Abstract. The goal of this paper is to present a complete characterization of points of order continuity in abstract Cesàro function spaces CX for X being a symmetric function space. Under some additional assumptions mentioned result takes the form (CX)a = C(Xa). We also find simple equivalent condition for this equality which in the case of I = [0, 1] comes to X = L ∞ . Furthermore, we prove that X is order continuous if and only if CX is, under assumption that the Cesàro operator is bounded on X. This result is applied to particular spaces, namely: Cesàro-Orlicz function spaces, Cesàro-Lorentz function spaces and Cesàro-Marcinkiewicz function spaces to get criteria for OC-points.
We characterize Cesàro-Orlicz function spaces Cesϕ containing order isomorphically isometric copy of l ∞ under some mild assumption imposed on the Orlicz function ϕ. We discuss also some useful applicable sufficient conditions for the existence of such a copy.
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