Weakly almost periodic elements in $L_ \infty (G)$ of a locally compact group
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- by Anthony To Ming Lau and James C. S. Wong PDF
- Proc. Amer. Math. Soc. 107 (1989), 1031-1036 Request permission
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
- R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963
- Paul Civin and Bertram Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847–870. MR 143056, DOI 10.2140/pjm.1961.11.847
- Henry W. Davis, An elementary proof that Haar measurable almost periodic functions are continuous, Pacific J. Math. 21 (1967), 241–248. MR 212121, DOI 10.2140/pjm.1967.21.241
- J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 3-4, 309–325. MR 559675, DOI 10.1017/S0308210500017170 N. Dunford and J. T. Schwartz, Linear operators I, Interscience, 1958.
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628, DOI 10.24033/bsmf.1607
- R. R. Goldberg and A. B. Simon, Characterization of some classes of measures, Acta Sci. Math. (Szeged) 27 (1966), 157–161. MR 209772 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). E. Hewitt and K. Ross, Abstract harmonic analysis I, Springer-Verlag New York, 1963.
- Gregory A. Hively, The representation of norm-continuous multipliers on $L^{\infty }$-spaces, Trans. Amer. Math. Soc. 184 (1973), 343–353. MR 346425, DOI 10.1090/S0002-9947-1973-0346425-2
- A. Ionescu Tulcea and C. Ionescu Tulcea, On the existence of a lifting commuting with the left translations of an arbitrary locally compact group, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 63–97. MR 0212122
- Anthony To Ming Lau, Closed convex invariant subsets of $L_{p}(G)$, Trans. Amer. Math. Soc. 232 (1977), 131–142. MR 477604, DOI 10.1090/S0002-9947-1977-0477604-5
- Anthony To Ming Lau, The second conjugate algebra of the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 267 (1981), no. 1, 53–63. MR 621972, DOI 10.1090/S0002-9947-1981-0621972-9
- Anthony To Ming Lau and Viktor Losert, Weak$^\ast$-closed complemented invariant subspaces of $L_\infty (G)$ and amenable locally compact groups, Pacific J. Math. 123 (1986), no. 1, 149–159. MR 834144, DOI 10.2140/pjm.1986.123.149
- Ali Ülger, Continuity of weakly almost periodic functionals on $L^1(G)$, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 495–497. MR 868624, DOI 10.1093/qmath/37.4.495
- N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. (2) 24 (1973), 59–62. MR 320756, DOI 10.1093/qmath/24.1.59
Additional Information
- © Copyright 1989 American Mathematical Society
- 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