A Hilbert bundle characterization of Hilbert C*-modules
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- by George A. Elliott and Katsunori Kawamura PDF
- Trans. Amer. Math. Soc. 360 (2008), 4841-4862 Request permission
Abstract:
The category of Hilbert C*-modules over a given C*-algebra is shown to be equivalent to a certain simply described category of Hilbert bundles (i.e., continuous fields of Hilbert spaces) over the space of pure states of the C*-algebra with the zero functional adjoined.References
- Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
- Vasile Brînzănescu, Holomorphic vector bundles over compact complex surfaces, Lecture Notes in Mathematics, vol. 1624, Springer-Verlag, Berlin, 1996. MR 1439504, DOI 10.1007/BFb0093696
- Lawrence G. Brown, Complements to various Stone-Weierstrass theorems for $C^*$-algebras and a theorem of Shultz, Comm. Math. Phys. 143 (1992), no. 2, 405–413. MR 1145802
- Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
- J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233–280. MR 164248, DOI 10.1007/BF02545788
- Roger Godement, Théorie générale des sommes continues d’espaces de Banach, C. R. Acad. Sci. Paris 228 (1949), 1321–1323 (French). MR 29102
- R. Godement, Sur la théorie des représentations unitaires, Ann. of Math. (2) 53 (1951), 68–124 (French). MR 38571, DOI 10.2307/1969343
- Kjeld Knudsen Jensen and Klaus Thomsen, Elements of $KK$-theory, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991. MR 1124848, DOI 10.1007/978-1-4612-0449-7
- Irving Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839–858. MR 58137, DOI 10.2307/2372552
- G. G. Kasparov, Hilbert $C^{\ast }$-modules: theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), no. 1, 133–150. MR 587371
- Katsunori Kawamura, Serre-Swan theorem for non-commutative $C^*$-algebras, J. Geom. Phys. 48 (2003), no. 2-3, 275–296. MR 2007596, DOI 10.1016/S0393-0440(03)00044-5
- 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
- V. M. Manuilov and E. V. Troitsky, Hilbert $C^*$-modules, Translations of Mathematical Monographs, vol. 226, American Mathematical Society, Providence, RI, 2005. Translated from the 2001 Russian original by the authors. MR 2125398, DOI 10.1090/mmono/226
- William L. Paschke, Inner product modules over $B^{\ast }$-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. MR 355613, DOI 10.1090/S0002-9947-1973-0355613-0
- Gert K. Pedersen, Applications of weak$^{\ast }$ semicontinuity in $C^{\ast }$-algebra theory, Duke Math. J. 39 (1972), 431–450. MR 315463
- Marc A. Rieffel, Induced representations of $C^{\ast }$-algebras, Advances in Math. 13 (1974), 176–257. MR 353003, DOI 10.1016/0001-8708(74)90068-1
- Frederic W. Shultz, Pure states as a dual object for $C^{\ast }$-algebras, Comm. Math. Phys. 82 (1981/82), no. 4, 497–509. MR 641911
- Jun Tomiyama and Masamichi Takesaki, Applications of fibre bundles to the certain class of $C^{\ast }$-algebras, Tohoku Math. J. (2) 13 (1961), 498–522. MR 139025, DOI 10.2748/tmj/1178244253
Additional Information
- George A. Elliott
- Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4
- MR Author ID: 62980
- Email: elliott@math.toronto.edu
- Katsunori Kawamura
- Affiliation: Department of Mathematics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
- Email: kawamura@kurims.kyoto-u.ac.jp
- Received by editor(s): August 28, 2006
- Published electronically: April 24, 2008
- Additional Notes: The work of the first author was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 4841-4862
- MSC (2000): Primary 46L05
- DOI: https://doi.org/10.1090/S0002-9947-08-04600-X
- MathSciNet review: 2403706