The isomorphism problem for complete Pick algebras: a survey
Guy Salomon, Orr Shalit
TL;DR
The survey addresses the isomorphism problem for complete Pick algebras by showing irreducible complete Pick algebras are completely isometrically isomorphic to multiplier algebras M_V of varieties V inside the ball, via the Drury–Arveson framework. It develops a unified approach linking algebraic isomorphisms to multiplier biholomorphisms and ball automorphisms, clarifying when weak-* continuous, isometric, or completely isometric isomorphisms correspond to geometric equivalences. It highlights key results for finite/infinite dimensions, homogeneous varieties, and finite Riemann surfaces, and presents counterexamples illustrating the limits of biholomorphic invariance, while outlining open problems and directions for further research. The work synthesizes function theory, complex geometry, and operator algebra techniques to illuminate how algebraic structure encodes and reflects the geometry of the underlying varieties, with implications for operator-algebraic geometry and interpolation theory.
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.