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

 

A description of Hilbert $\mathbf{\mathit{C}^*}$-modules in which all closed submodules are orthogonally closed


Author: Jürgen Schweizer
Journal: Proc. Amer. Math. Soc. 127 (1999), 2123-2125
MSC (1991): Primary 46L05
Published electronically: March 17, 1999
MathSciNet review: 1646207
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Abstract: Let $A$, $B$ be $C^*$-algebras and $E$ a full Hilbert $A$-$B$-bimodule such that every closed right submodule $E_{0}\subseteq E$ is orthogonally closed, i.e., $E_{0}=(E_{0}^{\perp })^{\perp }$. Then there are families of Hilbert spaces $\{\mathcal{H}_{i}\}$, $\{\mathcal{V}_{i}\}$ such that $A$ and $B$ are isomorphic to $c_{0}$-direct sums $\sum \!\mathcal{K}(\mathcal{V}_{i})$, resp. $\sum \!\mathcal{K}(\mathcal{H}_{i})$, and $E$ is isomorphic to the outer direct sum $\sum _{\!0}\mathcal{K}(\mathcal{H}_{i},\mathcal{V}_{i})$.


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Additional Information

Jürgen Schweizer
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Email: juergen.schweizer@uni-tuebingen.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05219-3
PII: S 0002-9939(99)05219-3
Keywords: Hilbert $C^*$-modules, complemented submodules
Received by editor(s): October 23, 1997
Published electronically: March 17, 1999
Additional Notes: The results of this paper are part of the author’s doctoral dissertation at the University of Tübingen, which was completed before we received the preprint \cite{5} by Magajna.
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society