On the Isomorphism Question for Complete Pick Multiplier Algebras

Abstract

Every multiplier algebra of an irreducible complete Pick kernel arises as the restriction algebra $${\mathcal{M}_V = \{f \big|_V : f \in \mathcal{M}_d\}}$$ , where d is some integer or $${\infty, \mathcal{M}_d}$$ is the multiplier algebra of the Drury-Arveson space $${H^2_d}$$ , and V is a subvariety of the unit ball. For finite dimensional d it is known that, under mild assumptions, every isomorphism between two such algebras $${\mathcal{M}_V}$$ and $${\mathcal{M}_W}$$ 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.