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Locally compact groups appearing as ranges of cocycles of ergodic ℤ-actions

Published online by Cambridge University Press:  19 September 2008

V. Ya. Golodets  and
S. D. Sinelshchikov
V. Ya. Golodets
Affiliation:
Institute of Low Temperature Physics and Engineering, UkrSSR Academy of Sciences, 47 Lenin Ave, Kharkov 310164, USSR
S. D. Sinelshchikov
Affiliation:
Institute of Low Temperature Physics and Engineering, UkrSSR Academy of Sciences, 47 Lenin Ave, Kharkov 310164, USSR
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Abstract

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The paper contains the proof of the fact that every solvable locally compact separable group is the range of a cocycle of an ergodic automorphism. The proof is based on the theory of representations of canonical anticommutation relations and the orbit theory of dynamical systems. The slight generalization of reasoning shows further that this result holds for amenable Lie groups as well and can be also extended to almost connected amenable locally compact separable groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 1 , March 1985 , pp. 47 - 57
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Connes, A. & Krieger, W.. Measure space automorphisms, the normalizers of their full groups, and the approximate finiteness. J. Funct. Anal. 24(1977), 336–352.CrossRefGoogle Scholar
[2]Connes, A., Feldman, J. & Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1 (1981), 431–450.CrossRefGoogle Scholar
[3]Feldman, J., Hahn, P. & Moore, C.. Orbit structure and countable sections for actions of continuous groups. Adv. in Math. 28 (1978), 186–230.CrossRefGoogle Scholar
[4]Golodets, V. Ya.. Description of representations of anticommutation relations (in Russian). Uspekhi Mat. nauk. 24 (1969), 3–64.Google Scholar
[5]Golodets, V. Ya. & Sinelshchikov, S. D.. Existence and uniqueness of cocycles of an ergodic automorphism with dense ranges in amenable groups. Preprint, 1983.Google Scholar
[6]Herman, M.. Constructions des diffeomorphismes ergodiques. Preprint.Google Scholar
[7]Ismagilov, R. S.. On irreducible cycles connected with a dynamical system (in Russian). Funktz. Anal, i Prilozhen 3 (1969), 92–93.Google Scholar
[8]Kirillov, A. A.. Dynamical systems, factors and representations of groups (in Russian). Uspekhi Mat. nauk 22 (1967), 67–80.Google Scholar
[9]Mackey, G. W.. Ergodic theory and virtual groups. Math. Ann. 166 (1966), 187–207.CrossRefGoogle Scholar
[10]Montgomery, D. & Zippen, L.. Topological Transformation Groups. Interscience: New York-London, 1955.Google Scholar
[11]Stepin, A. M.. On the cohomology of groups of automorphisms of a Lebesgue space (in Russian). Funktz. Analiz i Prilozhen 5 (1971), 91–92.Google Scholar
[12]Zimmer, R.. Extensions of ergodic group actions. Illinois J. Math. 20 (1976), 373–409.CrossRefGoogle Scholar
[13]Zimmer, R.. Random walks on compact groups and the existence of cocycles. Isr. J. Math. 26 (1977), 84–90.CrossRefGoogle Scholar
[14]Zimmer, R.. Cocycles and the structure of ergodic group actions. Isr. J. Math. 26 (1977), 214–220.CrossRefGoogle Scholar
[15]Zimmer, R.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Funct. Anal. 27 (1978), 350–372.CrossRefGoogle Scholar