## Abstract

Complete Pick algebras -these are, roughly, the multiplier algebras in which Pick’s interpolation theorem holds true -have been the focus of much research in the last twenty years or so. All (irreducible) complete Pick algebras may be realized concretely as the algebras obtained by restricting multipliers on Drury-Arveson space to a subvariety of the unit ball; to be precise: every irreducible complete Pick algebra has the form [Formula presented] where M_{d} denotes the multiplier algebra of the Drury-Arveson space [Formula presented], and V is the joint zero set of some functions in M_{d}. In recent years several works were devoted to the classification of complete Pick algebras in terms of the complex geometry of the varieties with which they are associated. The purpose of this survey is to give an account of this research in a comprehensive and unified way. We describe the array of tools and methods that were developed for this program, and take the opportunity to clarify, improve, and correct some parts of the literature.

Original language | English |
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Pages (from-to) | 167-198 |

Number of pages | 32 |

Journal | Operator Theory: Advances and Applications |

Volume | 255 |

DOIs | |

State | Published - 1 Jan 2016 |

Externally published | Yes |

## Keywords

- Complete Pick spaces
- Multiplier algebras
- Nonself-adjoint operator algebras
- Reproducing kernel Hilbert spaces

## ASJC Scopus subject areas

- Analysis