Funny Rank One and the Approximate Transitivity for Induced Actions

 Let H be a closed normal subgroup of a locally compact separable group G, and suppose that H acts ergodically on a Lebesgue space. If both the given H-action and the natural G-action on G/H have funny rank one, then the induced G-action has it also. For G solvable, the second condition is always true. A similar (but easier) theorem is also true for approximately transitive actions, even without the normality assumption on H. Received 23 February 1998     

Funny Rank One and the Approximate Transitivity for Induced Actions

 Let H be a closed normal subgroup of a locally compact separable group G, and suppose that H acts ergodically on a Lebesgue space. If both the given H-action and the natural G-action on G/H have funny rank one, then the induced G-action has it also. For G solvable, the second condition is always true. A similar (but easier) theorem is also true for approximately transitive actions, even without the normality assumption on H.     

Product Cocycles and the Approximate Transitivity

Some criteria of the approximate transitivity in the terms of Mackey actions and product cocycles are proved. The Mackey action constructed by an amenable type II or III transformation group G and a 1-cocycle × , where is the Radon–Nikodym cocycle while is an arbitrary 1-cocycle with values in a locally compact separable group A, is approximately transitive (AT) if and only if the pair (G,(, )) is weakly equivalent to a product odometer supplied with a product cocycle. Besides, in the case when the given AT action from the very beginning was a range of a type II action and a nontransient cocycle, then this cocycle turns out to be cohomologous to a -product cocycle. An example is constructed that shows that it is necessary to consider the double Mackey actions since they can not be reduced to the single ones.     

Ergodic actions of Abelian groups with a discrete spectrum,and approximate transitivity

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 51, pp. 117–122, 1989.     

Properties of a jointly ergodic action of the direct product of two groups

An ergodic action a of the direct product of and , not isomorphic to a product of actions of and G, is constructed, such that the actions of and G separately are not ergodic. The actions of on its ergodic components are metrically isomorphic if and only if these components are taken into one another by the action of G. Finally, the centralizerC (×G) is such thatC (×G)/(×G)₂.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 684–688, May, 1991.