FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 8, PAGES 193-212

On isomorphity of measure-preserving $$Z²-actions that have isomorphic Cartesian powers

A. E. Troitskaya

Abstract

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Assume that $$D and $$P are representations of the group $$Z² by operators on the space $L$₂(X, m) that are induced by measure-preserving automorphisms, and for some $d$, the representations $$D^Äd and $$P^Äd are conjugate to each other, $$D (Z² \ (0,0)) consists of weakly mixing operators, and there is a weak limit (over some subsequence in $$Z² of operators from $$D (Z²)) which is equal to a nontrivial, convex linear combination of elements of $$D (Z²) and of the projection onto constant functions. We prove that in this case, $$D and $$P are also conjugate to each other.

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Last modified: June 13, 2008