A Hilbert bundle characterization of Hilbert C*-modules

Elliott, George; Kawamura, Katsunori

doi:10.1090/S0002-9947-08-04600-X

A Hilbert bundle characterization of Hilbert C*-modules
HTML articles powered by AMS MathViewer

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