Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-08T02:29:02.035Z Has data issue: false hasContentIssue false
  • English
  • Français

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:
[email protected] für Technik, Wirtschaft und Kultur (HTWK) Leipzig, Fachbereich IMN, Gustav-Freytag-Strasse 42A, D-04277 Leipzig, Germany
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Arveson, W., An Invitation to C*-algebras, Springer, New York, 1976CrossRefGoogle Scholar
2.Bakić, D., Guljaš, B., Hilbert C*-modules over C*-algebras of compact operators, Acta Sci. Math. (Szeged) 68(2002), 249–269Google Scholar
3.Frank, M., Self-duality and C*-reflexivity of Hilbert C*-modules, Zeitschr. Anal. Anwendungen 9(1990), 165–176CrossRefGoogle Scholar
4.Frank, M., Troitsky, E. V., Lefschetz numbers and geometry of operators in W*-modules, Funkt. Anal. i Prilozh. 30(1996), no. 4, 45–57 / Funct. Anal. Appl. 30(1996), 257–266Google Scholar
5.Frank, M., Beiträge zur Theorie der Hilbert-C*-Moduln, Habilitation Thesis, (ISBN 3-8265-3217-1, Shaker Verlag, Aachen, 1997), Universität Leipzig, Leipzig, F.R.G., October 1997Google Scholar
6.Frank, M., Geometrical aspects of Hilbert C*-modules, Positivity 3(1999), 215–243CrossRefGoogle Scholar
7.Frank, M., Larson, D. R., Frames in Hilbert C*-modules and C*-algebras, J. Operator Theory 48(2002), 273–314Google Scholar
8.Frank, M., Paulsen, V. I., Injective and projective Hilbert C*-modules, and C*- algebras of compact operators, preprint math.OA/0611349 at www.arxiv.org, 2006Google Scholar
9.Frank, M., Sharifi, K., Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem, to appear in J. Operator Theory, 2008Google Scholar
10.Frank, M., Sharifi, K., Generalized inverses and polar decomposition of unbounded regular operators on Hilbert C*-modules, preprint, HTWK Leipzig, Germany, and Ferdowsi University, Mashhad, Iran, 2007Google Scholar
11.Jensen, K. K., Thomsen, K., Elements of KK-Theory, (Series: Mathematics: Theory & Applications), Birkhäuser, Boston-Basel-Berlin, 1991Google Scholar
12.Kasparov, G. G., Hilbert C*-modules: The theorems of Stinespring and Voiculescu, J. Operator Theory 4(1980), 133–150Google Scholar
13.Kusuda, M., Discrete spectra of C*-algebras and complemented submodules in Hilbert C*-modules, Proc. Amer. Math. Soc. 131(2003), 3075–3081CrossRefGoogle Scholar
14.Kusuda, M., Discrete spectra of C*-algebras and orthogonally closed submodules in Hilbert C*-modules, Proc. Amer. Math. Soc. 133(2005), 3341–3344CrossRefGoogle Scholar
15.Lance, E. C., Hilbert C*-modules – a Toolkit for Operator Algebraists, London Math. Soc. Lecture Notes Series 210, Cambridge University Press, Cambridge, England, 1995Google Scholar
16.Lin, Huaxin, Bounded module maps and pure completely positive maps, J. Operator Theory 26(1991), 121–138Google Scholar
17.Lin, Huaxin, Injective Hilbert C*-modules, Pacific J. Math. 154(1992), 131–164CrossRefGoogle Scholar
18.Magajna, B., Hilbert C*-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc. 125(1997), 849–852CrossRefGoogle Scholar
19.Manuilov, V. M., An example of a noncomplemented Hilbert W*-module, Vestn. Moskov. Univ., Ser. I: Mat. Mekh., no. 5, 2000, 58–59, translated: Moscow Univ. Math. Bull. 2000, 38–39Google Scholar
20.Paschke, W. L., Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182(1973), 443–468Google Scholar
21.Pedersen, G. K., C*-algebras and Their Automorphism Groups, Academic Press, London, 1979Google Scholar
22.Raeburn, I., Williams, D. P., Morita Equivalence and Continuous Trace C*- algebras, Math. Surveys and Monogr. v. 60, Amer. Math. Soc., Providence, R.I., 1998Google Scholar
23.Schweizer, J., A description of Hilbert C*-modules in which all closed submodules are orthogonally closed, Proc. Amer. Math. Soc. 127(1999), 2123–2125CrossRefGoogle Scholar
24.Wegge-Olsen, N. E., K-theory and C-algebras: a Friendly Approach, Oxford University Press, Oxford, England, 1993CrossRefGoogle Scholar