Variants of order semicontinuity in Banach lattices

In this article we consider various properties of a normed lattice, which are similar to semicontinuity (also known as the Fatou property), establish relations between these properties, and discuss stability of these properties under renorming. In particular, we show that a normed lattice $F$ is order continuous iff every renorming of $F$ is semicontinuous. We also prove that the weakly Fatou (we propose the term ``demicontinuous’') normed lattices are precisely the ones isomorphic to regular sublattices of monotonically complete Banach lattices. In order to do so we introduce the concept of the Lorentz completion of a demicontinuous normed lattice, which is somewhat analogous to the universal completion from the vector lattice theory. Furthermore, we unify and simplify the proofs of the characterizations of monotone completeness from \cite{aw} and \cite{taylor} and provide their quantitative versions. The semicontinuity-related properties in AM-spaces have some additional features. While the classical Kakutani theorem states that AM-spaces are precisely the closed sublattices of $\text{Co}\left(K\right)$-spaces, we show, using two different methods, that the semicontinuous AM-spaces are precisely the closed regular sublattices of $\text{Co}\left(K\right)$-spaces. Finally, we prove that for a normed space $E$, the AM-space of positively homogeneous weak* continuous functions on ${\text{Ba}}_{E^{\ast }}$ is semicontinuous iff it is a regular sublattice of $\text{Co}({\text{Ba}}_{E^{\ast }},{\mathrm{w}}^{\ast })$ iff $\dim E<\text{8}$.