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On groups of automorphisms of operator algebras

Published online by Cambridge University Press:  24 October 2008

J. Moffat
J. Moffat
Affiliation:
Ministry of Defence, London
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Extract

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R, say α(g) = ad Wg with Wg ∈ U(R). Then there is a weakly continuous unitary representation of G, by unitaries in R, implementing α if and only if the Wg's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now Gis a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R. In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G-invariant state on R. This generalizes a result of Hajian and Kakutani on invariant measures (2).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 2 , March 1977 , pp. 237 - 243
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Dixmier, J.Les Algèbres d'operateurs dans l'espace Hilbertien (Algèbres de von Neumann), 2nd ed., Cahiers Scientifiques, Fasc. XXV (Paris, Gauthier-Villars, 1969).Google Scholar
(2)Hajian, A. and Kakutani, S.Weakly wandering sets and invariant measures. Trans. Amer. Math Soc. 110 (1964), 136–151.CrossRefGoogle Scholar
(3)Kallman, R. R.Groups of inner automorphisms of von Neumann Algebras. J. Functional Analysis 7 (1971), 43–60.CrossRefGoogle Scholar
(4)Kallman, R. R.Spatially induced groups of automorphisme of certain von Neumann algebras. Trans. Amer. Math. Soc. 156 (1971), 505–515.CrossRefGoogle Scholar
(5)Mackey, G. W.Borel structure in groups and their duals. Trans. Amer. Math. Soc. 85 (1957), 134–165.CrossRefGoogle Scholar
(6)Mackey, G. W.Ensembles Boreliens et extensions des groupes. J. Math. Pure Appl. 36 (1957), 171–178.Google Scholar
(7)Moffat, J.Connected topological groups acting on von Neumann Algebras. J. London Math. Soc. 9 (1975), 411–417.CrossRefGoogle Scholar
(8)Takesaki, M.On the conjugate space of operator algebra. Tôhoku Math. J. 10 (1958), 194–203.CrossRefGoogle Scholar
(9)Takesaki, M.On the singularity of a positive linear functional on operator algebra. Proc. Japan Acad. 25 (1959), 365–366.Google Scholar