A description of Hilbert $C^\ast$-modules in which all closed submodules are orthogonally closed
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- by Jürgen Schweizer PDF
- Proc. Amer. Math. Soc. 127 (1999), 2123-2125 Request permission
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 $\sum _{0}\mathcal {K}(\mathcal {H}_{i},\mathcal {V}_{i})$.References
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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
- 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 [B. Magajna, Hilbert $C^*$-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc. 125 (1997), 849–852] by Magajna.
- Communicated by: David R. Larson
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 2123-2125
- MSC (1991): Primary 46L05
- DOI: https://doi.org/10.1090/S0002-9939-99-05219-3
- MathSciNet review: 1646207