Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-16T22:39:47.373Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodic actions of group extensions on von Neumann algebras

Published online by Cambridge University Press:  14 November 2011

Klaus Thomsen
Klaus Thomsen
Affiliation:
Matematisk Institut, Ny Munkegade, 8000 Aarhus C, Denmark
Get access Rights & Permissions [Opens in a new window]

Synopsis

We consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 112 , Issue 1-2 , 1989 , pp. 71 - 112
Copyright
Copyright © Royal Society of Edinburgh 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Albeverio, S. and Hoegh-Krohn, R.. Ergodic actions by compact groups on C *-algebras. Math. Z. 173 (1980), 1–17.CrossRefGoogle Scholar
2Brenken, B.. Representations and automorphisms of the irrational rotation algebra. Pac. J. Math. 111 (1984), 257–282.CrossRefGoogle Scholar
3Dixmier, J.. Dual et quasi-dual d'une algébre de Banach involutive. Trans. Amer. Math. Soc. 104 (1963), 273–283.Google Scholar
4Elliott, G. A.. The diffeomorphism group of the irrational rotation C *-algebra. C.R. Math. Rep. Acad. Sci. Canada VIII (1986), 329–333.Google Scholar
5Hoegh-Krohn, R., Landstad, M. B. and Stormer, E.. Compact ergodic groups of automorphisms. Ann. Math. 114 (1981), 75–86.CrossRefGoogle Scholar
6Jones, V.. Actions of finite groups on the hyperfinite II 1 factor. Mem. Amer. Math. Soc. 237 (1980).Google Scholar
7Jones, V.. Prime actions of compact abelian groups on the hyperfinite type II 1 factor. J. Oper. Theory 9 (1983), 181–186.Google Scholar
8Jones, V. and Takesaki, M.. Actions of compact abelian groups on semifinite injective factors. Ada Math. 153 (1984), 212–258.Google Scholar
9Kadison, R. V. and Ringrose, J. R.. Fundamentals of the Theory of Operator Algebras, Vols I and II (New York: Academic Press, 1986).Google Scholar
10Kirillov, A. A.. Elements of the theory of Representations, Grundlehren der mathematischen Wissenschaften 220 (Berlin, Heidelberg, New York: Springer Verlag, 1976).CrossRefGoogle Scholar
11Kovács, I. and Szücs, J.. Ergodic type theorems in von Neumann algebras. Ada Sci. Math. 27 (1966), 233–246.Google Scholar
12Kumjian, A.. On C*-diagonals. Can. J. Math. 38 (1986), 969–1008.CrossRefGoogle Scholar
13Mackey, G. W.. Les ensembles borelien et des extensions des groups. J. Math. Pures Appl. (9), 36 (1957), 171–178.Google Scholar
14Moore, C. C.. Group extensions and cohomology for locally compact groups. III. Trans. Amer. Math. Soc. 113 (1976), 1–33.CrossRefGoogle Scholar
15Olesen, D., Pedersen, G. K. and Takesaki, M.. Ergodic actions of compact abelian groups. J. Oper. Theory 3 (1980), 237–269.Google Scholar
16Pedersen, G. K.. C*-Algebras and Their Automorphism Groups. London Mathematical Society Monographs, Vol. 14 (London: Academic Press, 1979).Google Scholar
17Renault, J.. A groupoid approach to C *-algebras. Lecture Notes in Mathematics 793 (Berlin, Heidelberg, New York: Springer Verlag, 1980).CrossRefGoogle Scholar
18Rudin, W.. Fourier Analysis on Groups, Interscience Tracts in Pure and Applied Mathematics 12 (New York, London: Interscience, 1962).Google Scholar
19Slawny, J.. On factor representations and the C *-algebra of canonical commutation relations. Comm. Math. Phys. 24 (1972), 151–170.CrossRefGoogle Scholar
20Stormer, E.. Automorphism groups and invariant states. Proc. Symp. Pure Math. 38 (1982), (2).Google Scholar
21Sutherland, C. E. and Takesaki, M.. Actions of discrete amenable groups and groupoids on von Neumann algebras. R.I.M.S. Kyoto University 21 (1985), 1087–1120.CrossRefGoogle Scholar
22Sutherland, C. E. and Takesaki, M.. Actions of discrete amenable groups on injective factors of type III λ, λ ≠ 1 (Preprint, 1987).Google Scholar
23Takesaki, M.. Theory of Operator Algebras (New York, Heidelberg, Berlin: Springer, 1979).CrossRefGoogle Scholar
24Thomsen, K.. Compact abelian prime actions on von Neumann Algebras. Trans. Amer. Math. Soc. (to appear).Google Scholar
25Westman, J.. Cohomology for ergodic groupoids. Trans. Amer. Math. Soc. 146 (1969), 465–471.CrossRefGoogle Scholar