TY - JOUR

T1 - The isomorphism problem for complete pick algebras

T2 - A survey

AU - Salomon, Guy

AU - Shalit, Orr Moshe

N1 - Publisher Copyright:
© 2016 Springer International Publishing.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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 Md denotes the multiplier algebra of the Drury-Arveson space [Formula presented], and V is the joint zero set of some functions in Md. 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.

AB - 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 Md denotes the multiplier algebra of the Drury-Arveson space [Formula presented], and V is the joint zero set of some functions in Md. 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.

KW - Complete Pick spaces

KW - Multiplier algebras

KW - Nonself-adjoint operator algebras

KW - Reproducing kernel Hilbert spaces

UR - http://www.scopus.com/inward/record.url?scp=85006756504&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-31383-2_9

DO - 10.1007/978-3-319-31383-2_9

M3 - Article

AN - SCOPUS:85006756504

VL - 255

SP - 167

EP - 198

JO - Operator Theory: Advances and Applications

JF - Operator Theory: Advances and Applications

SN - 0255-0156

ER -