On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces

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 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.

58 pages; small changes

References in corpus (6)

• The isomorphism problem for some universal operator algebras
• Topological isomorphisms for some universal operator algebras
• Multipliers of embedded discs
• Spaces of Dirichlet series with the complete Pick property
• Nevanlinna-Pick Kernels and Localization
• The isomorphism problem for complete Pick algebras: a survey

Cited by in corpus (21)

• Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball
• Multiplier algebras of complete Nevanlinna-Pick spaces: dilations, boundary representations and hyperrigidity
• The Smirnov class for spaces with the complete Pick property
• Interpolating sequences in spaces with the complete Pick property
• Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem
• Spaces of Dirichlet series with the complete Pick property
• von Neumann's inequality for row contractive matrix tuples
• A Beurling-Lax-Halmos theorem for spaces with a complete Nevanlinna-Pick factor
• Cyclicity preserving operators on spaces of analytic functions in ${\mathbb{C}}^n$
• The classification problem for operator algebraic varieties and their multiplier algebras
• Sub-Bergman Hilbert spaces on the unit disk III
• Tensor algebras of subproduct systems and noncommutative function theory
• Three Classification Results In The Theory Of Weighted Hardy Spaces On The Ball
• Tetrahedra in complex hyperbolic space and Hilbert spaces with Pick kernels
• Shift-cyclicity in analytic function spaces
• Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball
• Embedding dimension of the Dirichlet space
• Complex Hyperbolic Geometry and Hilbert Spaces with the Complete Pick Property
• de Branges-Rovnyak spaces which are complete Nevanlinna-Pick spaces
• Jointly cyclic polynomials and maximal domains
• A Gleason-Kahane-Żelazko theorem for reproducing kernel Hilbert spaces