Spaces of $CD_0$-functions and $CD_0$-sections of Banach bundles

We first briefly expose some crucial phases in studying the space $CD_0(Q)=C(Q)+c_0(Q)$ whose elements are the sums of continuous and “discrete” functions defined on a compact Hausdorff space $Q$ without isolated points. In this part, special emphasis is on describing the compact space $\widetilde Q$ representing the Banach lattice $CD_0(Q)$ as $C(\widetilde Q)$. The rest of the article is dedicated to the analogous frame related to the space $CD_0(Q,\chi)$ of “continuous-discrete” sections of a Banach bundle $\chi$ and the space of $CD_0$-homomorphisms of Banach bundles.