Skip to main content Accessibility help

$C^*$ -algebra structure on certain Banach algebra products

  • Fatemeh Abtahi (a1)


Let $\mathcal A$ and $\mathcal B$ be commutative and semisimple Banach algebras and let $\theta \in \Delta (\mathcal B)$ . In this paper, we prove that $\mathcal A\times _{\theta }\mathcal B$ is a type I-BSE algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are so. As a main application of this result, we prove that $\mathcal A\times _{\theta }\mathcal B$ is isomorphic with a $C^*$ -algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are isomorphic with $C^* $ -algebras. Moreover, we derive related results for the case where $\mathcal A$ is unital.


Corresponding author


Hide All
[1] Abtahi, F., Kamali, Z., and Toutounchi, M., The Bochner-Schoenberg-Eberlein property for vector-valued Lipschitz algebras . J. Math. Anal. Appl. 479(2019), 11721181.
[2] Bochner, S., A theorem on Fourier-Stieltjes integrals . Bull. Amer. Math. Soc. 40(1934), 271276.
[3] Eberlein, W. F., Characterizations of Fourier-Stieltjes transforms . Duke Math. J. 22(1955), 465468.
[4] Inoue, J. and Takahasi, S. E., On characterizations of the image of Gelfand transform of commutative Banach algebras . Math. Nachr. 280(2007), 105126.
[5] Kamali, Z. and Lashkarizadeh Bami, M., Bochner-Schoenberg-Eberlein property for abstract Segal algebras . Proc. Jpn. Acad. Ser A. Math. Sci. 89(2013), 107110.
[6] Kaniuth, E., The Bochner-Schoenberg-Eberlein property and spectral synthesis for certain Banach algebra products . Canad. J. Math. 67(2015), 827847.
[7] Kaniuth, E., A course in commutative Banach algebras. Graduate Texts in Mathematics, 246, Springer, New York, 2009.
[8] Kaniuth, E., Lau, A. T., and Ülger, A., Homomorphisms of commutative Banach algebras and extensions to multiplier algebras with applications to Fourier algebras . Studia Math. 183(2007), 3562.
[9] Kaniuth, E. and Ülger, A., The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier-Stieltjes algebras . Trans. Amer. Math. Soc. 362(2010), 43314356.
[10] Larsen, R., An introduction to the theory of multipliers . Die Grundlehren der mathematischen Wissenchaften, 175, Springer-Verlag, New York-Heidelberg, 1971.
[11] Monfared, M. S., On certain products of Banach algebras with applications to harmonic analysis . Studia Math. 178(2007), 277294.
[12] Monfared, M. S., Character amenability of Banach algebras . Math. Proc. Camb. Philos. Soc. 144(2008), 697706.
[13] Schoenberg, I. J., A remark on the preceding note by Bochner . Bull. Amer. Math. Soc. 40(1934), 277278.
[14] Takahasi, S. E. and Hatori, O., Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein-type theorem . Proc. Amer. Math. Soc. 110(1990), 149158.
[15] Takahasi, S. E. and Hatori, O., Commutative Banach algebras and BSE-inequalities . Math. Japonica 37(1992), 4752.
[16] Ülger, A., Multipliers with closed range on commutative Banach algebras . Studia Math. 153(2002), 5980.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

$C^*$ -algebra structure on certain Banach algebra products

  • Fatemeh Abtahi (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.