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

 

Discrete spectra of $C^{*}$-algebras and orthogonally closed submodules in Hilbert $C^{*}$-modules


Author: Masaharu Kusuda
Journal: Proc. Amer. Math. Soc. 133 (2005), 3341-3344
MSC (2000): Primary 46L05, 46L08
Published electronically: May 9, 2005
MathSciNet review: 2161158
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Abstract: Let $A$ and $B$ be $C^{*}$-algebras and let $X$ be an $A$-$B$-imprimitivity bimodule. Then it is shown that if the spectrum $\widehat A$ of $A$ (resp. $\widehat B$ of $B$) is discrete, then every closed $A$-$B$-submodule of $X$ is orthogonally closed in $X$, and conversely that if $\widehat A$(resp. $\widehat B$) is a $T_{1}$-space and if every closed $A$-$B$-submodule of $X$ is orthogonally closed in $X$, then $\widehat A$(resp. $\widehat B$) is discrete.


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Additional Information

Masaharu Kusuda
Affiliation: Department of Mathematics, Faculty of Engineering, Kansai University, Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan
Email: kusuda@ipcku.kansai-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07909-8
PII: S 0002-9939(05)07909-8
Keywords: Hilbert $C^{*}$-modules, orthogonally closed
Received by editor(s): December 3, 2003
Received by editor(s) in revised form: June 23, 2004
Published electronically: May 9, 2005
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.