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Continuity of automorphic representations

Published online by Cambridge University Press:  24 October 2008

J. Moffat
J. Moffat
Affiliation:
University of Newcastle upon Tyne
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Extract

Let ℛ be a von Neumann algebra, with predual ℛ*, acting on a Hilbert space ℋ; G a locally compact group with left Haar measure m, and α a representation of G on aut (ℛ), the group of all *-automorphisms of ℛ, i.e. α is a group homomorphism from G to aut (ℛ). We shall show that if ℋ is separable, then very weak measurability assumptions on the representation α produce strong continuity properties. This will be used to obtain results on the extension of representations from a C*-algebra to its weak closure, giving a much simpler proof of a result of Aarnes ((1), theorem 8, p. 31), and on continuity of tensor products of representations. The main result was suggested by the analogous theory concerning unitary representations of locally compact groups, and its proof employs ideas frequently used in that context. (See, for example, (5), theorem 22.20 (b), p. 347.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 3 , November 1973 , pp. 461 - 465
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Aarnes, J. F.Continuity of group representations with applications to C*-algebras. J. Func. Anal. 5 (1970), 14–36.CrossRefGoogle Scholar
(2)Dell'antonko, G.On some groups of automorphisms of physical observables. Comm. Math. Phys. 2 (1966), 384–397.CrossRefGoogle Scholar
(3)Dixmier, J.Les algébres d'opérateurs dans l'espace Hilbertien, 2nd edn. (Gauthier-Villars, Paris, 1969).Google Scholar
(4)Dixmier, J.Les C*-algébres et leurs représentations (Paris, Gauthier-Villars, 1964).Google Scholar
(5)Hewitt, E. and Ross, K. A.Abstract harmonic analysis, vol. I (Berlin, Gottingen, Heidelberg, 1963, Springer-Verlag).Google Scholar
(6)Hille, E. and Phillips, R. S.Functional analysis and semigroups. Amer. Math. Soc., Colloquium Publications, vol. XXXI.Google Scholar