The Product Between Three Banach Algebras

Document Type : Original Article

Author
Ali Khotanloo
Department of Mathematics, Faculty of Sciences, Malayer University, Malayer, Hamedan, Iran.
10.29252/maco.1.2.2
Abstract
Abstract. Let A, B, and C be Banach algebras, α ∈ Hom(A, B) and β ∈ Hom(C, B), and k α k≤ 1, kβ k≤ 1. IN this paper we define the Banach algebra A×α B×β C by new product on A×B×C which is a strongly splitting extension of C by B. Then we show that these products from a large class of Banach algebras which contains all module extensions and triangular Banach algebras. Finally we consider spectrum, Arens regularity, amenability and weak amenability of these products.

Keywords

[1] W. G. Bade, H. G. Dales, and Z. A. Lykova, Algebraic and strong splittings of extensions of Banach algebras. Mem. Amer. Math. Soc. 656.
[2] H. G. Dales, Banach Algebras and Automatic Continuity, Oxford University Press, 2000.
[3] H.G. Dales, F. Ghahramani, N. Gronbaek, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1998) 19-54.
[4] G. H. Esslamzadeh, A. Khotanloo, and B. Tabatabaie Shourijeh, Structure of certain Banach algebra products, J. Sci. I. R. Iran, 259 (2014),265–271.
[5] G. H. Esslamzadeh and B. Shojaee, Approximate weak amenability of Banach algebras, Bull. Belg. Math. Soc. 18 (2011), 415-429.
[6] H. Farhadi, H. Ghahramani, Some notes on semidirect products of Banach algebras, Results in Mathematics, 74, (2019) 107.
[7] B. E. Forrest and L. W. Marcoux, Weak amenability of triangular Banach algebras, Trans. Amer. Math. Soc. 354 (2002), 1435-1452.
[8] F. Ghahramani, R.J. Loy, Generalized notions of amenability, J. Funct. Anal. 208 (1) (2004) 229-260.
[9] F. Gourdeau, Amenability and the second dual of a Banach algebras, Studia Math. 125 (1997), 75-81.
[10] E. Kaniuth, A Course in Commutative Banach Algebras, Springer Verlag, New York, 2009.
[11] A. T.-M. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups, Fund. Math. 118 (1983), 161-175.
[12] M. S. Monfared, On certain products of Banach algebras with application to Harmonic analysis, Studia Math. 178 (3) (2007), 277-294.
[13] T. W. Palmer, Banach algebras and the general theory of ∗ -algebras. Vol. 1: Algebras and Banach algebras, Cambridge University Press, 1994.
[14] A. Pourabbas, N. Razi, Some homological properties of T-Lau product algebras, Acta Mathematica Hungarica 149, no. 1 (2016), 31–49.
[15] V. Runde, Lectures on Amenability, Springer, New York, 2002.
[16] Y. Zhang, Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (2002) 4131-4151.
Mathematical Analysis and Convex Optimization
Volume 1, Issue 2
December 2020
Pages 15-24