The Bochner-Schoenberg-Eberlein module property for amalgamated duplication of Banach algebras

Year 2024, Volume: 53 Issue: 4, 915 - 925, 27.08.2024

Mohammad Ali Abolfathi , Ali Ebadian , Ali Jabbari

https://doi.org/10.15672/hujms.1261110

Abstract

The Bochner-Schoenberg-Eberlein module property on commutative Banach algebras is a property related to extensions of multipliers on Banach algebras to module morphisms from Banach algebras into Banach modules. In this paper, we answer the problem (1) raised in [J. Algebra Appl., 21(8) (2022), 2250155, DOI: 10.1142/S0219498822501559]. We show that the Banach $\mathcal{A}\rtimes\mathfrak{A}$-module $X\times Y$ ($X$ is a Banach $\mathcal{A},\mathfrak{A}$-module and $Y$ is a Banach $\mathfrak{A}$-module) has a BSE-module property if and only if $X$ is a BSE Banach $\mathcal{A},\mathfrak{A}$-module and $Y$ is a BSE Banach $\mathfrak{A}$-module.

Keywords

amalgamated Banach algebra, BSE-Banach algebra, BSE module property, character, multiplier

References

• [1] F. Abtahi, Z. Kamali and M. Toutounchi, The Bochner-Schoenberg-Eberlein property for vector-valued Lipschitz algebras, J. Math. Anal. Appl. 479 (1), 1172-1181, 2019.
• [2] N. Alizadeh, S. Ostadbashi, A. Ebadian and A. Jabbari, The BochnerSchoenbergEberlain property of extensions of Banach algebras and Banach modules, Bull. Aust. Math. Soc. 105 (1), 134-145, 2022.
• [3] P.A. Dabhi, Multipliers of perturued Cartesian product with an application to BSEproperty, Acta Math. Hungar. 149 (1), 58-66, 2016.
• [4] P.A. Dabhi and R.S. Upadhyay, The Semigroup Algebra $\ell^1(\mathbb{Z}^2, \max)$ is a Bochner- SchoenbergEberlein (BSE) Algebra, Mediterr. J. Math. 16 (1), 12, 2019.
• [5] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: The basic properties, J. Algebra Appl. 6, 443-459, 2007.
• [6] M. Daws, Multipliers, self-induced and dual Banach algebras, Dissert. Math. 470, 1-62, 2010.
• [7] A. Ebadian and A. Jabbari, Biprojectivity and biflatness of amalgamated duplication of Banach algebras, J. Algebra Appl. 19 (7), 2050132, 2020.
• [8] A. Ebadian and A. Jabbari, $C^*$-algebras defined by amalgamated duplication of $C^*$- algebras, J. Algebra Appl. 20 (2), 2150019, 2021.
• [9] A. Ebadian and A. Jabbari, The BochnerSchoenbergEberlein property for amalgamated duplication of Banach algebras, J. Algebra Appl. 21 (8), 2250155, 2022.
• [10] M. Essmaili, A. Rejali and A. Salehi Marzijarani, Biprojectivity of generalized module extension and second dual of Banach algebras, J. Algebra Appl. 21 (4), 2250070, 2022.
• [11] M. Fozouni and M. Nemati, BSE-property for some certain Segal and Banach algebras, Mediterr. J. Math. 16 (2), 38, 2019.
• [12] J. Inoue and S.E. Takahasi, Segal algebras in commutative Banach algebras, Rocky Mountain J. Math. 44, 539-589, 2014.
• [13] H. Javanshiri and M. Nemati, Amalgamated duplication of the Banach algebra $\mathfrak{A}$ along a $\mathfrak{A}$-bimodule $\mathcal{A}$, J. Algebra Appl. 17 (9), 1850169-1-1850169-21, 2018.
• [14] Z. Kamali and F. Abtahi, The BochnerSchoenbergEberlein property for vector-valued $\ell^p$-spaces, Mediterr. J. Math. 17 (3), 94, 2020.
• [15] Z. Kamali and M. Lashkarizadeh Bami, Bochner-Schoenberg-Eberlein property for abstract Segal algebras, Proc. Jpn. Acad. (Ser A) 89, 107-110, 2013.
• [16] Z. Kamali and M. Lashkarizadeh Bami, The multiplier algebra and BSE property of the direct sum of Banach algebras, Bull. Aust. Math. Soc. 88, 250-258, 2013.
• [17] Z. Kamali and M. Lashkarizadeh, The BochnerSchoenbergEberlein property for $L^1(\mathbb{R}^+)$, J. Fourier Anal. Appl. 20 (2), 225-233, 2014.
• [18] Z. Kamali and M. Lashkarizadeh, A characterization of the $L^\infty$-representation algebra $R(S)$ of a foundation semigroup and its application to BSE algebras, Proc. Jpn. Acad. Ser. A Math. Sci. 92 (5), 59-63, 2016.
• [19] Z. Kamali and M. Lashkarizadeh, The BochnerSchoenbergEberlein property for totally ordered semigroup algebras, J. Fourier Anal. Appl. 22 (6), 1225-1234, 2016.
• [20] E. Kaniuth, The Bochner-Schoenberg-Eberlein property and spectral synthesis for certain Banach algebra products, Canad. J. Math. 67, 827-847, 2015.
• [21] E. Kaniuth and A. Ülger, The BochnerSchoenbergEberlein property for commutative Banach algebras, especially Fourier and FourierStieltjes algebras, Trans. Amer. Math. Soc. 362, 4331-4356, 2010.
• [22] R. Larsen, An Introduction to the Theorey of Multipliers, Springer, New York, 1971.
• [23] W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962.
• [24] S.-E. Takahasi, BSE Banach modules and multipliers, J. Funct. Anal. 125, 67-68, 1994.
• [25] S.-E. Takahasi and O. Hatori, Commutative Banach algebras which satisfy a BochnerSchoenbergEberlein-type theorem, Proc. Amer. Math. Soc. 110, 149158, 1990.
• [26] S.-E. Takahasi and O. Hatori, Commutative Banach algebras and BSE-inequalities, Math. Japonica 37, 47-52, 1992.