# Connectness of $K$ and the existence of non-trivial $M$-summands on $C_0(K)$

For a locally compact Hausdorff space $$K$$, we denote by $$C_0(K)$$ the Banach space of continuous functions from $$K$$ to $$\mathbb R$$ which vanish at infinity equipped with the usual supremum norm.

For any Banach space $$X$$, a closed subspace $$E$$ of $$X$$ is said to be an $$M$$-summand of $$X$$ if there exists a subspace $$F$$ such that $$X=E\oplus F$$ and for any $$x=u+v$$, with $$u\in E$$ and $$v \in F$$, we have $$\|x\|=\max\{\|u\|,\|v\|\}.$$ We say that $$E$$ is a non-trivial M-summand if $$\emptyset \subsetneqq E \subsetneqq X$$.

Considering the space $$C_0(K)$$, when $$K$$ is disconnected, fixing two disjointed non-empty clopens $$U, V\subset K$$ such that $$U\cup V=K$$, the closed subspace $$E_U=\{f\in C_0(K): f|_{U^c}\equiv0\}$$ is a non-trivial $$M$$-summand of $$C_0(K)$$, for the closed subspace $$F_U=\{f\in C_0(K): f|_{V^c}\equiv0\}$$ complements $$E_U$$ and the condition on the norm is clearly satisfied.

My question is: what about the reverse? That is, if $$K$$ is connected, is it possible that $$C_0(K)$$ contains a non-trivial $$M$$-summand?

I approached this problem as follows. Pick $$E$$ a non-trivial $$M$$-summand of $$C_0(K)$$ and $$F$$ its complement. Consider the set $$K_E=\{k\in K | \exists f\in E: |f(k)|>\|f\|/2\}.$$ It surely is an open set of $$K$$. Picking $$K_F$$ the similar definition for $$F$$, it is also an open set of $$K$$. It is immediate to see that $$K_E\cap K_F=\emptyset$$, for if $$k\in K_E\cap K_F$$, then it would exist $$f\in K_E$$, $$g\in K_F$$ with $$\|f\|=\|g\|=c$$ and a scalar $$|\lambda|=1$$ such that $$\|f+\lambda g\|=\max\{\|f\|,\|\lambda g\|\}=c,$$ but $$|f(k)+\lambda g(k)|=|f(k)|+|g(k)|> c/2 + c/2 = c = \|f+\lambda g\|,$$ an absurd.

I want to prove that $$K_E$$ and $$K_F$$ are clopens by concluding that $$K=K_E\cup K_F$$. However, I didn't find a good approach. My intuition says that it is not true, but I have no idea or repertory to conclude so.

The converse is indeed true; in particular $$C_0(K)$$ does not have nontrivial $$M$$-summands if $$K$$ is connected. The argument to show this in [P. Harmand, yours truly and W. Werner, $$M$$-Ideals in Banach Spaces and Banach Algebras, Lect. Notes in Math. 1547] is based on the fact that the $$L$$-projection in the dual splits the set of extreme functionals.