Some properties of \(LUC({\mathcal X},{\mathcal G})^*\) as a banach left \(LUC({\mathcal G})^*\)-module

Research Article

Abstract

Associated with a locally compact group \(\mathcal G\) and a \(\mathcal G\)-space \(\mathcal X\) there is a Banach subspace \(LUC({\mathcal X},{\mathcal G})\) of \(C_b({\mathcal X})\), which has been introduced and studied by Chu and Lau (Math Z 268:649–673, 2011). In this paper, we study some properties of the first dual space of \(LUC({\mathcal X},{\mathcal G})\). In particular, we introduce a left action of \(LUC({\mathcal G})^*\) on \(LUC({\mathcal X},{\mathcal G})^*\) to make it a Banach left module and then we investigate the Banach subalgebra \({{\mathfrak {Z}}({\mathcal X},{\mathcal G})}\) of \(LUC({\mathcal G})^*\), as the topological centre related to this module action, which contains \(M({\mathcal G})\) as a closed subalgebra. Also, we show that the faithfulness of this module action is related to the properties of the action of \(\mathcal G\) on \(\mathcal X\) and we prove an analogue of the main result of Lau (Math Proc Cambridge Philos Soc 99:273–283, 1986) for \({\mathcal G}\)-spaces. Sufficient and/or necessary conditions for the equality \({{\mathfrak {Z}}({\mathcal X},{\mathcal G})}=M({\mathcal G})\) or \(LUC({\mathcal G})^*\) are given. Finally, we apply our results to some special cases of \(\mathcal G\) and \(\mathcal X\) for obtaining various examples whose topological centres \({{\mathfrak {Z}}({\mathcal X},{\mathcal G})}\) are \(M({\mathcal G})\), \(LUC({\mathcal G})^*\) or neither of them.

Keywords

\(\mathcal G\)-space Left uniformly continuous function Complex radon measure Measure algebra Left module action Topological centre 

Notes

Acknowledgements

We are indebted to the referee for the careful reading of the paper, the detailed criticism and their numerous remarks and suggestions which greatly improved the presentation of the paper. In particular, we corrected our mis-statements in Theorem 3.1 by the ones suggested by him/her.

References

  1. 1.
    Abel, M., Arhippainen, J., Kauppi, J.: Stone-Weierstrass type theorems for algebras containing continuous unbounded functions. Sci. Math. Jpn. 71, 1–10 (2010)MathSciNetMATHGoogle Scholar
  2. 2.
    Arens, R.: The adjoint of a bilinear operation. Proc. Am. Math. Soc. 2, 839–848 (1951)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chan, P.K.: Topological centers of module actions induced by unitary representations. J. Funct. Anal. 259, 2193–2214 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chu, C.H., Leung, C.W.: Harmonic functions on homogeneous spaces. Monatsh. Math. 128, 227–235 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chu, C.H., Lau, A.T.-M.: Harmonic functions on topological groups and symmetric spaces. Math. Z. 268, 649–673 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dales, H.G., Rodrigues-Palacios, A., Velasco, M.V.: The second transpose of a derivation. J. Lond. Math. Soc. 64, 707–721 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Daws, M.: Arens-regularity of algebras arising from tensor norms. N. Y. J. Math. 13, 215–270 (2007)MathSciNetMATHGoogle Scholar
  8. 8.
    Eshaghi, M.: Gordji and M Filali, Arens regularity of module actions. Studia Math. 181, 237–254 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Eshaghi, M.: Gordji and M Filali, Weak amenability of the second dual of a Banach algebra. Studia Math. 182, 205–213 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Filali, M., Salmi, P.: One-sided ideals and right cancellation in the second dual of the group algebra and similar algebras. J. London Math. Soc. 75, 35–46 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995)MATHGoogle Scholar
  12. 12.
    Ghahramani, F.: Weighted group algebra as an ideal in its second dual space. Proc. Am. Math. Soc. 90, 71–76 (1984)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ghahramani, F., Lau, A.T.-M., Losert, V.: Isometric isomorphisms between Banach algebras related to locally compact groups. Trans. Am. Math. Soc. 321, 273–283 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Grosser, M., Losert, V.: The norm-strict bidual of a Banach algebra and the dual of \(C_u(\cal{G})^*\). Manuscripta Math. 45, 127–146 (1984)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lashkarizadeh, M.: Bami, on the multipliers of the pair \((M_a(S), L^\infty (S;M_a(S)))\) of a foundation semigroup \(S\). Math. Nachr. 181, 73–80 (1996)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lau, A.T.-M.: Uniformly continuous functionals on the Fourier algebra of any locally compact group. Trans. Am. Math. Soc. 251, 39–59 (1979)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lau, A.T.-M.: The second conjugate algebra of the Fourier algebra of a locally compact group. Trans. Am. Math. Soc. 267, 53–63 (1981)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lau, A.T.-M.: Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups. Math. Proc. Cambridge Philos. Soc. 99, 273–283 (1986)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lau, A.T.-M., Pym, J.: The topological centre of a compactification of a locally compact group. Math. Z. 219, 567–579 (1995)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mohammadzadeh, S., Vishki, H.R.E.: Arens regularity of module actions and the second adjoint of a derivation. Bull. Aust. Math. Soc. 77, 465–476 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Veech, W.A.: Topological dynamics. Bull. Am. Math. Soc. 83, 775–830 (1977)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zappa, A.: The centre of the convolution algebra \(C_u(\cal{G})^*\). Rend. Sem. Mat. Univ. Padova. 52, 71–83 (1975)MATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsYazd UniversityYazdIran
  2. 2.Department of Mathematics, School of Mathematics and Computer ScienceDamghan UniversityDamghanIran

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