BSE-Property for Some Certain Segal and Banach Algebras

  • Mohammad FozouniEmail author
  • Mehdi Nemati


For a commutative semi-simple Banach algebra A which is an ideal in its second dual, we give a necessary and sufficient condition for an essential abstract Segal algebra in A to be a BSE-algebra. We show that a large class of abstract Segal algebras in the Fourier algebra A(G) of a locally compact group G are BSE-algebra if and only if they have bounded weak approximate identities. In addition, in the case that G is discrete, we show that \(A_{\mathrm{cb}}(G)\) is a BSE-algebra if and only if G is weakly amenable. We study the BSE-property of some certain Segal algebras implemented by local functions that were recently introduced by J. Inoue and S.-E. Takahasi. Finally, we give a similar construction for the group algebra implemented by a measurable and sub-multiplicative function.


Banach algebra Segal algebra BSE-algebra locally compact group Fourier algebra 

Mathematics Subject Classification

Primary 46H05 Secondary 46J10 22D05 43A25 



The authors would like to thank the referee for his/her suggestions and comments which improved the presentation of the paper especially, giving a shorter proof for Theorem 2.1. The first named author of the paper supported partially by a grant from Gonbad Kavous University.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsGonbad Kavous UniversityGonbad-e KavousIran
  2. 2.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

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