Weak and cyclic amenability of certain function algebras

Document Type: Research Paper


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



We consider $C^{b\varphi}(K)$ to be the space $C^b(K)$ equipped with the product $f\cdot g=f\varphi g$ for all $f, g\in C^b(K)$ where, $K=\overline{B_1^{(0)}}$ is the closed unit ball of a non-zero normed vector space $A$ and $\varphi$ is a non-zero element of $A^*$ such that $\Vert \varphi \Vert\leq 1$. We define $\Vert f \Vert_\varphi=\Vert f\varphi \Vert_\infty$ for all $f\in C^{b\varphi}(K)$. Some relations between the dual spaces of $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$ are investigated. Also we characterize the derivations from $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$ into $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)^*$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)^*$ respectively. Finally we investigate the weak and cyclic amenability of $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$.


[1] H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. Ser., 24, The Clarendon Press, Oxford University Press, New York, 2000.
[2] N. Gr{o}nb{ae}k, Weak and cyclic amenability for non-commutative Banach algebras, Proc. Edinb. Math. Soc., 35 (1992), 315-328.
[3] R.A. Kamyabi-Gol and M. Janfada, Banach algebras related to the elements of the unit ball of a Banach algebra, Taiwanese J. Math., 12(7) (2008),  1769-1779.
[4] A.R. Khoddami, Biflatness, biprojectivity, $varphi-$amenability and $varphi-$contractibility of a certain class of Banach algebras, Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar, 80 (2018), 169-178.
[5] A.R. Khoddami, Bounded and continuous functions on the closed unit ball of a normed vector space equipped with a new product, Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar, 81 (2019), 81-86.
[6] A.R. Khoddami, Non equivalent norms on $C^b(K)$, Sahand Commun. Math. Anal., 2020, Articles in Press.
[7] V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer-Verlag, Berlin, 2002.