Abstract
The purpose of this paper is to describe a method of using flow equivalence for homeomorphisms to construct certain projections in the corresponding transformation group C*-algebras, by following a program which was outlined in our recent paper [15]. Using methods stemming from work of Connes [3] and Rieffel and Green [21] we will explicitly construct the strong Morita equivalence bimodule relating two C -algebras associated to flow equivalent transformations, and the corresponding projections; these projections will be related to the isomorphism of Connes between KO(C*(Y,Z)) and K1(M(Y)) where (Y,Z) is a dynamical system, C*(Y,Z) the associated C*-algebra, and M(Y) is the mapping torus for (Y.Z).
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References
Bowen, R.: On axiom A diffeomorphisms, Regional Conference Series in Mathematics, No.35, A.M.S., Providence, 1978.
Connes, A.: An analogue of the Thom isomorphism for crossed products of a C*-algebra by an action of R, Adv. in Math. 39(1981), 31–55.
Connes, A.: A survey of foliations and operator algebras, in Proc. Symp. Pure Math., v.38, pp.521–628.
Dixmier, J.: C*-algebras, North-Holland Publishing Co., Amsterdam, 1982.
Devaney, R.: Linked twist mappings are almost Anosov, in Springer Lecture Notes in Math., 819, pp.121–145.
Exel, R.: Rotation numbers for automorphisms of C*-algebras, Ph. D. Thesis, University of California at Berkeley, 1985.
Fathi, A.; Shub, M.: Some dynamics of pseudo-Anosov diffeomorphisms, in Travaux de Thurston sur les surfaces, (A. Fathi, F. Laudenbach and V. Poenaru, Eds.), Astérisque 66–67,1979.
Fried, D.: Geometry of cross-section to flows, Topology 21(1982), 353–371.
Fried, D.: Fibrations with pseudo-Anosov monodromy, in Travaux de Thurston sur les surfaces, (A. Fathi, F.Laudenbach and V.Poenaru, Eds.), Astérisque 66–67, 1979.
Fried, D.: Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory and Dynamical Systems 5(1985), 539–563.
Friedman, N.: Introduction to ergodic theory,Van Nostrand-Reinhold, Princeton, N.J., 1970.
Green, P.: C*-algebras of transformation groups with smooth orbit space, Pacific J. Math. 72(1977), 71–97.
Green, P.: The structure of imprimitivity algebras, J. Funct. Anal. 36(1980), 88–104.
Muhly, P.: Renault, J.; Williams, D.: Equivalence and isomorphism for groupoid C*-algebras, preprint.
Packer, J.: K-theoretic invariants for C*-algebras associated to transformations and induced flows, J. Func. Anal. 67(1986), 25–59.
Packer, J.: Strong Morita equivalence and the positive cones of the KO - groups of Heisenberg C*-algebras, preprint.
Paschke, W.: On the mapping torus of an automorphism, Proc. Amer. Math. Soc. 88(1983), 481–485.
Pimsner, M.: Range of traces on KO of reduced crossed products by free groups, in Springer Lecture Notes in Math., 1132, pp.374–408.
Raeburn, I.: On the Picard group of a continuous trace C*-algebra, Trans. Amer. Math. Soc.263(1981), 183–205.
Rieffel, M.: C*-algebras associated with irrational rotations, Pacific J. Math. 93(1981), 415–429.
Rieffel, M: Applications of strong Morita equivalence to transformation group C*-algebras, Proc. Symp. Pure Math., v.38I, pp.289–310.
Rieffel, M.: The cancellation theorem for projective modules over irrational rotation C*-algebras, Proc. London Math. Soc.(3), 17(1983), 285–302.
Rieffel, M.: Projective modules over higher dimensional non-commutative tori, preprint.
Schwartzman, S.: Asymptotic cycles, Ann. of Math.66(1957), 270–284.
Schwartzman, S.: Global cross sections of compact dynamical systems, Proc. Nat. Acad. Sci. 48(1962), 786–791.
Smale, S.: Differentiable dynamical systems, Bull. Amer. Math. Soc. 73(1967), 747–817.
Thurston, W.: On the geometry and dynamics of diffeomorphisms of surface, preprint.
Walters, P.: Ergodic theory, Springer Lecture Notes in Math, v.458, 1975.
Effros, E.: Hahn, F.: Locally compact transformation groups and C*-algebras, Mem. Amer. Math. Soc., 75, 1967.
Pimsner, M.; Voiculescu, D.: Exact sequences for K-groups and Ext-groups of certain crossed product C*-algebras, J. Operator Theory 4(1980), 93–118.
Anderson, J.; Paschke, W.: The rotation algebra. M.S.R.I. preprint 04111-85.
Petersen, K.; Shapiro, L.: Induced flows, Trans. Amer. Math. Soc. 177(1973), 375–391.
Pun, Y. P.: A K-theoretic invariant for dynamical systems, preprint.
Putnam, I.: The C*-algebras associated with minimal homeomorphisms of the Cantor set, preprint.
Ji, R: On the crossed product C*-algebras associated with Furstenberg transformations on tori, preprint.
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© 1988 Birkhäuser Verlag Basel
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Packer, J.A. (1988). Flow Equivalence for Dynamical Systems and the Corresponding C*-Algebras. In: Arsene, G. (eds) Special Classes of Linear Operators and Other Topics. Operator Theory: Advances and Applications, vol 28. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9164-6_16
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DOI: https://doi.org/10.1007/978-3-0348-9164-6_16
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