Ergodic actions of abelian groups and properties of their joint actions
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The concept of joint action is introduced. The joint action of two ergodynamic, measure preserving actions of an arbitrary locally compact Abelian group on Lebesgue spaces is investigated. It is proved that it is ergodic with a pure point spectrum. The spectrum of the joint action is computed. For the proof the individual Emerson—Greenleaf theorem for amenable groups has been used. The results generalize the known theorem of T. Hamachi and M. Osikawa for the actions of groups of real numbers.
KeywordsReal Number Abelian Group Joint Action Measure Preserve Lebesgue Space
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- 1.T. Hamachi, Y. Oka, and M. Osikawa, “Flows associated with ergodic nonsingular transformation groups,” Publ. Res. Inst. Math. Sci., Kyoto Univ.,11, No. 1, 31–50 (1975).Google Scholar
- 2.T. Hamachi and M. Osikawa, “Ergodic groups of automorphisms and Krieger's theorems,” Sem. Math. Sci.,3, Keio University, Yokohama (1981).Google Scholar
- 3.V. A. Rokhlin, “On the fundamental concepts of measure theory,” Mat. Sb.,25 (67), 107–150 (1949).Google Scholar
- 4.E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer, Berlin (1963).Google Scholar
- 5.O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vols. 1 and 2, Springer, New York (1979 and 1981).Google Scholar
- 6.W. R. Emerson, “The pointwise ergodic theorem for amenable groups,” Am. J. Math.,96, No. 3, 472–487 (1974).Google Scholar