Ela Stable

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 on S is called a subnorm if f(a) > 0 for all 0 = a ∈ S, and f(αa) = |α|f(a) for all a ∈ S and α ∈ F. If in addition, S is closed under raising to powers, then a subnorm f is said to be stable if there exists a positive constant σ so that f(a) ≤ σf(a) for all a ∈ S and k = 1, 2, 3, . . . . 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 aim is to review and discuss some of the results in several previous papers, dealing with both continuous and discontinuous subnorms.