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

 

A characterization and a generalization of $ W^*$-modules


Authors: David P. Blecher and Upasana Kashyap
Journal: Trans. Amer. Math. Soc. 363 (2011), 345-363
MSC (2010): Primary 46L08, 47L30, 47L45; Secondary 16D90, 47L25
Published electronically: August 18, 2010
MathSciNet review: 2719685
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Abstract: We give a new Banach module characterization of $ W^*$-modules, also known as self-dual Hilbert $ C^*$-modules over a von Neumann algebra. This leads to a generalization of the notion, and the theory, of $ W^*$-modules, to the setting where the operator algebras are $ \sigma$-weakly closed algebras of operators on a Hilbert space. That is, we find the appropriate weak* topology variant of our earlier notion of rigged modules, and their theory, which in turn generalizes the notions of a $ C^*$-module and a Hilbert space, successively. Our $ w^*$-rigged modules have canonical `envelopes' which are $ W^*$-modules. Indeed, a $ w^*$-rigged module may be defined to be a subspace of a $ W^*$-module possessing certain properties.


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Additional Information

David P. Blecher
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: dblecher@math.uh.edu

Upasana Kashyap
Affiliation: Department of Mathematics and Computer Science, The Citadel, 171 Moultrie Street, Charleston, South Carolina 29409
Email: ukashyap1@citadel.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05153-0
PII: S 0002-9947(2010)05153-0
Received by editor(s): December 7, 2007
Received by editor(s) in revised form: December 10, 2007, and March 23, 2009
Published electronically: August 18, 2010
Additional Notes: The first author was supported by grant DMS 0400731 from the National Science Foundation
Article copyright: © Copyright 2010 American Mathematical Society