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.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez: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.
📊
Temel Bulgular & Veri:In this paper, we answer the problem (1) raised in [J.
💡
Sonuç & Katkı: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.
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.
Abolfathi, M. A., Ebadian, A., Jabbari, A. (2024). The Bochner-Schoenberg-Eberlein module property for amalgamated duplication of Banach algebras. Hacettepe Journal of Mathematics and Statistics, Volume: 53 Issue: 4. https://doi.org/10.15672/hujms.1261110
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.
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.
P.A. Dabhi, Multipliers of perturued Cartesian product with an application to BSEproperty, Acta Math. Hungar. 149 (1), 58-66, 2016.
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.
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.
M. Daws, Multipliers, self-induced and dual Banach algebras, Dissert. Math. 470, 1-62, 2010.
A. Ebadian and A. Jabbari, Biprojectivity and biflatness of amalgamated duplication of Banach algebras, J. Algebra Appl. 19 (7), 2050132, 2020.
A. Ebadian and A. Jabbari, $C^*$-algebras defined by amalgamated duplication of $C^*$- algebras, J. Algebra Appl. 20 (2), 2150019, 2021.
A. Ebadian and A. Jabbari, The BochnerSchoenbergEberlein property for amalgamated duplication of Banach algebras, J. Algebra Appl. 21 (8), 2250155, 2022.
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.
M. Fozouni and M. Nemati, BSE-property for some certain Segal and Banach algebras, Mediterr. J. Math. 16 (2), 38, 2019.
J. Inoue and S.E. Takahasi, Segal algebras in commutative Banach algebras, Rocky Mountain J. Math. 44, 539-589, 2014.
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.
Z. Kamali and F. Abtahi, The BochnerSchoenbergEberlein property for vector-valued $\ell^p$-spaces, Mediterr. J. Math. 17 (3), 94, 2020.
Z. Kamali and M. Lashkarizadeh Bami, Bochner-Schoenberg-Eberlein property for abstract Segal algebras, Proc. Jpn. Acad. (Ser A) 89, 107-110, 2013.
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.
Z. Kamali and M. Lashkarizadeh, The BochnerSchoenbergEberlein property for $L^1(\mathbb{R}^+)$, J. Fourier Anal. Appl. 20 (2), 225-233, 2014.
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.
Z. Kamali and M. Lashkarizadeh, The BochnerSchoenbergEberlein property for totally ordered semigroup algebras, J. Fourier Anal. Appl. 22 (6), 1225-1234, 2016.
E. Kaniuth, The Bochner-Schoenberg-Eberlein property and spectral synthesis for certain Banach algebra products, Canad. J. Math. 67, 827-847, 2015.
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.
R. Larsen, An Introduction to the Theorey of Multipliers, Springer, New York, 1971.
W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962.
S.-E. Takahasi, BSE Banach modules and multipliers, J. Funct. Anal. 125, 67-68, 1994.
S.-E. Takahasi and O. Hatori, Commutative Banach algebras which satisfy a BochnerSchoenbergEberlein-type theorem, Proc. Amer. Math. Soc. 110, 149158, 1990.
S.-E. Takahasi and O. Hatori, Commutative Banach algebras and BSE-inequalities, Math. Japonica 37, 47-52, 1992.