Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T14:56:15.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous Ergodic Extensions and Fibre Bundles

Published online by Cambridge University Press:  20 November 2018

Robert J. Zimmer
Robert J. Zimmer*
Affiliation:
United States Naval Academy, Annapolis, Maryland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If a locally compact group G acts as a measure preserving transformation group on a Lebesgue space X, then there is a naturally induced unitary representation of G on L2(X), and one can study the action on X by means of this representation. The situation in which the representation has discrete spectrum (i.e., is the direct sum of finite dimensional representations) and the action is ergodic was examined by von Neumann and Halmos when G is the integers or the real line [7], and by Mackey for general non-abelian G [10].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 2 , 01 April 1978 , pp. 373 - 391
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Auslander, L., Green, L. and Hahn, F., Flows on homogeneous spaces, Annals of Math. Studies 53 (1963).Google Scholar
2. Bredon, G., Introduction to compact transformation groups (Academic Press, New York, 1972).Google Scholar
3. Ellis, R., Lectures on topological dynamics (Benjamin, New York, 1969).Google Scholar
4. Furstenberg, H., The structure of distal flows, Amer. J. Math. 85 (1963), 477–515.Google Scholar
5. Glasner, S., Relatively invariant measures, Pac. J. Math. 58 (1975) 393–410.Google Scholar
6. Halmos, P. R., Lectures on ergodic theory (Chelsea Pub. Co., New York, 1956).Google Scholar
7. Halmos, P. R. and von Neumann, J., Operator methods in classical mechanics II, Annals of Math. 43 (1942), 235–247.Google Scholar
8. Hewitt, E. and Ross, K., Abstract harmonic analysis, Vol. II (Springer-Verlag, Berlin, 1970).Google Scholar
9. Knapp, A. W., Distal functions on groups, Trans. Amer. Math. Soc. 128 (1967), 1–40.Google Scholar
10. Mackey, G. W., Ergodic transformation groups with a pure point spectrum, Illinois J. Math. 6 (1962), 327–335.Google Scholar
11. Zimmer, R. J., Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373–409.Google Scholar
12. Zimmer, R. J., Ergodic actions with generalized discrete spectrum, Illinois J. Math. 20 (1976), 555–588.Google Scholar
13. Zimmer, R. J., Distal transformation groups and fibre bundles, Bull. Amer. Math. Soc. 81 (1975), 959–960.Google Scholar