On the Isomorphism Question for Complete Pick Multiplier Algebras |
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Authors: | Matt Kerr John E McCarthy Orr Moshe Shalit |
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Institution: | 1. Washington University, St. Louis, MO, 63130, USA 2. Ben-Gurion University of the Negev, Be’er-Sheva, Israel
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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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