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Transactions of the American Mathematical Society

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Weakly almost periodic functions and almost convergent functions on a group


Author: Ching Chou
Journal: Trans. Amer. Math. Soc. 206 (1975), 175-200
MSC: Primary 43A60; Secondary 43A07
DOI: https://doi.org/10.1090/S0002-9947-1975-0394062-8
MathSciNet review: 0394062
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Abstract: Let $ G$ be a locally compact group, $ UC(G)$ the space of bounded uniformly continuous complex functions on $ G,{C_0}(G)$ the subspace of $ UC(G)$ consisting of functions vanishing at infinity. Let $ W(G)$ be the space of weakly almost periodic functions on $ G$ and $ {W_0}(G)$ the space of functions in $ W(G)$ such that their absolute values have zero invariant mean. If $ G$ is amenable let $ F(G)$ be the space of almost convergent functions in $ UC(G)$ and $ {F_0}(G)$ the space of functions in $ F(G)$ such that their absolute values are almost convergent to zero. The inclusive relations among the above-mentioned spaces are studied. It is shown that if $ G$ is noncompact and satisfies certain conditions, e.g. $ G$ is nilpotent, then each of the quotient Banach spaces $ UC(G)/W(G),{W_0}(G)/{C_0}(G),{F_0}(G)/{W_0}(G)$ contains a linear isometric copy of $ {l^\infty }$. On the other hand, an example of a noncompact group $ G$ is given which satisfies the condition that $ {C_0}(G) = {W_0}(G)$.


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  • [1] R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach, New York, 1970. MR 41 #8562. MR 0263963 (41:8562)
  • [2] C. Chou, On the size of the set of left invariant means on a semigroup, Proc. Amer. Math. Soc. 23 (1969), 199-205. MR 40 #710. MR 0247444 (40:710)
  • [3] -, On topologically invariant means on a locally compact group, Trans. Amer. Math. Soc. 151 (1970), 443-456. MR 42 #4675. MR 0269780 (42:4675)
  • [4] C. Chou and J. P. Duran, Multipliers for the space of almost-convergent functions on a semigroup, Proc. Amer. Math. Soc, 39 (1973), 125-128. MR 0315356 (47:3905)
  • [5] M. M. Day, Semigroups and amenability, (Proc Sympos. Semigroups, Wayne State Univ., Detroit, Michigan, 1968), Academic Press, New York, 1969, pp. 5-53. MR 42 #411. MR 0265502 (42:411)
  • [6] K. deLeeuw and I. Glicksberg, The decomposition of certain group representations, Analyse Math. 15 (1965), 135-192. MR 32 #4211. MR 0186755 (32:4211)
  • [7] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [8] W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217-240. MR 12, 112. MR 0036455 (12:112a)
  • [9] E. E. Granirer, Exposed points of convex sets and weak sequential convergence, Mem. Amer. Math. Soc. No. 123, 1972. MR 0365090 (51:1343)
  • [10] F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand-Reinhold Math. Studies, no. 16, Van Nostrand-Reinhold, New York, 1969. MR 40 #4776. MR 0251549 (40:4776)
  • [11] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. 1: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissen-schaften, Band 115, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
  • [12] K. Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507-558. MR 10, 679. MR 0029911 (10:679a)
  • [13] D. Montgomery and L. Zippin, Topological transformation groups, Interscience, New York, 1955. MR 17, 383. MR 0073104 (17:383b)
  • [14] W. Rudin, Weak almost periodic functions and Fourier-Stieltjes transforms, Duke Math. J. 26 (1959), 215-220. MR 21 #1492. MR 0102705 (21:1492)
  • [15] J. C. S. Wong, Topologically stationary locally compact groups and amenability, Trans. Amer. Math. Soc. 144 (1969), 351-363. MR 40 #2781. MR 0249536 (40:2781)

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DOI: https://doi.org/10.1090/S0002-9947-1975-0394062-8
Article copyright: © Copyright 1975 American Mathematical Society

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