Monotone Norms on C(Ω) and Multiplicative Factors

Abstract

Let $C(\Omega)$ be the algebra of continuous complex-valued functions on a topological space $\Omega$ and let $\rho$ be a function norm on $C(\Omega).$ We give necessary and sufficient conditions on the set $A_{\rho}=\{f\in C(\Omega)\:rho(f)<\infty\}$ to be an algebra. Also, we prove that every complete function norm is quasi-submultiplicative provided $A_{\rho}$ is an algebra and we give a characterization of the best multiplicative factor of $\rho.$ Finally we characterize the infinity norm and we prove that every quasi-submultiplicative function norm on $C(\Omega)$ is equivalent to the infinity norm.