Abstract

Let A be a finite-dimensional, power-associative algebra over a field F, either R or C, and let S, a subset of A, be closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0≠a in S, and f(αa) = |α|f(a) for all a in S and α in F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant σ > 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 F. Our goal is to review and extend several of our results in two previous papers, dealing mostly with continuous subnorms on closed sets.

Authors