Multiplicativity factors for function seminorms (Q1910011)

Let \((X,{\mathcal A},\mu)\) be a \(\sigma\)-finite measure space, and let \(\mathbb{R}\), \(\mathbb{C}\), denote the set of real numbers and the set of complex numbers, respectively. \({\mathcal M}= {\mathcal M}(X,{\mathcal A},\mu)\) denotes the equivalence classes of measurable functions which differ on sets of measure 0. Let \(L_\infty\) denote the space of essentially bounded functions on \(X\). If \(\rho:{\mathcal M}\to (0,\infty)\) is a function seminorm which is \(\sigma\)-subadditive and \(L_\rho=\{f\in{\mathcal M}:\rho(f)<\infty\}\), then the results of this paper show that \(L_\rho\) is an algebra if and only if \(L_\rho\subseteq L_\infty+N\), where \(N=\{f\in{\mathcal M}:\rho(f)=0\}\). It is stated in the paper that results of \textit{R. Arens}, \textit{M. Goldberg} and \textit{W. A. J. Luxemburg} [J. Math. Anal. Appl. 177, No. 2, 368-385 (1993; Zbl 0801.46026)], show that: if \(\rho\) is a \(\sigma\)-subadditive norm, then \(L_\rho\) is an algebra if and only if \(L_\rho\subseteq L_\infty\).