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The C*-algebras of Morse—Smale flows on two-manifolds

Published online by Cambridge University Press:  19 September 2008

Xiaolu Wang
Xiaolu Wang
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
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Abstract

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We give a classification of all the C*-algebras of Morse-Smale flows on closed two-manifolds, and determine the relation between the invariants of dynamical systems and the topological invariants of the C*-algebras.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 3 , September 1990 , pp. 565 - 597
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[C1]Connes, A.. Sur la théorie noncommutative de I'intégration. Lecture Notes in Mathematics 725, pp. 19–43. Springer-Verlag: New York, 1979.Google Scholar
[C2]Connes, A.. A survey of foliations and operator algebras. Proc. Symp. Pure Math. 38 (1982), 521–628.CrossRefGoogle Scholar
[C3]Connes, A.. Noncommutative differential geometry. Publ. Math. I.H.E.S. 62 (1986), 41–144.CrossRefGoogle Scholar
[Cun]Cuntz, J.. K-theory and C*-algebras. Lect. Notes Math. No. 1046. Springer-Verlag: New York, 1984.Google Scholar
[C-V]Culler, M. & Vogtmann, K.. Moduli of graphs and automorphisms of free groups. Invent. Math. 84 (1986), 91–119.CrossRefGoogle Scholar
[E-H]Effros, E. G. & Hahn, F.. Locally compact transformation groups and C*-algebras. Mem. Amer. Math. Soc. 75 (1967).Google Scholar
[F]Fleitas, G.. Classification of gradient-like flows on dimensions two and three. Bol. Soc. Bras. Mat. 6 (1975), 155–183.CrossRefGoogle Scholar
[F-W]Fack, T. & Wang, X.. The C*-algebras of Reeb foliations are not AF-embeddable. Proc. Amer. Math. Soc. 108 (4) (1990), 941–947.Google Scholar
[Gr]Green, P.. C*-algebras of transformation groups with smooth orbit spaces. Pacif. J. Math. 72 (1977), 71–97.CrossRefGoogle Scholar
[H-S]Hilsum, M. & Skandalis, G.. Stabilité des C*-algèbres de feuilletages. Ann. Inst. Fourier 33 (1983), 201–208.CrossRefGoogle Scholar
[H-W1]Hirsch, Morris & Wang, X.. Foliations of planar regions and CCR C*-algebras with infinite composition length. Amer. J. Math.Google Scholar
[H-W]Hirsch, Michael & Wang, X.. Links and Morse flows in four-manifolds. Preprint.Google Scholar
[Gt]Gootman, E. C.. The type of some C*-and W*-algebras associated with transformation groups. Pacif. J. Math. 48 (1) (1973), 98–106.CrossRefGoogle Scholar
[K]Kasparov, G. G.. The operator K-functor and extensions of C*-algebras. Izv. Akad. Nauk SSSR Ser. Mat. 16 No. (3) (1981), 513–572.Google Scholar
[Lev1]Levitt, G.. Foliations and laminations on hyperbolic surfaces. Topology 22 (2) (1983), 119–135.CrossRefGoogle Scholar
[Lev2]Levitt, G.. Pantalons et feuilletages des surfaces. Topology 21 (1) (1982), 9–33.CrossRefGoogle Scholar
[Mar]Martinez-Maure, Y.. Feuilletages des surfaces et décompositions en pantalons. Bull. Soc. Math. France 112 (1984), 387–396.Google Scholar
[M-Sc]Moore, C. C. & Schochet, C.. Global Analysis on Foliated Spaces, M.S.R.I. Monographs. Springer-Verlag: New York, 1988.CrossRefGoogle Scholar
[M-S]Morgan, J. & Shalen, P.. Valuations, trees, and degenerations of hyperbolic structures. I. Ann. Math. 120 (1984), 401–476;CrossRefGoogle Scholar
II, Valuations, trees, and degenerations of hyperbolic structures. I. Ann. Math. 127.Google Scholar
[P-M]Palis, J. Jr, & Melo, W.. Geometric Theory of Dynamical Systems. An Introduction. Springer-Verlag: New York, 1982.CrossRefGoogle Scholar
[P1]Peixoto, M. M.. Structural stability on two-manifolds. Topology 1 (1962), 101–120.CrossRefGoogle Scholar
[P2]Peixoto, M. M.. On the classification of flows on two-manifolds. Dynamical Systems Proc. Symp., Salvador, pp. 389–419. Academic Press: New York, 1973.Google Scholar
[P-P]Peixoto, M. C. & Peixoto, M. M.. Structural stability in the plane with enlarged boundary conditions. Ann. Acad. Bras. Ci. 31 (1959), 135–160.Google Scholar
[R1]Rieffel, M.. Applications of strong Morita equivalence to transformation group C*-algebras. Proc. Symp. Pure Math. 38 Part I (1982), 299–310.CrossRefGoogle Scholar
[R2]Rieffel, M.. Dimension and stable rank in the K-theory of C*-algebras. Proc. Bull. Lond. Math. Soc. 46 (1983), 301–333.CrossRefGoogle Scholar
[R-S]Rosenberg, J. & Schochet, C.. The Künneth and the universal coefficient theorem for Kasparov's generalized K-functor. Duke Math. J. 55 (1987), 431–474.CrossRefGoogle Scholar
[Ser]Serre, J. P.. Trees. Springer-Verlag: New York, 1980.CrossRefGoogle Scholar
[S]Smale, S.. Differentiate dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[Tor]Torpe, A. M.. K-theory for the leaf space of foliations by Reeb components. J. Func. Anal. 61 (1985), 15–71.CrossRefGoogle Scholar
[W1]Wang, X.. On the C*-algebras of a family of solvable Lie groups. Pitman Research Notes in Math. No. 199. Longman: London, 1989.Google Scholar
[W2]Wang, X.. On the C*-algebras of foliations in the plane. Lect. Notes Math., No. 1257. Springer-Verlag: New York, 1987.Google Scholar
[W3]Wang, X.. Noncommutative CW-complexes. Contemp. Math., Amer. Math. Soc. 70 (1988), 303–322.CrossRefGoogle Scholar
[W4]Wang, X.. On the relation between C*-algebras of foliations and those of their coverings. Proc. Amer. Math. Soc. 102 (2) (1988), 355–360.Google Scholar
[Wil]Williams, D.. The topology on the primitive ideal space of transformation group C*-algebras and CCR transformation group C*-algebras. Trans. Amer. Math. Soc. 266 (1981), 335–359.Google Scholar