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

Kusuda, Masaharu

doi:10.1090/S0002-9939-05-07909-8

Discrete spectra of $C^{}$-algebras and orthogonally closed submodules in Hilbert $C^{}$-modules
HTML articles powered by AMS MathViewer

by Masaharu Kusuda PDF

Proc. Amer. Math. Soc. 133 (2005), 3341-3344 Request permission

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.

References

Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
Masaharu Kusuda, Three-space problems in discrete spectra of $C^*$-algebras and dual $C^*$-algebras, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 3, 701–707. MR 1838507, DOI 10.1017/S0308210500001050
Masaharu Kusuda, Discrete spectra of $C^*$-algebras and complemented submodules in Hilbert $C^*$-modules, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3075–3081. MR 1993216, DOI 10.1090/S0002-9939-03-06855-2
E. C. Lance, Hilbert $C^*$-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694, DOI 10.1017/CBO9780511526206
Bojan Magajna, Hilbert $C^*$-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc. 125 (1997), no. 3, 849–852. MR 1346981, DOI 10.1090/S0002-9939-97-03551-X
Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace $C^*$-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. MR 1634408, DOI 10.1090/surv/060
Jürgen Schweizer, A description of Hilbert $C^\ast$-modules in which all closed submodules are orthogonally closed, Proc. Amer. Math. Soc. 127 (1999), no. 7, 2123–2125. MR 1646207, DOI 10.1090/S0002-9939-99-05219-3