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Continuous-trace algebras from the bundle theoretic point of view

Part of: Selfadjoint operator algebras Fiber spaces and bundles Methods of category theory in functional analysis $K$-theory and operator algebras

Published online by Cambridge University Press:  09 April 2009

Jonathan Rosenberg
Jonathan Rosenberg
Affiliation:
Department of Mathematics, University of Maryland College Park, Maryland 20742, U.S.A.
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Abstract

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Using various facts about principal bundles over a space, we give a unified treatment of several theorems about the structure of stable separable continuous-trace algebras, their automorphisms, and their K-theory. We also present a classification of real continuous-trace algebras from the same point of view.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 46L35: Classifications of $C^*$-algebras 46L80: $K$-theory and operator algebras (including cyclic theory) 46M20: Methods of algebraic topology (cohomology, sheaf and bundle theory, etc.) 55R10: Fiber bundles 19K99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 3 , December 1989 , pp. 368 - 381
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Adams, J. F., Stable Homotopy and Generalised Homology, (Chicago Lectures in Math., Univ. of Chicago Press, Chicago and London, 1974).Google Scholar
[2]Akemann, C. A., Pedersen, G. K. and Tomiyama, J., ‘Tomiyama, ‘Multipliers of C *-algebras’, J. Funct. Anal. 13 (1973), 277–301.CrossRefGoogle Scholar
[3]Atiyah, M. F. and Segal, G., ‘Equivariant K-theory and completion,’ J. Differential Geom. 3 (1969), 1–18.CrossRefGoogle Scholar
[4]Dixmier, J., C*-algebras, (North-Holland, Amsterdam and New York, 1977).Google Scholar
[5]Dixmier, J. and Douady, A., ‘Champs continus d'espaces Hilbertiens et de C *-algèbres,’ Bull. Soc. Math. France 91 (1963), 227–284.CrossRefGoogle Scholar
[6]Donovan, P. and Karoubi, M., ‘Graded Brauer groups and K-theory with local coefficients’, Publ. Math. I.H.É.S. 38 (1970), 5–25.CrossRefGoogle Scholar
[7]Grothendieck, A., Le Groupe de Brauer, I: Algèbres d 'Azumaya et Interprétations Diverses, Séminaire Bourbaki, Paris, 17e année (1964/1964), exposé 290.Google Scholar
[8]Karoubi, M., K-Theory: An Introduction, (Springer, Berlin, Heidelberg, and New York, 1978).CrossRefGoogle Scholar
[9]Kuiper, N. H., ‘The homotopy type of the unitary group of Hilbert space’, Topology 3 (1965), 19–30.CrossRefGoogle Scholar
[10]Madsen, I. and Rosenberg, J., ‘The universal coefficient theorem for equivariant K-theory of real and complex C *-algebras’, Index Theory of Elliptic Operators, Foliations, and Operator Algebras, edited by Kaminker, J., Millett, K., and Schochet, C., (Contemp. Math., vol. 70, Amer. Math. Soc., Providence, R.I., 1988, pp. 145–173).CrossRefGoogle Scholar
[11]Mingo, J. A., K-Theory and Multipliers of Stable c *Algebras (Ph. D. thesis, Dalhousie univ., Halifax, N.s., 1982).Google Scholar
[12]Parker, E. M., ‘The Brauer group of graded continuous-trace C*-algebras’, preprint, 1987.CrossRefGoogle Scholar
[13]Phillips, J. and Raeburn, I., ‘Automorphisms of C *-algebras and second Čech cohomology’, Indiana Univ. Math. J. 29 (1980), 799–822.CrossRefGoogle Scholar
[14]Rosenberg, J., ‘Homological invariants of extensions of C *-algebras’, Operator Algebras and Applications, edited by Kadison, R. V., (Proc. Sympos. Pure Math., vol. 38, Part 1, Amer. Math. Soc., Providence, R.I., 1982, pp. 35–75).CrossRefGoogle Scholar
[15]Rosenberg, J., ‘The role of K-theory in non-commutative algebraic topology’, Operator Algebras and K-Theory, edited by Douglas, R. G. and Schochet, C., (Contemp. Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1982, pp. 155–182).Google Scholar
[16]Rosenberg, J., ‘C*-algebras, positive scalar curvature, and the Novikov conjecture, III,’ Topology 25 (1986), 319–336.CrossRefGoogle Scholar
[17]Segal, G., ‘Classifying spaces and spectral sequences’, Publ. Math. I.H.É.S. 34 (1968), 105–112.CrossRefGoogle Scholar