Abstract
This article is intended to answer the question “Why do you guys always want to twist everything?” We review the various ways in which twists, twisted actions and twisted crossed products arise, and then discuss some cohomological obstructions to the existence and triviality of twisted actions.
This research was supported by the Australian Research Council.
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References
A. Adem and R.J. Milgram. Cohomology of finite groups, Springer-Verlag, Berlin, 1994.
J. Bellissard, A. van Elst and H. Schulz-Baldes. The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35 (1994), 5373–5451.
H. Baumgärtel and F. Lledo. Dual actions on C*-algebras and Hilbert extensions, Preprint, U. of Potsdam, 1999.
R.C. Busby and H.A. Smith. Representations of twisted group algebras, Trans. Amer. Math. Soc. 149 (1970), 503–537.
G. Cassinelli, E. de Vito, P.J. Lahti and A. Levrero. Symmetry groups in quantum mechanics and the theorem of Wigner on the symmetry transformations, Rev. Math. Phys. 9 (1997), 921–941.
A.L. Carey, H. Grundling, C.A. Hurst and E. Langmann. Realising 3-cocycles as obstructions, J. Math. Phys. 36 (1995), 2605–2620.
A.L. Carey, H. Grundling, I. Raeburn and C.E. Sutherland. Group actions on C*-algebras, 3-cocycles and quantum field theory, Comm. Math. Phys. 168 (1995), 389–416.
A.L. Carey, K.C. Hannabuss, V. Mathai and P. McCann. Quantum Hall effect on the hyperbolic plane, Comm. Math. Phys. 190 (1997), 629–673.
D. Crocker, A. Kumjian, I. Raeburn and D.P. Williams. An equivariant Brauer group and actions of groups on C*-algebras, J. Funct. Anal. 146 (1997), 151–184.
S. Echterhoff. Morita equivalent twisted actions and a new version of the Packer-Raeburn stabilisation trick, J. London. Math. Soc. 50 (1994), 170–186.
P. Green. The local structure of twisted covariance algebras,Acta Math. 140 (1978), 191–250.
P. Green. The structure of imprimitivity algebras. J. Funct. Anal. 36 (1980), 88–104.
R.H. Herman and J. Rosenberg. Norm-close group actions on C*-algebras, J. Operator Theory 6 (1981), 25–37.
J. Huebschmann. Group extensions, crossed pairs and an eight term exact sequence, J. reine angew. Math. 321 (1981), 150–172.
V.F.R. Jones. Actions of finite groups on the hyperfinite type II1-factor, Mem. Amer. Math. Soc. 280 (1980).
Y. Kawahigashi, C.E. Sutherland, M. Takesaki. The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions. Acta Math. 169 (1992), 105–130.
M.B. Landstad and I. Raeburn. Twisted dual-group algebras: equivariant deformations of C0(G), J. Funct. Anal. 132 (1995), 43–85.
M.B. Landstad and I. Raeburn. Equivariant deformations of homogeneous spaces, J. Funct Anal. 148 (1997), 480–507.
J.-L. Loday. Cohomologie et groupe de Steinberg relatif, J. Algebra 54 (1978), 178–202.
G.W. Mackey. Unitary representations of group extensions I, Acta Math. 99 (1958), 265–311.
S. Mac Lane. Homology, Springer-Verlag, Berlin, 1993.
C.C. Moore. Extensions and low-dimensional cohomology theory of locally compact groups I, II, Trans. Amer. Math. Soc. 113 (1964), 40–63 and 64–86.
C.C. Moore. Group extensions and cohomology for locally compact groups III, IV, Trans. Amer. Math. Soc. 221 (1976), 1–33 and 35–58.
J.A. Packer. Transformation group C*-algebras: A selective survey, in C*-Algebras: 1943-1993: A Fifty Year Celebration (R.S. Doran, ed.), Contemporary Math., vol. 167, Amer. Math. Soc, Providence, 1994, pages 183–217.
J.A. Packer and I. Raeburn. Twisted crossed products of C*-algebras, Math. Proc. Camb. Phil. Soc. 106 (1989), 293–311.
J.A. Packer and I. Raeburn. Twisted crossed products of C*-algebras, II, Math. Ann. 287 (1990), 595–612.
J.A. Packer and I. Raeburn. On the structure of twisted group C*-algebras, Trans. Amer. Math. Soc. 334 (1992), 685–718.
I. Raeburn and J. Rosenberg. Crossed products of continuous-trace C*-algebras by smooth actions, Trans. Amer. Math. Soc. 305 (1988), 1–45.
I. Raeburn and D.P. Williams. Moore cohomology, principal bundles, and group actions on C*-algebras, Indiana Univ. Math. J. 40 (1991), 707–740.
I. Raeburn and D.P. Williams. Morita Equivalence and Continuous-Trace C* -Algebras, Math. Surveys & Monographs, vol. 60, Amer. Math. Soc, Providence, 1998.
J.G. Ratcliffe. Crossed extensions, Trans. Amer. Math. Soc. 257 (1980), 73–89.
M.A. Rieffel. Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (1989), 531–562.
J. Rosenberg. Some results on cohomology with Borel cochains, with applications to group actions on C*-algebras, Operator Theory: Adv. & Appl. 17 (1986), 301–330.
B. Simon. Representations of finite and compact groups, Grad. Studies in Math., vol. 10, Amer. Math. Soc, Providence, 1996.
C.E. Sutherland. Cohomology and extensions of von Neumann algebras, II, Proc. RIMS, Kyoto Univ. 16 (1980), 135–174.
C.E. Sutherland and M. Takesaki. Actions of discrete amenable groups and groupoids on von Neumann algebras. Publ. RIMS, Kyoto Univ. 21 (1985), 1087–1120.
M. Takesaki. Covariant representations of C* -algebras and their locally compact automorphism groups, Acta Math. 119 (1967), 273–303.
V.S. Varadarajan. Geometry of Quantum Theory, second edition, Springer-Verlag, New York, 1985.
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Raeburn, I., Sims, A., Williams, D.P. (2000). Twisted Actions and Obstructions in Group Cohomology. In: Cuntz, J., Echterhoff, S. (eds) C*-Algebras. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57288-3_9
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DOI: https://doi.org/10.1007/978-3-642-57288-3_9
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