Derivations on Lau product of Banach algebras

Subject Areas : Statistics

Saeid Shams ¹

1 - Assistant professor, Department of Mathematics, Faculty of Basic Science, Urmia University, Urmia, Iran

Received: 2020-02-06 Accepted : 2020-07-19 Published : 2021-09-23

Keywords: اشتقاق, گروه همانستگی, ضرب لائو, جبر باناخ,

Abstract :

Extension of Banach algebras defined by Cartesian product by a linear functional such as are called -Lau Banach algebras. Recently, this type of Banach algebras are interested by many researchers. Derivations play an important role in algebraic structures. By using this role, one can discover some of the properties of algebraic structures on which a derivation is defined, such as their semi simplicity. In this paper, we consider derivations that are defined on Lau product of Banach algebras and we characterize these derivations in some various cases. As a main characterization, for two Banach algebra , and , we show that is a derivation if and only if there are derivations , and a -derivation such that for all . We investigate the converse case of the above obtained result in some various cases and according to the obtained results related to characterization of derivations, we investigate and characterize the first cohomology of Banach algebras obtained by this product.

References:

[1] A. T.-M. Lau. Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups. Fundamenta Mathematicae 118: 161-175(1983)

 

[2] M. Sangani-Monfared. On certain products of Banach algebras with applications to harmonic analysis. Studia Mathematica. 178(3): 277-294(2007)

 

[3] S. J. Bhatt, P. A. Dabhi. Arens regularity and amenability of Lau product of Banach algebras", Bulletin of the Australiain Mathematical Society. 87: 195–206(2013)

 

[4] Y. Choi.,Triviality of the generalised Lau product associated to a Banach algebra homomorphism. Bulletin of the Australiain Mathematical Society. 94(2): 286-289(2016)

 

[5] P. A. Dabhi, S. K. Patel. Spectral properties of the Lau product of Banach algebras.  Annals of Functional Analysis. 9(2): 246-257(2018)

 

[6] P. A. Dabhi, A.  Jabbari, K. Haghnejad Azar.  Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism. Acta Mathematica Sinica, English Series. 31(9): 1461–1474(2015)

 

[7] E. Ghaderi, R. Nasr-Isfahani, M. Nemati. Some notions of amenability for certain products of Banach algebras. Colloquium Mathematicum. 130(2):147-157 (2013)

 

[8] A. R. Khoddami, H. R. Ebrahimi Vishki.  Biflatness and biprojectivity of Lau product of Banach algebras.  Bulletin of the Iranian Mathematical Society. 39(3): 559-568(2013)

 

[9] H. Pourmahmood Agababa. Derivations and generalized semidirect products of Banach algebras. Banach Journal of Mathematical Analysis. 10(3): 1735-8787(2016)

 

[11] H. G. Dales. Banach algebras and automatic continuity. London Mathematical Society (2000)

 

[12] V. Losert. The derivation problem for group algebras. Annals of Mathematics. 168(2): 221-246 (2008)

 

[13] I. M. Singer, J. Wermer. Derivations on commutative normed algebras. Mathematische Annalen.  129: 260-264(1955)

 

[14] W. G. Bade, P. C. Curtis, H. G. Dales. Amenability and weak amenability for Beurling and Lipschitz algebras. Proceedings of the London Mathematical Society. 55(3): 359-377(1987)

 

[15] M. Eshaghi Gordji, T. Yazdanpanah. Derivations into duals of ideals of Banach algebras.  Proceedings of the Indian Academy of Sciences. 114(4): 399-408(2004)