On Minimal Norms on M n

Abstract

We show that for each minimal norm $N(\cdot )$ on the algebra ${ℳ}_n$ of all $n\times n$ complex matrices, there exist norms ${\Vert \cdot \Vert }_1$ and ${\Vert \cdot \Vert }_2$ on ${ℂ}^n$ such that $N\left(A\right)=max\left\{{\Vert Ax\Vert }_2:{\Vert x\Vert }_1=1,x\in {ℂ}^n\right\}$ for all $A\in {ℳ}_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.