The isomorphism problem for complete Pick algebras: a survey

TL;DR

This survey reviews the classification of complete Pick algebras through complex geometry, highlighting tools, methods, and recent progress in understanding their isomorphism problem.

Contribution

It provides a comprehensive, unified overview of the classification of complete Pick algebras and clarifies existing literature and methods.

Findings

Complete Pick algebras can be realized as restrictions of multipliers on Drury-Arveson space.

Recent work connects algebra isomorphisms to geometric properties of associated varieties.

The survey clarifies and improves existing classification methods.

Abstract

Complete Pick algebras - these are, roughly, the multiplier algebras in which Pick's interpolation theorem holds true - have been the focus of much research in the last twenty years or so. All (irreducible) complete Pick algebras may be realized concretely as the algebras obtained by restricting multipliers on Drury-Arveson space to a subvariety of the unit ball; to be precise: every irreducible complete Pick algebra has the form $M_V = \{f|_V : f \in M_d\}$, where $M_d$ denotes the multiplier algebra of the Drury-Arveson space $H^2_d$, and $V$ is the joint zero set of some functions in $M_d$. In recent years several works were devoted to the classification of complete Pick algebras in terms of the complex geometry of the varieties with which they are associated. The purpose of this survey is to give an account of this research in a comprehensive and unified way. We describe the array of tools and methods that were developed for this program, and take the opportunity to clarify, improve, and correct some parts of the literature.