Weak and cyclic amenability of certain function algebras

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_۱^{(۰)}} 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 ۱. 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).