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Integral Equations and Operator Theory

, Volume 76, Issue 1, pp 39–53 | Cite as

On the Isomorphism Question for Complete Pick Multiplier Algebras

  • Matt Kerr
  • John E. McCarthy
  • Orr Moshe ShalitEmail author
Article

Abstract

Every multiplier algebra of an irreducible complete Pick kernel arises as the restriction algebra \({\mathcal{M}_V = \{f \big|_V : f \in \mathcal{M}_d\}}\) , where d is some integer or \({\infty, \mathcal{M}_d}\) is the multiplier algebra of the Drury-Arveson space \({H^2_d}\) , and V is a subvariety of the unit ball. For finite dimensional d it is known that, under mild assumptions, every isomorphism between two such algebras \({\mathcal{M}_V}\) and \({\mathcal{M}_W}\) is induced by a biholomorphism between W and V. In this paper we consider the converse, and obtain positive results in two directions. The first deals with the case where V is the proper image of a finite Riemann surface. The second deals with the case where V is a disjoint union of varieties.

Mathematics Subject Classification (2010)

30H50 47B32 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Matt Kerr
    • 1
  • John E. McCarthy
    • 1
  • Orr Moshe Shalit
    • 2
    Email author
  1. 1.Washington UniversitySt. LouisUSA
  2. 2.Ben-Gurion University of the NegevBe’er-ShevaIsrael

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