A concrete description of $CD_{0}(K)$-spaces as $C(X)$-spaces and its applications

We prove that for a compact Hausdorff space $K$ without isolated points, $CD_{0}(K)$and $C(K\times \{0,1\})$ are isometrically Riesz isomorphic spaces under a certain topology on $K\times \{0,1\}$. Moreover, $K$ is a closed subspace of $K\times \{0,1\}$. This provides concrete examples of compact Hausdorff spaces $X$ such that the Dedekind completion of $C(X)$ is $B(S)$ (= the set of all bounded real-valued functions on $S$) since the Dedekind completion of $CD_{0}(K)$ is $B(K)$ ( $CD_{0}(K,E)$ and $CD_{w}(K,E)$ spaces as Banach lattices).