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Dixmier-Douady Classes of Dynamical Systems and Crossed Products

Published online by Cambridge University Press:  20 November 2018

Iain Raeburn  and
Dana P. Williams
Iain Raeburn
Affiliation:
Department of Mathematics University of Newcastle Newcastle, New South Wales 2308 Australia
Dana P. Williams
Affiliation:
Department of Mathematics Dartmouth College Hanover, New Hampshire 03755-3551 USA
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Abstract

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Continuous-trace C*-algebras A with spectrum T can be characterized as those algebras which are locally Monta equivalent to C0(T). The Dixmier-Douady class δ(A) is an element of the Čech cohomology group Ȟ3(T, ℤ) and is the obstruction to building a global equivalence from the local equivalences. Here we shall be concerned with systems (A, G, α) which are locally Monta equivalent to their spectral system (C0(T),G, τ), in which G acts on the spectrum T of A via the action induced by α. Such systems include locally unitary actions as well as N-principal systems. Our new Dixmier-Douady class δ (A, G, α) will be the obstruction to piecing the local equivalences together to form a Monta equivalence of (A, G, α) with its spectral system. Our first main theorem is that two systems (A, G, α) and (B, G, β) are Monta equivalent if and only if δ (A, G, α) = δ (B, G, β). In our second main theorem, we give a detailed formula for δ (A ⋊α G) when (A, G, α) is N-principal.

Keywords

46L5546L05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 5 , 01 October 1993 , pp. 1032 - 1066
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Walter Beer, On Morita equivalence of Nuclear C*-algebras, J. Pure and Appl. Algebra 26(1982), 249–267.Google Scholar
2. Combes, F., Crossed products and Morita equivalence, Proc. London Math. Soc. (3) 49(1984), 289–306.Google Scholar
3. Curto, Raul E., Muhly, Paul and Dana Williams, P., Crossed products of strongly Morita equivalent C*- algebras, Proc. Amer. Math. Soc. 90(1984), 528–530.Google Scholar
4. Dixmier, Jacques, C*-algebras, North-Holland, New York, 1977.Google Scholar
5. Jacques Dixmier and Douady, A., Champs continus d'espaces hilbertiens et de C*-algebras, Bull. Soc. Math. Franc 91(1963), 227–284.Google Scholar
6. Green, Philip, The local structure of twisted covariance algebras, Acta. Math. 140(1978), 191–250.Google Scholar
7. Green, Philip, The Brauer group of a commutative C*-algebra, unpublished seminar notes, University of Pennsylvania, (1978).Google Scholar
8. Alexander Kumjian, On equivariant sheaf cohomology and elementary C* -bundles, J. Operator Theory 20(1988), 207–240.Google Scholar
9. Ellen Maycock Parker, The Brauer group of graded continuous trace C*-algebras, Trans. Amer. Math. Soc. 308(1988), 115–132.Google Scholar
10. John Phillips and Iain Raeburn, Crossed products by locally unitary automorphism groups and principal bundles, J. Operator Theory 11(1984), 215–241.Google Scholar
11. Raeburn, Iain, On the Picard group of a continuous-trace C*-algebra, Trans. Amer. Math. Soc. 263(1981), 183–205.Google Scholar
12. Raeburn, Iain, Induced C* -algebras and a symmetric imprimitivity theorem, Math. Ann. 280(1988), 369–387.Google Scholar
13. Raeburn, Iain and Jonathan Rosenberg, Crossed products of continuous-trace C*-algebras by smooth actions, Trans. Amer. Math. Soc. 305(1988), 1–45.Google Scholar
14. Raeburn, Iain and Taylor, Joseph L., Continuous-trace C*-algebras with given Dixmier-Douady class, J. Austral. Math. Soc. (A) 38(1985), 394–407.Google Scholar
15. Raeburn, Iain and Williams, Dana P., Pull-backs of C*-algebras and crossed products by certain diagonal actions, Trans. Amer. Math. Soc. 287(1985), 755–777.Google Scholar
16. Raeburn, Iain and Williams, Dana P., Moore cohomology, principal bundles, and actions of groups on C*-algebras, Indiana U. Math. J. 40(1991),707-740.Google Scholar
17. Raeburn, Iain and Williams, Dana P., Topological invariants associated to the spectrum of crossed product C* -algebras, J. Funct. Anal. to appear.Google Scholar
18. Raeburn, Iain, Equivariant cohomology and a Gysin sequence for principal bundles, preprint.Google Scholar
19. Rieffel, MarcA., Induced representations of C*-algebras, Adv. in Math. 13(1974), 176–257.Google Scholar
20. Rieffel, MarcA., Unitary representations of group extensions: an algebraic approach to the theory ofMackey and Blattner, Adv. in Math. Supplementary Studies 4(1979), 43–81.Google Scholar
21. Jonathan Rosenberg,/foraofog/ca/ invariants of extensions of C* -algebras, Proc. Symp. Pure Math., Amer. Math. Soc. 38(1982), part I, 35–75.Google Scholar
22. Jonathan Rosenberg , Continuous-trace algebras from the bundle-theoretic point of view, J. Aust. Math. Soc. (A) 47 (1989), 368–381.Google Scholar
23. Warner, FrankW., Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Company, Glenview, Illinois, 1971.Google Scholar