Non-Equivalent Norms on $C^b(K)$

Abstract

‎Let $A$ be a non-zero normed vector space and let $K=\overline{B_1^{(0)}}$ be the closed unit ball of $A$. Also, let $\varphi$ be a non-zero element of $ A^*$ such that $\Vert \varphi \Vert\leq 1$. We first define a new norm $\Vert \cdot \Vert_\varphi$ on $C^b(K)$, that is a non-complete, non-algebraic norm and also non-equivalent to the norm $\Vert \cdot \Vert_\infty$. We next show that for $0\neq\psi\in A^*$ with $\Vert \psi \Vert\leq 1$, the two norms $\Vert \cdot \Vert_\varphi$ and $\Vert \cdot \Vert_\psi$ are equivalent if and only if $\varphi$ and $\psi$ are linearly dependent. Also by applying the norm $\Vert \cdot \Vert_\varphi $ and a new product `` $\cdot$ '' on $C^b(K)$, we present the normed algebra $ \left( C^{b\varphi}(K), \Vert \cdot \Vert_\varphi \right)$. Finally we investigate some relations between strongly zero-product preserving maps on $C^b(K)$ and $C^{b\varphi}(K)$.

Keywords

• normed vector space
• equivalent norm
• zero-product preserving map
• strongly zero-product preserving map