Amenable pairs of groups and ergodic actions and the associated von Neumann algebras

Zimmer, Robert J.

doi:10.1090/S0002-9947-1978-0502907-6

Amenable pairs of groups and ergodic actions and the associated von Neumann algebras
HTML articles powered by AMS MathViewer

by Robert J. Zimmer PDF

Trans. Amer. Math. Soc. 243 (1978), 271-286 Request permission

Abstract:

If X and Y are ergodic G-spaces, where G is a locally compact group, and X is an extension of Y, we study a notion of amenability for the pair $(X,Y)$. This simultaneously generalizes and expands upon previous work of the author concerning the notion of amenability in ergodic theory based upon fixed point properties of affine cocycles, and the work of Eymard on the conditional fixed point property for groups. We study the relations between this concept of amenability, properties of the von Neumann algebras associated to the actions by the Murray-von Neumann construction, and the existence of relatively invariant measures and conditional invariant means.

References

A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
John Ernest, Hopf-von Neumann algebras, Functional Analysis (Proc. Conf., Irvine, Calif., 1966) Academic Press, London; Thompson Book Co., Washington, D.C., 1967, pp. 195–215. MR 0223909
Pierre Eymard, Moyennes invariantes et représentations unitaires, Lecture Notes in Mathematics, Vol. 300, Springer-Verlag, Berlin-New York, 1972 (French). MR 0447969, DOI 10.1007/BFb0060750
Shmuel Glasner, Relatively invariant measures, Pacific J. Math. 58 (1975), no. 2, 393–410. MR 380756, DOI 10.2140/pjm.1975.58.393
Hui Hsiung Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR 0461643
George W. Mackey, Point realizations of transformation groups, Illinois J. Math. 6 (1962), 327–335. MR 143874
Arlan Ramsay, Nontransitive quasi-orbits in Mackey’s analysis of group extensions, Acta Math. 137 (1976), no. 1, 17–48. MR 460531, DOI 10.1007/BF02392411
Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
I. E. Segal, Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc. 81 (1956), 106–134. MR 76317, DOI 10.1090/S0002-9947-1956-0076317-8
I. E. Segal, Ergodic subgroups of the orthogonal group on a real Hilbert space, Ann. of Math. (2) 66 (1957), 297–303. MR 89382, DOI 10.2307/1970001
Jun Tomiyama, On the projection of norm one in $W^{\ast }$-algebras, Proc. Japan Acad. 33 (1957), 608–612. MR 96140
Robert J. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), no. 3, 373–409. MR 409770
Robert J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Functional Analysis 27 (1978), no. 3, 350–372. MR 0473096, DOI 10.1016/0022-1236(78)90013-7
Robert J. Zimmer, On the von Neumann algebra of an ergodic group action, Proc. Amer. Math. Soc. 66 (1977), no. 2, 289–293. MR 460599, DOI 10.1090/S0002-9939-1977-0460599-3
Robert J. Zimmer, Hyperfinite factors and amenable ergodic actions, Invent. Math. 41 (1977), no. 1, 23–31. MR 470692, DOI 10.1007/BF01390162
Robert J. Zimmer, Orbit spaces of unitary representations, ergodic theory, and simple Lie groups, Ann. of Math. (2) 106 (1977), no. 3, 573–588. MR 466406, DOI 10.2307/1971068
Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (algèbres de von Neumann), Cahiers Scientifiques, Fasc. XXV, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition, revue et augmentée. MR 0352996