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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References
Burnham, J.T.: Closed ideals in subalgebras of Banach algebras I. Proc. Am. Math. Soc. 32, 551–555 (1972)
Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, New York (1990)
Dales, H.G.: Banach Algebras and Automatic Continuity. Clarendon Press, Oxford (2000)
Duncan, J., Hosseiniun, S.A.R.: The second dual of a Banach algebra. Proc. R. Soc. Edinb. Sect. 84, 309–325 (1979)
Effros, E .G., Ruan, Z.-J.: Operator Spaces. Clarendon Press, Oxford (2000)
Forrest, B.: Arens regularity and discrete groups. Pacific J. Math. 151(2), 217–227 (1991)
Forrest, B.: Some Banach algebras without discontinuous derivations. Proc. Am. Math. Soc. 114(4), 965–970 (1992)
Forrest, B.E.: Completely bounded multipliers and ideals in \(A(G)\) vanishing on closed subgroups. Contemp. Math. 363, 89–94 (2004)
Forrest, B.E., Runde, V., Spronk, N.: Operator amenability of the Fourier algebra in the \(cb\)-multiplier norm. Can. J. Math. 59, 966–980 (2007)
Inoue, J., Takahasi, S.-E.: Constructions of bounded weak approximate identities for Segal algebras on LCA groups. Acta Sci. Math. (Szeged) 66, 257–271 (2000)
Inoue, J., Takahasi, S.-E.: Segal algebras in commutative Banach algebras. Rocky Mt. J. Math. 44(2), 539–589 (2014)
Jones, C.A., Lahr, C.D.: Weak and norm approximate identities are different. Pacific J. Math. 72(1), 99–104 (1977)
Kamali, Z., Bami, M.L.: The Bochner–Schoenberg–Eberlein property for \({L}^{1}(\mathbb{R}^{+})\). J. Fourier Anal. Appl. 20(2), 225–233 (2014)
Kaniuth, E.: A Course in Commutative Banach Algebras. Graduate texts in mathematics. Springer, New York (2009)
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)
Laali, J., Fozouni, M.: On \(\Delta \)-weak \(\phi \)-amenable Banach algebras, U. P. B. Sci. Bull. Ser. A 77(4), 165–176 (2015)
Nemati, M.: On generalized notions of amenability and operator homology of the Fourier algebra. Quart. J. Math. 68(3), 781–789 (2017)
Sakai, S.: Weakly compact operators on operators algebra. Pacific J. Math. 14(2), 659–664 (1964)
Samei, E., Spronk, N., Stokke, R.: Biflatness and pseudo-amenability of Segal algebras. Can. J. Math. 62, 845–869 (2010)
Takahasi, S.-E., Hatori, O.: Commutative Banach algebras which satisfy a Bochner–Schoenberg–Eberlein-type theorem. Proc. Am. Math. Soc. 110, 149–158 (1990)
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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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