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 non-zero normed vector space $A$ and $varphi$ is a non-zero 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 $(C^{bvarphi}(K)‎, ‎Vert cdot Vert_varphi)$‎.

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