Journal of Fourier Analysis and Applications

, Volume 22, Issue 6, pp 1225–1234 | Cite as

The Bochner–Schoenberg–Eberlein Property for Totally Ordered Semigroup Algebras

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Abstract

The concepts of BSE property and BSE algebras were introduced and studied by Takahasi and Hatori in 1990 and later by Kaniuth and Ülger. This abbreviation refers to a famous theorem proved by Bochner and Schoenberg for \(L^1({\mathbb {R}})\), where \({\mathbb {R}}\) is the additive group of real numbers, and by Eberlein for \(L^1(G)\) of a locally compact abelian group G. In this paper we investigate this property for the Banach algebra \(L^p(S,\mu )\;(1\le p<\infty )\) where S is a compact totally ordered semigroup with multiplication \(xy=\max \{x,y\}\) and \(\mu \) is a regular bounded continuous measure on S. As an application, we have shown that \(L^1(S,\mu )\) is not an ideal in its second dual.

Keywords

BSE algebra Totally ordered semigroup Cantor function 

Mathematics Subject Classification

Primary 46Jxx Secondary 22A20 

Notes

Acknowledgments

The authors would like to thank the referees of the paper for the invaluable comments and suggestions which serve to improve the paper. The first author’s research was supported in part by a grant from IAU, Isfahan branch and IPM (No. 93470066). The second author acknowledge that this research was supported by the Center of Excellence for Mathematics and the office of Graduate Studies of the University of Isfahan.

References

  1. 1.
    Baker, J.W., Pym, J.S., Vasudea, H.L.: Totally ordered measure spaces and their \(L^p\) algebras. Mathematika 29, 42–54 (1982)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bochner, S.: A theorem on Fourier–Stieltjes integrals. Bull. Am. Math. Soc. 40, 271–276 (1934)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dovgoshey, O., Martio, O., Ryazanov, V., Vuorinen, M.: The Cantor function. Expo. Math. 24, 1–37 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dunford, N., Schwartz, J.T.: Linear operators, Part I. Wiley, New York (1958)MATHGoogle Scholar
  5. 5.
    Eberlein, W.F.: Characterizations of Fourier–Stieltjes transforms. Duke Math. J. 22, 465–468 (1955)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gorin, E.A., Kukushkin, B.N.: Integrals related to the Cantor function. St. Petersb. Math. J. 15, 449–468 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kamali, Z., Lashkarizadeh Bami, M.: The Bochner–Schoenberg–Eberlein property for \(L^1({\mathbb{R}}^+)\). J. Fourier Anal. Appl. 20, 225–233 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kaniuth, E., Ülger, A.: The Bochner–Schoenberg–Eberlein property for commutative Banach algebras, especially Fourier and Fourier–Stieltjes algebras. Trans. Am. Math. Soc. 362, 4331–4356 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Larsen, R.: An Introduction to the Theory of Multipliers. Springer, New York (1971)CrossRefMATHGoogle Scholar
  10. 10.
    Newman, S.E.: Measure algebras and functions of bounded variation on idempotent semigroups. Trans. Am. Math. Soc. 163, 189–205 (1972)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Royden, H.L.: Real Analysis, 2nd edn. Coller Macmilan International Editions, New York (1968)MATHGoogle Scholar
  12. 12.
    Rudin, W.: Fourier Analysis on Groups. Wiley Interscience, New York (1984)MATHGoogle Scholar
  13. 13.
    Sapounakis, A.: Properties of measures on topological spaces, Thesis. University of Liverpool (1980)Google Scholar
  14. 14.
    Sapounakis, A.: Measures on totally ordered spaces. Mathematika 27, 225–235 (1980)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Schoenberg, I.J.: A remark on the preceding note by Bochner. Bull. Am. Math. Soc. 40, 277–278 (1934)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Takahasi, S.-E., Hatori, O.: Commutative Banach algebras which satisfy a Bochner–Schoenberg–Eberlein-type theorem. Proc. Am. Math. Soc. 110, 149–158 (1990)MathSciNetMATHGoogle Scholar
  17. 17.
    Takahasi, S.-E., Hatori, O.: Commutative Banach algebras and BSE-inequalities. Math. Japonica 37, 47–52 (1992)MathSciNetMATHGoogle Scholar
  18. 18.
    Takahasi, S.-E., Takahashi, Y., Hatori, O., Tanahashi, K.: Commutative Banach algebras and BSE-norm. Math. Jpn. 46, 273–277 (1997)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Isfahan (Khorasgan) BranchIslamic Azad UniversityIsfahanIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran
  3. 3.Department of MathematicsUniversity of IsfahanIsfahanIran

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