## Abstract

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions M_{V} of the multiplier algebra M of Drury-Arveson space to a holomorphic subvariety V of the unit ball B_{d}. We find that M_{V} is completely isometrically isomorphic to 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 strengthened to show that when d < ∞ every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When V andW are each a finite union of irreducible varieties and a discrete variety, when d < ∞, an isomorphism between M_{V} and 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.

Original language | English |
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Pages (from-to) | 1121-1150 |

Number of pages | 30 |

Journal | Transactions of the American Mathematical Society |

Volume | 367 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2015 |

## Keywords

- Non-Selfadjoint operator algebras
- Reproducing kernel Hilbert spaces

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics