Abstract
This paper is an outgrowth of joint work with Richard Herman [5] and Iain Raeburn [13]. In both of these projects, questions concerning group actions on operator algebras naturally led to a study of obstruction classes in H2(G,U(A)), where U(A) is the (suitably topologized) unitary group of an abelian operator algebra A. The appropriate cohomology theory here is the “Borel cochain” theory of C.C.Moore, as developed and systematized in [8]. In case A is a von Neumann algebra, U(A) is essentially what Moore calls U(X,T) (X here is some standard measure space), and machinery for computing the relevant cohomology groups is developed and applied in [8] and [9].
Research supported in part by NSF Grants DMS-8400900 and 8120790.
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References
Borel, A.; Wallach, N.: Continuous cohomology, discrete subgroups, and repre- sentations of reductive groups, Annals of Math. Studies, Vol. 94, Princeton University Press, Princeton, 1980.
Dixmier, J.; Douady, A.: Champs continus d’espaces hilbertiens et de C*-algé- bres, Bull. Soc. Math. France 91 (1963), 227 – 284.
Dold,A.: Partitions of unity in the theory of fibrations, Ann. of Math. 78 (1963), 223 – 225.
Guichardet, A.: Cohomologie des groupes topologiques et des algèbres de Lie, Textes Mathématiques, vol. 2, CEDIC/Nathan, Paris, 1980.
Herman, R.H.; Rosenberg, J.: Norm-close group actions on C*-algebras, J. Operator Theory 6(1981), 25–37.
Michael, E.: Convex structures and continuous selections, Canad. J. Math. 11 (1959), 556 – 575.
Moore, C.C.: Extensions and low dimensional cohomology theory of locally compact groups. I, Trans. Amer. Math. Soc. 113 (1964), 40 – 63.
Moore, C.C.: Group extensions and cohomology for locally compact groups. III, Trans. Amer. Math. Soc. 221 (1976), 1 – 33.
Moore, C.C.: Group extentions and cohomology for locally compact groups. IV, Trans. Amer. Math. Soc. 221 (1976), 35 – 58.
Palais, R.: On the existence of slices for actions of non-compact Lie groups, Ann. of Math. 73 (1961), 295 – 323.
Pedersen, G.K.: C * -algebras and their automorphism groups, London Math. Soc. Monographs, vol. 14, Academic Press, London, 1979.
Phillips, J.; Raeburn, I.: Crossed products by locally unitary automorphism groups and principal bundles, J. Operator Theory 11(1984),215–241.
Raeburn, I.; Rosenberg, J.: Crossed products of continuous-trace C*-algebras by smooth actions, Preprint 05711–85, MSRI, Berkeley, 1985.
Raeburn, I.; Williams, D.P.: Pull-backs of C*-algebras and crossed products by certain diagonal actions, Trans. Amer. Math. Soc., 287 (1985), 755 – 777.
Vogan Jr., D.A.: Representations of real reductive groups, Progress in Math., vol. 15, Birkhäuser, Boston, 1981.
Whitehead, G.W.: Elements of homotopy theory, Graduate Texts in Math., vol. 61, Springer, New York, 1978.
Wigner, D.: Algebraic cohomology of topological groups, Trans. Amer. Math. Soc. 178 (1973), 83 – 93.
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Rosenberg, J. (1986). Some Results on Cohomology with Borel Cochains, with Applications to Group Actions on Operator Algebras. In: Douglas, R.G., Pearcy, C.M., Sz.-Nagy, B., Vasilescu, FH., Voiculescu, D., Arsene, G. (eds) Advances in Invariant Subspaces and Other Results of Operator Theory. Operator Theory: Advances and Applications, vol 17. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7698-8_22
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