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On the Isomorphism Problem for Multiplier Algebras of Nevanlinna-Pick Spaces

Published online by Cambridge University Press:  20 November 2018

Michael Hartz
Michael Hartz*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1 e-mail: mphartz@uwaterloo.ca
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Abstract

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We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with the restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic.This generalizes results of Davidson, Ramsey,Shalit, and the author.

Keywords

47L3046E2247A13non-selfadjoint operator algebrasreproducing kernelHilbert spacesmultiplier algebraNevanlinna-Pick kernelsisomorphism problem.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 1 , 01 February 2017 , pp. 54 - 106
Copyright
Copyright © Canadian Mathematical Society 2017

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