Discrete spectra of 𝐶*-algebras and complemented submodules in Hilbert 𝐶*-modules

Kusuda, Masaharu

doi:10.1090/S0002-9939-03-06855-2

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

Proc. Amer. Math. Soc. 131 (2003), 3075-3081

DOI: https://doi.org/10.1090/S0002-9939-03-06855-2

Published electronically: February 6, 2003

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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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