Let $X$ be a locally compact Hausdorff space, such that $C\_0(X)$ is an injective Banach space, *i.e.* a $\mathfrak{P}\_\lambda$ space for some $\lambda\geq 1$.


* Is it true that $X$ is compact?
* If additionally $X$ is a locally compact group, is it true that $X$ is finite?



On injectivity of the Banach space $C_0(X)$
$C\_0(G)$ is not injective for infinite $G$, because it is not complemented in $L\_\infty(G)$. 


See [Lau, Anthony To Ming; Losert, Viktor. Complementation of certain subspaces of $L\_\infty(G)$ of a locally compact group. Pacific Journal of Mathematics 141 (1990), no. 2, 295--310](http://projecteuclid.org/download/pdf_1/euclid.pjm/1102646608), corollary 3.


