Weakly almost periodic elements in of a locally compact group
Authors:
Anthony To Ming Lau and James C. S. Wong
Journal:
Proc. Amer. Math. Soc. 107 (1989), 10311036
MSC:
Primary 43A15; Secondary 22D25, 43A07
MathSciNet review:
991701
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Abstract 
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Abstract: Let be a locally compact abelian group with dual group acting on by pointwise multiplication. We show that if contains a nonzero element such that is relatively compact in the weak (or norm) topology of , then is discrete. In this case is relatively compact in the weak or norm topology of if and only if vanishes at infinity. A related result when acts on the von Neumann algebra is also determined.
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 R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach, New York, 1970. MR 0263963 (41:8562)
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 P. Civin and B. Yood, The second conjugate of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847870. MR 0143056 (26:622)
 [3]
 H. W. Davis, An elementary proof that Haar measurable almost periodic functions are continuous, Pacific J. Math. 21 (1967), 241248. 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), 309325. 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), 181236. MR 0228628 (37:4208)
 [7]
 R. R. Goldberg and A. B. Simon, Characterization of some classes of measures, Acta Sci. Math. (Szeged) 27 (1966), 157161. MR 0209772 (35:668)
 [8]
 C. C. Graham, A. T. Lau, and M. Leinert, Continuity of translation in the dual of and related spaces, Trans. Amer. Math. Soc. (to appear).
 [9]
 E. Hewitt and K. Ross, Abstract harmonic analysis I, SpringerVerlag New York, 1963.
 [10]
 G. A Hively, The representation of normcontinuous multipliers on spaces, Trans. Amer. Math. Soc. 184 (1973), 343353. 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. 6397. MR 0212122 (35:2997)
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 A. T. Lau, Closed convex invariant subsets of Trans. Amer. Math. Soc. 232 (1977), 131142. MR 0477604 (57:17122)
 [13]
 , The second conjugate algebra of the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 267 (1981), 5363. MR 621972 (83e:43009)
 [14]
 A. T. Lau and V. Losert, Weak*closed complemented invariant subspaces of and amenable locally compact groups, Pacific J. Math. 123 (1986), 149159. MR 834144 (87g:43001)
 [15]
 A. Ulger, Continuity of weakly almost periodic functional on , Quart. J. Math. Oxford 37 (1986), 495497. MR 868624 (88b:43005)
 [16]
 N. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. (2) 24 (1973), 5962. MR 0320756 (47:9290)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198909917010
PII:
S 00029939(1989)09917010
Article copyright:
© Copyright 1989
American Mathematical Society
