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A note on cocycles of unitary representations


Authors: W. Parry and K. Schmidt
Journal: Proc. Amer. Math. Soc. 55 (1976), 185-190
DOI: https://doi.org/10.1090/S0002-9939-1976-0393336-0
MathSciNet review: 0393336
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Abstract | References | Additional Information

Abstract: Given a unitary representation $ U$ of a locally compact abelian group $ G$, we investigate the relationship between two cocycles $ {a_1},{a_2}:V{a_1} = {a_2} + b$ for some unitary operator $ V$ commuting with $ U$ and some coboundary $ b$. A necessary and sufficient condition is given in terms of canonical-finite measures defined on $ G - 1$. These results are applied to the representation of $ Z$ defined by the shift of a stationary Markov chain.


References [Enhancements On Off] (What's this?)

  • [1] R. Fellgett and W. Parry, Endomorphisms of a Lebesgue space. II, Bull. London Math. Soc. 7 (1975), 151-158. MR 0382590 (52:3473a)
  • [2] F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Math. Studies, no. 16, Van Nostrand Reinhold, New York, 1969. MR 40 #4776. MR 0251549 (40:4776)
  • [3] K. R. Parthasarathy and K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, Lecture Notes in Math., vol. 272, Springer-Verlag, Berlin and New York, 1972. MR 0622034 (58:29849)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0393336-0
Keywords: Unitary representations, cocycles, coboundaries
Article copyright: © Copyright 1976 American Mathematical Society

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