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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quotients of standard Hilbert modules


Author: William Arveson
Journal: Trans. Amer. Math. Soc. 359 (2007), 6027-6055
MSC (2000): Primary 46L07, 47A99
Published electronically: June 13, 2007
MathSciNet review: 2336315
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Abstract: We initiate a study of Hilbert modules over the polynomial algebra $ \mathcal A=\mathbb{C}[z_1,\dots,z_d]$ that are obtained by completing $ \mathcal A$ with respect to an inner product having certain natural properties. A standard Hilbert module is a finite multiplicity version of one of these. Standard Hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra, and their quotients by submodules give rise to universal solutions of nonlinear relations. Essentially all of the basic Hilbert modules that have received attention over the years are standard, including the Hilbert module of the $ d$-shift, the Hardy and Bergman modules of the unit ball, modules associated with more general domains in $ \mathbb{C}^d$, and those associated with projective algebraic varieties.

We address the general problem of determining when a quotient $ H/M$ of an essentially normal standard Hilbert module $ H$ is essentially normal. This problem has been resistant. Our main result is that it can be ``linearized'' in that the nonlinear relations defining the submodule $ M$ can be reduced, appropriately, to linear relations through an iteration procedure, and we give a concrete description of linearized quotients.


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Additional Information

William Arveson
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: arveson@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04209-2
PII: S 0002-9947(07)04209-2
Received by editor(s): July 19, 2005
Received by editor(s) in revised form: November 5, 2005
Published electronically: June 13, 2007
Additional Notes: The author was supported by NSF grant DMS-0100487
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.