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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On function and operator modules


Authors: David Blecher and Christian Le Merdy
Journal: Proc. Amer. Math. Soc. 129 (2001), 833-844
MSC (2000): Primary 47L30, 47L25; Secondary 46H25, 46J10, 46L07
Published electronically: August 30, 2000
MathSciNet review: 1802002
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Abstract: Let $A$ be a unital Banach algebra. We give a characterization of the left Banach $A$-modules $X$ for which there exists a commutative unital $C^{*}$-algebra $C(K)$, a linear isometry $i\colon X\to C(K)$, and a contractive unital homomorphism $\theta \colon A\to C(K)$ such that $i(a\cdotp x) =\theta (a)i(x)$ for any $a\in A, x\in X$. We then deduce a ``commutative" version of the Christensen-Effros-Sinclair characterization of operator bimodules. In the last section of the paper, we prove a $w^{*}$-version of the latter characterization, which generalizes some previous work of Effros and Ruan.


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Additional Information

David Blecher
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3476
Email: dblecher@math.uh.edu

Christian Le Merdy
Affiliation: Département de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05866-4
PII: S 0002-9939(00)05866-4
Received by editor(s): May 24, 1999
Published electronically: August 30, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society