On the relative injectivity of the $C_0(S)$-module $C_0(S)$

Let $S$ be a locally compact Hausdorff space, and consider the algebra $C_0(S)$ of continuous, vanishing at infinity functions on $S$. In this talk, we give a strong necessary condition of the relative injectivity of $C_0(S)$ as a module over itself. As a corollary, we show that $S$ cannot be an infinite metric space. Also we discuss a module analog of Sobczyk's theorem which asserts that $c_0$ is relatively injective as an $\ell_\infty$-module.