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Weakly almost periodic elements in $ L\sb \infty(G)$ of a locally compact group


Authors: Anthony To Ming Lau and James C. S. Wong
Journal: Proc. Amer. Math. Soc. 107 (1989), 1031-1036
MSC: Primary 43A15; Secondary 22D25, 43A07
DOI: https://doi.org/10.1090/S0002-9939-1989-0991701-0
MathSciNet review: 991701
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Abstract: Let $ G$ be a locally compact abelian group with dual group $ G$ acting on $ {L_\infty }(G)$ by pointwise multiplication. We show that if $ {L_\infty }(G)$ contains a nonzero element $ f$ such that $ O(f) = \left\{ {x \cdot f:\chi \in \hat G} \right\}$ is relatively compact in the weak (or norm) topology of $ {L_\infty }(G)$, then $ G$ is discrete. In this case $ O(f)$ is relatively compact in the weak or norm topology of $ {L_\infty }(G)$ if and only if $ f$ vanishes at infinity. A related result when $ G$ acts on the von Neumann algebra $ VN(G)$ is also determined.


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

  • [1] R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach, New York, 1970. MR 0263963 (41:8562)
  • [2] P. Civin and B. Yood, The second conjugate of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870. MR 0143056 (26:622)
  • [3] H. W. Davis, An elementary proof that Haar measurable almost periodic functions are continuous, Pacific J. Math. 21 (1967), 241-248. MR 0212121 (35:2996)
  • [4] J. Duncan and S. A. Hosseiniun, The second dual of a Banach algebra, Proc. Roy Soc. Edinburgh Sect. A 84 (1979), 309-325. MR 559675 (81f:46057)
  • [5] N. Dunford and J. T. Schwartz, Linear operators I, Interscience, 1958.
  • [6] P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. MR 0228628 (37:4208)
  • [7] R. R. Goldberg and A. B. Simon, Characterization of some classes of measures, Acta Sci. Math. (Szeged) 27 (1966), 157-161. MR 0209772 (35:668)
  • [8] C. C. Graham, A. T. Lau, and M. Leinert, Continuity of translation in the dual of $ {L_\infty }(G)$ and related spaces, Trans. Amer. Math. Soc. (to appear).
  • [9] E. Hewitt and K. Ross, Abstract harmonic analysis I, Springer-Verlag New York, 1963.
  • [10] G. A Hively, The representation of norm-continuous multipliers on $ {L^1}$-spaces, Trans. Amer. Math. Soc. 184 (1973), 343-353. MR 0346425 (49:11150)
  • [11] A. Ionescu Tulcea and C. Ionescu Tulcea, On the existence of a lifting commuting with the left translations of a locally compact group, Proc. Fifth Berkeley Sympos. Math. Statist. and Probab. (Berkeley, California, 1965/66), Vol. II: Contributions to Probability Theory, Part I, Univ. of California Press, Berkeley, California, 1967, pp. 63-97. MR 0212122 (35:2997)
  • [12] A. T. Lau, Closed convex invariant subsets of $ {L_p}(G)$ Trans. Amer. Math. Soc. 232 (1977), 131-142. MR 0477604 (57:17122)
  • [13] -, The second conjugate algebra of the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 267 (1981), 53-63. MR 621972 (83e:43009)
  • [14] A. T. Lau and V. Losert, Weak*-closed complemented invariant subspaces of $ {L_\infty }(G)$ and amenable locally compact groups, Pacific J. Math. 123 (1986), 149-159. MR 834144 (87g:43001)
  • [15] A. Ulger, Continuity of weakly almost periodic functional on $ {L_1}(G)$, Quart. J. Math. Oxford 37 (1986), 495-497. MR 868624 (88b:43005)
  • [16] N. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. (2) 24 (1973), 59-62. MR 0320756 (47:9290)

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DOI: https://doi.org/10.1090/S0002-9939-1989-0991701-0
Article copyright: © Copyright 1989 American Mathematical Society

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