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

Author: Masaharu Kusuda
Journal: Proc. Amer. Math. Soc. 131 (2003), 3075-3081
MSC (2000): Primary 46L05, 46L08
Published electronically: February 6, 2003
MathSciNet review: 1993216
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Abstract: Let $A$ be a $C^{*}$-algebra and let $X$ be a full (right) Hilbert $A$-module. It is shown that if the spectrum $\widehat A$ of $A$ is discrete, then every closed $\mathcal{K}(X)$-$A$-submodule of $X$ is complemented in $X$, and conversely that if $\widehat A$ is a $T_{1}$-space and if every closed $\mathcal{K}(X)$-$A$-submodule of $X$ is complemented in $X$, then $\widehat A$ 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

Keywords: Hilbert $C^{*}$-module, complemented subspace
Received by editor(s): November 6, 2001
Received by editor(s) in revised form: April 23, 2002
Published electronically: February 6, 2003
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society