$n$-weak amenability of a certain class of function spaces

Document Type : Research Paper

Author

Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran.

10.22072/wala.2023.547710.1360

Abstract

Let $A$ be a non-zero normed vector space and let $\varphi$ be a non-zero element of $A^*$ such that $\Vert \varphi \Vert \leq 1$. Assume that $K=\overline{B_1^{(0)}}$ is the closed unit ball of $A$. According to the our recent studies on the spaces of $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$, generated by $C^b(K)$ and equipped with a new product `` $ \cdot $ '' and different norms $\Vert \cdot \Vert_\infty $ and $\Vert \cdot \Vert_\varphi$, the $n-$weak amenability of $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$ are investigated. 

Keywords


[1] G.W. Bade, P.C. Curtis and H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. 
      London Math. Soc., 55 (1987), 359-377.
[2] H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. Ser., 24, The Clarendon Press, 
     Oxford University Press, New York, 2000.
[3] H.G. Dales, F. Ghahramani and N. Gr{\o}nb{\ae}k, Derivations into iterated duals of Banach algebras, Studia Math., 128 
     (1998), 19-54.   
[4] N. Gr{\o}nb{\ae}k, Weak and cyclic amenability for non-commutative Banach algebras, Proc. Edinburgh Math. Soc., 35 
     (1992), 315-328.      
[5] R.A. Kamyabi-Gol and M. Janfada, Banach algebras related to the elements of the unit ball of a Banach algebra, 
     Taiwan. J. Math., 12 (2008), 1769-1779.       
[6] A.R. Khoddami, Biflatness, biprojectivity, $\varphi-$amenability and $\varphi-$contractibility of a certain class of 
     Banach algebras, U.P.B. Sci. Bull., Series A, 80 (2018), 169-178.
[7] A.R. Khoddami, Bounded and continuous functions on the closed unit ball of a normed vector space equipped with a 
     new product, U.P.B. Sci. Bull., Series A, 81 (2019), 81-86.
[8] A.R. Khoddami, Non equivalent norms on $C^b(K)$, Sahand Commun. Math. Anal., 17 (2020), 1-11.
[9] A.R. Khoddami, Weak and cyclic amenability of certain function algebras, Wavelets and Linear Algebr., 7 (2020), 31-41.
[10] V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer-Verlag, Berlin, 2002.