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A Problem of Gelfand on Rings of Operators and Dynamical Systems

Published online by Cambridge University Press:  20 November 2018

Robert R. Kallman
Robert R. Kallman*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts; Yale University, New Haven, Connecticut
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Let G be a separable locally compact group (separable in the sense that the topology of G has a countable base). Let S be a standard Borel space on which G acts on the right such that:

(1) s · g1g2 = (s · g1) · g2;

(2) s · e = s;

(3) (s, g) → s · g is a Borel function from S × G to S.

If μ is a Borel measure on S, let μg be the Borel measure on S defined by μg(E) = μ(E · g).

Let μ be a Borel measure on S which is quasi-invariant under the action of G; i.e., μg and μ are absolutely continuous (g ∈ G). The triple (G, S, μ) is called a dynamical system [11; 8].

Consider the following general problem. Let (G, S, μ) be a dynamical system.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 514 - 517
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Blattner, R. J., On a theorem oJG. W. Mackey, Bull. Amer. Math. Soc. 68 (1962), 585–587.Google Scholar
2. Blattner, R. J., Positive definite measures, Proc. Amer. Math. Soc. 14 (1963), 423–428.Google Scholar
3. Dixmier, J., Algebres quasi-unitaires, Comment. Math. Helv. 26 (1952), 275–322.Google Scholar
4. Dixmier, J., Les algèbres d]'opérateurs dans Vespace hilbertien (Algèbres de von Neumann); Cahiers scientifiques, Fasc. XXV (Gauthier-Villars, Paris, 1957).Google Scholar
5. Gelfand, I. M., Some questions of analysis and differential equations, Amer. Math. Soc. Transi. (2) 26 (1963), 201–219.Google Scholar
6. Kadison, R. V. and Singer, I. M., Some remarks on representations of connected groups, Proc. Nat. Acad. Sci. U.S.A. 88 (1952), 419–423.Google Scholar
7. Kaplansky, I., Group algebras in the large, Töhoku Math. J. (2) 8 (1951), 249–256.Google Scholar
8. Kirillov, A. A., Dynamical systems, factors, and representations of groups, Russian Math. Surveys, No. 5, 22 (1967), 63–75.Google Scholar
9. Mackey, G. W., Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139.Google Scholar
10. Mackey, G. W., Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311.Google Scholar
11. Mackey, G. W., Infinite-dimensional group representations, Bull. Amer. Math. Soc. 69 (1963), 628–686.Google Scholar
12. Murray, F. J. and Neumann, J. von, On rings of operators, Ann. of Math. (2) 87 (1936), 116–229.Google Scholar
13. Murray, F. J. and Neumann, J. von, On rings of operators. IV, Ann. of Math. (2) 44 (1943), 716–808.Google Scholar
14. von Neumann, J., On rings of operators. III, Ann. of Math. (2) 41 (1940), 94–161.Google Scholar