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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

  • Zeinab Kamali
  • Mahmood Lashkarizadeh Bami
Article
  • 142 Downloads

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.

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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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