$\rm BSE$ norm for Abstract Segal algebras

Document Type : Research Paper

Authors

1 Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, P.O.Box 81746-73441, Isfahan, Iran

2 Department of pure mathematics, Faculty of mathematics and statistics, University of Isfahan, Isfahan, Iran.

Abstract

Let $(\mathcal A,\|\cdot\|_{\mathcal A})$ be a commutative and semisimple Banach algebra and $(\mathcal B,\|\cdot\|_{\mathcal B})$ be an abstract Segal algebra with respect to $\mathcal A$. In this paper, we first recall and study three important and practical mappings ${}_{\mathcal A}L$, $\Gamma_1$ and $\Gamma_2$. Then we investigate
whenever these mappings have closed ranges. In fact, we research and study the conditions, under which having closed range of one of these mappings implies having the closed range of the another mapping. After that, using these results, we give a necessary and sufficient condition for $(\mathcal B,\|\cdot\|_{\mathcal B})$, to be an algebra with $\rm BSE$ norm. Finally, we generalize some general results about abstract Segal algebras with respect to natural Banach functional algebras for abstract Segal algebras with respect to arbitrary Banach algebras. Also, throughout the paper, we provide examples to clarify the stated content.

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Main Subjects

References

[1] F. Abtahi, Z. Kamali, M. Toutounchi, The Bochner-Schoenberg-Eberlein property for vector-valued Lipschitz algebras, J. Math. Anal. Appl., 479 (2019) 1172–1181.
[2] F. Abtahi, Z. Kamali and M. Toutounchi, The BSE concepts for vector-valued Lipschitz algebras, Commun. Pure Appl. Anal., 20 no. 3 (2021) 1171–1186.
[3] M. Alaghmandan, R. Nasr Isfahani and M. Nemati, Character amenability and contractibility of abstract Segal algebras, Bull. Aust. Math. Soc., 82 (2010) 274–281.
[4] S. Bochner, A theorem on Fourier- Stieltjes integrals, Bull. Amer. Math. Soc., 40 (1934) 271–276.
[5] J. T. Burnham, Closed ideals in subalgebras of Banach algebras. I, Proc. Amer. Math. Soc., 32 no. 2 (1972) 551–555.
[6] H. G. Dales and A. Ülger, Approximate identities and BSE norms for Banach functin algebras, Fields Institute, Toronto, 2014.
[7] H. G. Dales and A. Ülger, Banach function algebras and BSE norms, Graduate course during 23rd Banach algebra conference, Oulu, Finland, 2017.
[8] W. F. Eberlein, Characterizations of Fourier-Stieltjes transforms, Duke Math. J., 22 (1955) 465–468.
[9] J. Inoue, Jyunji and Sin-Ei. Takahasi, Constructions of bounded weak approximate identities for Segal algebras on LCA groups, Acta Sci. Math. (Szeged), 66 (2000) 257–271.
[10] J. Inoue, Jyunji and Sin-Ei. Takahasi, Segal algebras in commutative Banach algebras, Rocky Mt. J. Math., 44 no. 2 (2014) 539–589.
[11] C. A. Jones and C. D. Lahr, Weak and norm approximate identities are different, Pacific J. Math., 72 (1977) 99–104.
[12] E. Kaniuth, A course in commutative Banach algebras, Springer, New York, 2009.
[13] E. Kaniuth and A. Ülger, The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier-Stieltjes algebras, Trans. Amer. Math. Soc., 362 (2010) 4331–4356.
[14] E. Kaniuth, The Bochner-Schoenberg-Eberlein Property and Spectral Synthesis for Certain Banach Algebra Products, Canad. J. Math., 67 (2015) 827–847.
[15] Yu. N. Kuznetsova, Weighted Lp −algebras on groups, Funktsional. Anal. i Prilozhen., 40 no. 3 (2006) 82–85. Translation in Funct. Anal. Appl., 40 no. 3 (2006) 234–236.
[16] Yu. N. Kuznetsova, Invariant weighted algebras Lp (G, ω), Mat. Zametki, 84 no. 4 (2008) 567–576.
[17] Yu. N. Kuznetsova, Example of a weighted algebra Lp (G, ω) on an uncountable discrete group, J. Math. Anal. Appl., 353 (2009) 660–665.
[18] R. Larsen, An introduction to the theory of multipliers, Springer-Verlag, New York, 1971.
[19] W. Rudin, Fourier analysis on groups, Interscience, New York, 1962.
[20] I. J. Schoenberg, A remark on the preceding note by Bochner, Bull. Amer. Math. Soc., 40 (1934) 277–278.
[21] S. E. Takahasi and O. Hatori, Commutative Banach algebras which satisfy a Bochner-Schoenberg Eberlein-type theorem, Proc. Amer. Math. Soc., 110 (1990) 149–158.
[22] S. E. Takahasi and O. Hatori, Commutative Banach algebras and BSE-inequalities, Math. Japonica, 37 (1992) 47–52.
[23] S. E. Takahasi, Y. Takahashi, O. Hatori and K. Tanahashi, Commutative Banach algebras and BSE-norm, Math. Japonica, 46 no. 2 (1997) 273–277.
Mathematics and Society
Volume 9, Issue 4 - Serial Number 4
December 2024
Pages 87-102

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History

  • Receive Date: 27 March 2024
  • Revise Date: 05 June 2024
  • Accept Date: 12 June 2024
  • Publish Date: 19 February 2025

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