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Application of the group algebra of the problem of the tail \(\sigma\)- algebra of a random walk on a group and the problem of ergodicity of a skew-product action. (English. Russian original) Zbl 0659.60022
Math. USSR, Izv. 31, No. 1, 209-222 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 4, 893-907 (1987).
Simple conditions of triviality of the tail \(\sigma\)-algebra of (nonhomogeneous) random walks, generated by the products of independent random elements of a group from some class are obtained. This class includes Abel, compact, nilpotent, some solvable groups and the groups of motions of Euclidean space.
It is also proved that a skew product action generated by a cocycle of a special type is ergodic, iff the tail \(\sigma\)-algebra of a (nonhomogeneous) random walk constructed by this cocycle is trivial. In particular, a simple proof of the result of V. Y. Golodets and S. D. Sineljshtshikov [The existence and uniqueness of cocycles of an ergodic action with dense actions in amenable groups. Preprint, Fiz.- Tekhn. Inst. Nizkikh Temperatur, Akad. Nauk Ukrain. SSR (1983)] on the existence of the cocycle of ergodic action for amenable groups is given.
The author’s corrections: In definition 1 the condition \(K=K^{-1}\) is omitted. The following is to be added to theorem 5: “the absolute continuous component of the measure \(\tau^{*n}\) is positive on some essentially generic set”.
Reviewer: A.Grincevičius

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
22D40 Ergodic theory on groups
22D15 Group algebras of locally compact groups
43A20 \(L^1\)-algebras on groups, semigroups, etc.
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
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