A class of seminorms on function algebras

Abstract

Let A be a function algebra on a set T. In this paper we study seminorms on A of the form Sc(x)={norm of matrix}cx{norm of matrix} where c, 0 ≠ c ∈ A, is a fixed element and {norm of matrix}·{norm of matrix} is the sup norm on T. We begin by proving that under suitable assumptions, elements c, d ∈ A satisfy c ≤ d on T, if and only if for some p, 0 < p < ∞, Sc(xp) ≤ Sd(xp) for all x in a subset B of A. These results are then used in order to study multiplicativity and quadrativity factors for Sc on B, i.e., constants μ > 0 and λ > 0 for which Sc(xy) ≤ μSc(x) Sc(y) and Sc(x2) ≤ λSc(x)2 for all x, y ∈ B. Finally, for a family T of functions in A, we define the seminorm SF(x)=sup{Sf(x):f ∈ F}, and provide conditions under which SF has multiplicativity and quadrativity factors by exhibiting an element c ∈ A such that SF=S c on A. © 1991.