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.
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Kerr, M., McCarthy, J.E. & Shalit, O.M. On the Isomorphism Question for Complete Pick Multiplier Algebras. Integr. Equ. Oper. Theory 76, 39–53 (2013). https://doi.org/10.1007/s00020-013-2048-2
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DOI: https://doi.org/10.1007/s00020-013-2048-2