Abstract

Let $\mathcal{A}$ be a finite-dimensional, power-associative algebra over a field $\mathbb{F}$, either $ℝ$ or $ℂ$, and let $\mathcal{S}$, a subset of $\mathcal{A}$, be closed under scalar multiplication. A real-valued function $f$ defined on $\mathcal{S}$, shall be called a subnorm if $f\left(a\right)>0$ for all $0\ne a\in \mathcal{S}$, and $f\left(\alpha a\right)=\left|\alpha \right|f\left(a\right)$ for all $a\in \mathcal{S}$ and $\alpha \in \mathbb{F}$. If in addition, $\mathcal{S}$ is closed under raising to powers, then a subnorm $f$ shall be called stable if there exists a constant $\sigma >0$ so that

The purpose of this paper is to provide an updated account of our study of stable subnorms on subsets of finite-dimensional, power-associative algebras over $\mathbb{F}$. Our goal is to review and extend several of our results in two previous papers, dealing mostly with continuous subnorms on closed sets.

Milestones