Remarks on the Imprimitivity Theorem for nonlocally compact Polish groups

In tills paper we analyze the possibility of estabishing a Theorem of Impri mitivity in the case of nonlocally compact Polish groups. We Drove that sys tems of imprimitivity for a Polish group G based on a locally compact homog eneous G-space M = G/H equipped with a quasi-invariant probability measure mu, are in one-to-one correspondence with elements of the space H-1(G, F-mu (M, U(K))) of the first cohomology of the group G of equivalence classes of continuous cocycles. As a corollary, we have the complete Imprimitivity Th eorem H-1(G, F-mu(M, U(K))) similar to Hom(H, U(K)) in the case of discrete countable homogeneous G-spaces equipped with a quasi-invariant measure. Fi nally, we outline the possibility of estabishing the complete Imprimitivity Theorem for particular classes of Polish groups. These examples cover the case of (separable) Frechet spaces, for which it is shown that the complete Imprimitivity Theorem holds as well.