.

Contribution

It establishes a complete isometric isomorphism criterion for these algebras based on biholomorphic automorphisms, and explores algebraic isomorphisms for unions of varieties, with several counterexamples.

Findings

Isometric isomorphisms correspond to biholomorphic automorphisms.

Algebraic isomorphisms often induce biholomorphic maps between varieties.

Counterexamples show the converse implications do not always hold.

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-$*$ 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.