On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces

TL;DR

This paper investigates the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces, establishing conditions for algebraic and isometric isomorphisms directly from the Hilbert spaces and their kernels, especially on homogeneous varieties.

Contribution

It offers a new approach by working directly with Hilbert spaces and kernels, and provides a complete characterization of isomorphisms for spaces on homogeneous varieties.

Findings

Two multiplier algebras are equal iff their Hilbert spaces are equal.

Complete characterization of algebraic and isometric isomorphisms for spaces on homogeneous varieties.

Extends previous results by generalizing to broader classes of Nevanlinna-Pick spaces.

Abstract

We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic. This generalizes results of Davidson, Ramsey, Shalit, and the author.