Monotone norms on \(C(\Omega)\) and multiplicative factors (Q1978957)

Let \(\rho \) be a monotone function norm on the algebra \(C(\Omega)\) of continuous complex-valued functions on an arbitrary topological space \(\Omega \). It means that \(\rho :C(\Omega)\to [0,\infty ]\) has properties of a norm and \(\rho (f)\leq \rho (g)\) if \(|f|\leq |g|\). It is shown e.g. that, if \(\rho (\sum _{n=1}^{\infty }|f_n|)\leq \sum _{n=1}^{\infty }\rho (|f_n|)\) whenever \(\sum _{n=1}^{\infty }|f_n|\in C(\Omega)\), the set \(A_{\rho }=\{f\in C(\Omega); \rho (f) <\infty \}\) is an algebra if and only if all elements of \(A_{\rho }\) are bounded. Using the previous result it is proved that, if \((A_{\rho },\rho)\) is a complete subalgebra of \(C(\Omega)\), there is a \(K\) such that \(\|f\|_\infty \leq K\rho (f)\) for all \(f\in C(\Omega)\). The quasi-submultiplicativity property \(\rho (fg)\leq K\rho (f)\rho (g)\) is investigated and the smallest possible \(K\) is described e.g. if \(A_\rho \) is a complete algebra.