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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

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