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Operator algebras for analytic varieties


Authors: Kenneth R. Davidson, Christopher Ramsey and Orr Moshe Shalit
Journal: Trans. Amer. Math. Soc. 367 (2015), 1121-1150
MSC (2010): Primary 47L30, 47A13, 46E22
DOI: https://doi.org/10.1090/S0002-9947-2014-05888-1
Published electronically: July 17, 2014
MathSciNet review: 3280039
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Abstract: We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions $ \mathcal {M}_V$ of the multiplier algebra $ \mathcal {M}$ of Drury-Arveson space to a holomorphic subvariety $ V$ of the unit ball $ \mathbb{B}_d$.

We find that $ \mathcal {M}_V$ is completely isometrically isomorphic to $ \mathcal {M}_W$ if and only if $ W$ is the image of $ V$ under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that when $ d<\infty $ every isometric isomorphism is completely isometric.

The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When $ V$ and $ W$ are each a finite union of irreducible varieties and a discrete variety, when $ d<\infty $, an isomorphism between $ \mathcal {M}_V$ and $ \mathcal {M}_W$ determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-$ *$ continuous.

We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold--particularly, smooth curves and Blaschke sequences.

We also discuss the norm closed algebras associated to a variety, and point out some of the differences.


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Additional Information

Kenneth R. Davidson
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: krdavids@math.uwaterloo.ca

Christopher Ramsey
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Address at time of publication: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904-4137
Email: ciramsey@math.uwaterloo.ca, chramsey@gmail.com

Orr Moshe Shalit
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, 84105 Be’er Sheva, Israel
Email: oshalit@math.bgu.ac.il

DOI: https://doi.org/10.1090/S0002-9947-2014-05888-1
Keywords: Non-selfadjoint operator algebras, reproducing kernel Hilbert spaces
Received by editor(s): January 20, 2012
Received by editor(s) in revised form: January 13, 2013
Published electronically: July 17, 2014
Additional Notes: The first author was partially supported by an NSERC grant.
The third author was supported by ISF Research Grant no. 474/12 and by EU FP7/2007-2013 Grant no. 321749.
Dedicated: Dedicated to the memory of William B. Arveson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.