Operator algebras for analytic varieties
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- by Kenneth R. Davidson, Christopher Ramsey and Orr Moshe Shalit PDF
- Trans. Amer. Math. Soc. 367 (2015), 1121-1150 Request permission
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
- MR Author ID: 55000
- ORCID: 0000-0002-5247-5548
- 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
- MR Author ID: 842766
- 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
- MR Author ID: 829657
- Email: oshalit@math.bgu.ac.il
- 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. - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3280039
Dedicated: Dedicated to the memory of William B. Arveson