ON ISOMORPHITY OF MEASURE-PRESERVING Z~2-ACTIONS THAT HAVE ISOMORPHIC CARTESIAN POWERS

摘要

Assume that Δ and ∏ are representations of the group Z~2 by operators on the spaee L_2(X, μ) that are induced by measure-preserving automorphisms, and for some d, the representations Δ~(☉d) and ∏~(☉d) are conjugate to each other, Δ(Z~2 (0,0)) consists of weakly mixing operators, and there is a weak limit (over some subsequence in Z~2 of operators from Δ(Z~2)) which is equal to a nontrivial, convex linear combination of elements of Δ(Z~2) and of the projection onto constant functions. We prove that in this case, Δ and Π are also conjugate to each other.