A Problem of Gelfand on Rings of Operators and Dynamical Systems

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.