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

  Published:2016-05-10
 Printed: Feb 2017
  • Michael Hartz,
    Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
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Abstract

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 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: non-selfadjoint operator algebras, reproducing kernel Hilbert spaces, multiplier algebra, Nevanlinna-Pick kernels, isomorphism problem non-selfadjoint operator algebras, reproducing kernel Hilbert spaces, multiplier algebra, Nevanlinna-Pick kernels, isomorphism problem
MSC Classifications: 47L30, 46E22, 47A13 show english descriptions Abstract operator algebras on Hilbert spaces
Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
Several-variable operator theory (spectral, Fredholm, etc.)
47L30 - Abstract operator algebras on Hilbert spaces
46E22 - Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
47A13 - Several-variable operator theory (spectral, Fredholm, etc.)
 

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