Operator algebras for analytic varieties
  We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions $\mathcal M_V$ of the multiplier algebra $\mathcal M$ of Drury-Arveson space to a holomorphic subvariety $V$ of the unit ball $\mathbb{B}_d$.   We find that $\mathcal M_V$ is completely isometrically isomorphic to $\mathcal M_W$ if and only if $W$ is the image of $V$ under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthend to show that, when $d<\infty$, every isometric isomorphism is completely isometric.   The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When $V$ and $W$ are each a finite union of irreducible varieties and a discrete variety in $\mathbb{B}_d$ with $d<\infty$, then an isomorphism between $\mathcal M_V$ and $\mathcal M_W$ 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. 
Kenneth R. Davidson, Christopher Ramsey, Orr Shalit
math.OA