Stable seminorms revisited

A seminorm S on an algebra A is called stable if for some constant σ > 0 , S(x^k) ≤ σS(x)^k for all x ∈ A and all k = 1, 2, 3,.... We call S strongly stable if the above holds with σ = 1 . In this note we use several known and new results to shed light on the concepts of stability. In particular, the interrelation between stability and similar ideas is discussed.