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

The Banach space $CD_0(Q,\mathcal{X})=C(Q,\mathcal{X})+c_0(Q,\mathcal{X})$ is considered whose elements are the sums of continuous and “discrete” sections of a Banach bundle $\mathcal{X}$ over a compact Hausdorff space $Q$ without isolated points. As is known, $CD_0(Q,\mathcal{X})$ is isometric to the space $C(\tilde{Q},\tilde{\mathcal{X}})$ of continuous sections of a Banach bundle $\tilde{\mathcal{X}}$ over the set $\tilde{Q}=Q\times\{0,1\}$ endowed with a special topology. The connections are clarified between $\mathcal{X}$ and $\tilde{\mathcal{X}}$ related to subbundles as well as to bundles obtained by a continuous change of variable and by the restriction onto a topological subspace. In addition, we introduce and study the space $CD_0[\mathcal{X},\mathcal{Y}]$ of $CD_0$-homomorphisms of Banach bundles $\mathcal{X}$ and $\mathcal{Y}$ and demonstrate that it possesses certain properties analogous to those of the space of $CD_0$-sections.