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Adapted sets of measures and invariant functionals on $ L\sp p(G)$


Author: Rodney Nillsen
Journal: Trans. Amer. Math. Soc. 325 (1991), 345-362
MSC: Primary 43A15
DOI: https://doi.org/10.1090/S0002-9947-1991-1018576-1
MathSciNet review: 1018576
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Abstract: Let $ G$ be a locally compact group. If $ G$ is compact, let $ L_0^p(G)$ denote the functions in $ {L^p}(G)$ having zero Haar integral. Let $ {M^1}(G)$ denote the probability measures on $ G$ and let $ {\mathcal{P}^1}(G) = {M^1}(G) \cap {L^1}(G)$. If $ S \subseteq {M^1}(G)$, let $ \Delta ({L^p}(G),S)$ denote the subspace of $ {L^p}(G)$ generated by functions of the form $ f - \mu\ast f$, $ f \in {L^p}(G)$, $ \mu \in S$. If $ G$ is compact, $ \Delta ({L^p}(G),S) \subseteq L_0^p(G)$ . When $ G$ is compact, conditions are given on $ S$ which ensure that for some finite subset $ F$ of $ S$, $ \Delta ({L^p}(G),F) = L_0^p(G)$ for all $ 1 < p < \infty $. The finite subset $ F$ will then have the property that every $ F$-invariant linear functional on $ {L^p}(G)$ is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if $ 1 \leq p \leq \infty $, conditions are given upon $ G$, and upon subsets $ S$ of $ {M^1}(G)$ whose elements satisfy certain growth conditions, which ensure that $ {L^p}(G)$ has discontinuous, $ S$-invariant linear functionals. The results are applied to show that for $ 1 \leq p \leq \infty$, $ {L^p}(\mathbb{R})$ has an infinite, independent family of discontinuous translation invariant functionals which are not $ {\mathcal{P}^1}(\mathbb{R})$-invariant.


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

  • [1] J. Bourgain, Translation invariant forms on $ {L^p}(G)\;(1 < p < \infty)$, Ann. Inst. Fourier (Grenoble) 36 (1986), 97-104. MR 840715 (87h:43003)
  • [2] E. Granirer, Criteria for compactness and for discreteness of locally compact amenable groups, Proc. Amer. Math. Soc. 40 (1973), 615-624. MR 0340962 (49:5712)
  • [3] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. II, Springer-Verlag, 1970. MR 551496 (81k:43001)
  • [4] B. E. Johnson, A proof of the translation invariant form conjecture for $ {L^2}(G)$, Bull. Sci. Math. (2) 197 (1983), 301-310. MR 719270 (85k:43003)
  • [5] Y. Katznelson, An introduction to harmonic analysis, Wiley, New York, 1968. MR 0248482 (40:1734)
  • [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] G. H. Meisters, Some discontinuous translation-invariant linear forms, J. Funct. Anal. 12 (1973), 199-210. MR 0346418 (49:11143)
  • [8] -, Some problems and results on translation-invariant linear forms, Radical Banach Algebras and Automatic Continuity, (J. M. Bachar and W. G. Bade et al., Eds.), Lecture Notes in Math., vol. 975, Springer-Verlag, 1983. MR 697605 (85c:46035)
  • [9] R. Nillsen, Group actions and direct sum decompositions of $ {L^p}$ spaces, Proc. Amer. Math. Soc. 106 (1989), 975-985. MR 972237 (89m:43007)
  • [10] J.-P. Pier, Amenable locally compact groups, Wiley, New York, 1984. MR 767264 (86a:43001)
  • [11] J. Rosenblatt, Invariant means on the continuous bounded functions, Trans. Amer. Math. Soc. 236 (1978), 315-324. MR 0473714 (57:13377)
  • [12] -, Translation invariant linear forms on $ {L^p}(G)$, Proc. Amer. Math. Soc. 94 (1985), 226-228. MR 784168 (86e:43006)
  • [13] -, Ergodic group actions, Arch. Math. 47 (1986), 263-269. MR 861875 (88d:28024)
  • [14] W. Rudin, Invariant means on $ {L^\infty }$, Studia Math. 44 (1972), 219-227. MR 0304975 (46:4105)
  • [15] G. A. Willis, Continuity of translation invariant linear functionals on $ {C_0}(G)$ for certain locally compact groups $ G$, Monatsh. Math. 105 (1988), 161-164. MR 930434 (89c:43004)
  • [16] G. Woodward, Translation-invariant linear forms on $ {C_0}(G)$, $ C(G)$, $ {L^p}(G)$ for noncompact groups, J. Funct. Anal. 12 (1973), 205-220. MR 0344801 (49:9540)

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DOI: https://doi.org/10.1090/S0002-9947-1991-1018576-1
Article copyright: © Copyright 1991 American Mathematical Society

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