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 $Vert varphi Vertleq 1$. 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 $0neqpsiin A^*$ with $Vert psi Vertleq 1$, 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^{bvarphi}(K), Vert cdot Vert_varphi right)$. Finally we  investigate some relations between strongly zero-product preserving maps on $C^b(K)$ and $C^{bvarphi}(K)$.