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The von Neumann algebra of a foliation

  • Alain Connes
Main Lectures
Part of the Lecture Notes in Physics book series (LNP, volume 80)

Abstract

Every smooth foliation
of a manifold V gives rise very naturally to a von Neumann algebra
. The weights on M correspond exactly to operator valued forms on the “manifold” of leaves of
. We compute their modular automorphism group, this yields the continuous decomposition of M in terms of another foliation
of V and a one parameter group of automorphisms of
. We then illustrate this decomposition with a few examples.

Keywords

Operator Density Compact Manifold Random Operator Ergodic Decomposition Transverse Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. [1]
    R. BARRE — De quelques aspects de la théorie des Q-variétés différentielles et analytiques. Annales Inst. Fourier, tome 23 (1973).Google Scholar
  2. [2]
    R. BOWEN — Anosov foliations are hyperfinite (preprint).Google Scholar
  3. [3]
    A. CONNES — Une classification des facteurs de type III, Annales Scientifiques E.N.S., 4éme série tome 6, fasc. 2, (1973), p.133–252.Google Scholar
  4. [4]
    A. CONNES and D. SULLIVAN — (To appear).Google Scholar
  5. [5]
    A. CONNES and M. TAKESAKI — The flow of weights on type III factors. Tohoku Math. Journal. 29 (1977) p.473–575Google Scholar
  6. [6]
    W. KRIEGER — Ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976), p. 19–70.Google Scholar
  7. [7]
    D. RUELLE and D. SULLIVAN — Currents, flows and diffeomorphisms, Topology Vol. 14, p. 319–327.Google Scholar
  8. [8]
    M. TAKESAKI — Duality in cross products and the structure of von Neumann algebras of typr III, Acta Math. 131 (1973), p. 249–310.Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Alain Connes

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