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Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules

Published online by Cambridge University Press:  04 March 2008

Michael Frank
Michael Frank
Affiliation:
mfrank@imn.htwk-leipzig.deHochschule für Technik, Wirtschaft und Kultur (HTWK) Leipzig, Fachbereich IMN, Gustav-Freytag-Strasse 42A, D-04277 Leipzig, Germany
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Abstract

C*-algebras A of compact operators are characterized as those C*-algebras of coefficients of Hilbert C*-modules for which (i) every bounded A-linear operator between two Hilbert A-modules possesses an adjoint operator, (ii) the kernels of all bounded A-linear operators between Hilbert A-modules are orthogonal summands, (iii) the images of all bounded A-linear operators with closed range between Hilbert A-modules are orthogonal summands, and (iv) for every Hilbert A-module every Hilbert A-submodule is a topological summand. Thus, the theory of Hilbert C*-modules over C*-algebras of compact operators has similarities with the theory of Hilbert spaces. In passing, we obtain a general closed graph theorem for bounded module operators on arbitrary Hilbert C*-modules.

Keywords

Hilbert C*-modulesbounded module mapsC*-algebras of compact operators
Type
Research Article
Information
Journal of K-Theory , Volume 2 , Special Issue 3: In Memory of Yurii Petrovich Solovyev October 8, 1944–September 11, 2003 , December 2008 , pp. 453 - 462
Copyright
Copyright © ISOPP 2008

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