On the isomorphism question for complete Pick multiplier algebras

TL;DR

This paper investigates when isomorphisms between certain multiplier algebras of complete Pick kernels imply biholomorphic equivalences of their underlying varieties, focusing on cases involving Riemann surfaces and disjoint unions.

Contribution

It extends known results by establishing conditions under which algebra isomorphisms correspond to geometric biholomorphisms for specific classes of varieties.

Findings

Isomorphisms induced by biholomorphisms for Riemann surface images

Results for disjoint unions of varieties

Positive partial converse to known isomorphism conditions

Abstract

Every multiplier algebra of an irreducible complete Pick kernel arises as the restriction algebra $\mv = \{f\big|_V : f \in \cM_d\}$, where $d$ is some integer or $\infty$, $\cM_d$ is the multiplier algebra of the Drury-Arveson space $H^2_d$, and $V$ is a subvariety of the unit ball. For finite $d$ it is known that, under mild assumptions, every isomorphism between two such algebras $\mv$ and $\mw$ is induced by a biholomorphism between $W$ and $V$. In this paper we consider the converse, and obtain positive results in two directions. The first deals with the case where $V$ is the proper image of a finite Riemann surface. The second deals with the case where $V$ is a disjoint union of varieties.