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BSE-Property for Some Certain Segal and Banach Algebras

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Abstract

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.

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Acknowledgements

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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Correspondence to Mohammad Fozouni.

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Fozouni, M., Nemati, M. BSE-Property for Some Certain Segal and Banach Algebras. Mediterr. J. Math. 16, 38 (2019). https://doi.org/10.1007/s00009-019-1305-2

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  • DOI: https://doi.org/10.1007/s00009-019-1305-2

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