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Proceedings of the American Mathematical Society

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Translation-invariant linear forms on $ L\sb p(G)$


Author: Joseph Rosenblatt
Journal: Proc. Amer. Math. Soc. 94 (1985), 226-228
MSC: Primary 43A15
DOI: https://doi.org/10.1090/S0002-9939-1985-0784168-5
MathSciNet review: 784168
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Abstract: Let $ G$ be a compact group such that the identity representation of $ G$ is not contained in the regular representation on $ L_2^0(G,{\lambda _G})$ of $ G$ with the discrete topology. Then any left translation invariant linear form on $ {L_p}(G),1 < p < \infty $, is continuous and must be a constant times the Haar integral. This shows that many classical matrix groups $ G$ admit only continuous left translation invariant linear forms on $ {L_p}(G),1 < p < \infty $.


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

  • [1] J. Bourgain, Translation invariant forms on $ {L^p}(G)(1 < p < \infty )$, preprint.
  • [2] C. Chou, A. Lau and J. Rosenblatt, Approximation of compact operators by sums of translations, Illinois J. Math. (to appear). MR 784527 (86g:22007)
  • [3] V. G. Drinfeld, Solution of the Banach-Ruziewicz problem on $ {S^2}$ and $ {S^3}$, J. Funct. Anal. (to appear).
  • [4] G. A. Margulis, Some remarks on invariant means, Monatsh. Math. 90 (1980), 233-235. MR 596890 (82b:28034)
  • [5] G. H. Meisters, Some discontinuous translation-invariant forms, J. Funct. Anal. 12 (1973), 199-210. MR 0346418 (49:11143)
  • [6] G. H. Meisters and W. M. Schmidt, Translation-invariant linear forms on $ {L^2}(G)$ for compact abelian groups G. J. Funct. Anal. 11 (1972), 407-424. MR 0346417 (49:11142)
  • [7] J. Rosenblatt, Uniqueness of invariant means for measure-preserving transformations, Trans. Amer. Math. Soc. 265 (1981), 623-636. MR 610970 (83a:28026)
  • [8] -, Ergodic group actions, preprint (to appear).

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DOI: https://doi.org/10.1090/S0002-9939-1985-0784168-5
Article copyright: © Copyright 1985 American Mathematical Society

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