Wavelet and Linear Algebra

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. 

Abstract 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 $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$.