Weak and cyclic amenability of certain function algebras

We consider $C^{bvarphi}(K)$ to be the space $C^b(K)$ equipped with the product $fcdot g=fvarphi g$ for all $f, gin C^b(K)$ where, $K=overline{B_1^{(0)}}$ is the closed unit ball of a nonzero normed vector space $A$ and $varphi$ is a nonzero element of $A^*$ such that $Vert varphi Vertleq 1$. We define $Vert f Vert_varphi=Vert fvarphi Vert_infty$ for all $fin C^{bvarphi}(K)$. Some relations between the dual spaces of $(C^{bvarphi}(K), Vert cdot Vert_infty)$ and $(C^{bvarphi}(K), Vert cdot Vert_varphi)$ are investigated. Also we characterize the derivations from $(C^{bvarphi}(K), Vert cdot Vert_infty)$ and $(C^{bvarphi}(K), Vert cdot Vert_varphi)$ into $(C^{bvarphi}(K), Vert cdot Vert_infty)^*$ and $(C^{bvarphi}(K), Vert cdot Vert_varphi)^*$ respectively. Finally we investigate the weak and cyclic amenability of $(C^{bvarphi}(K), Vert cdot Vert_infty)$ and $(C^{bvarphi}(K), Vert cdot Vert_varphi)$.