Local approach to order continuity in Ces\`aro function spaces

Abstract

The goal of this paper is to present a complete characterisation of points of order continuity in abstract Ces\`aro function spaces $CX$ for $X$ being a symmetric function space. Under some additional assumptions mentioned result takes the form $(CX)_a = C(X_a)$. We also find simple equivalent condition for this equality which in the case of $I=[0,1]$ comes to $X\neq L^\infty$. Furthermore, we prove that $X$ is order continuous if and only if $CX$ is, under assumption that the Ces\`aro operator is bounded on $X$. This result is applied to particular spaces, namely: Ces\`aro-Orlicz function spaces, Ces\`aro-Lorentz function spaces and Ces\`aro-Marcinkiewicz function spaces to get criteria for OC-points.