Operator algebras for analytic varieties

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions M V \mathcal {M}_V of the multiplier algebra M \mathcal {M} of Drury-Arveson space to a holomorphic subvariety V V of the unit ball B d \mathbb {B}_d . We find that M V \mathcal {M}_V is completely isometrically isomorphic to M W \mathcal {M}_W if and only if W W is the image of V V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that when d > ∞ d>\infty every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When V V and W W are each a finite union of irreducible varieties and a discrete variety, when d > ∞ d>\infty , an isomorphism between M V \mathcal {M}_V and M W \mathcal {M}_W determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak- ∗ * continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold—particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences.