Is $c_0$ complemented in $C(K)$

Question

Suppose that $K$ is an infinite compact metric space. Define $c_0=\{ (x_n)_{n \in \mathbb{N}}| \lim_n{\| x_n \|}=0 \}$.

Is it true that $c_0$ complemented in $C(K)$, the set of continuous functions on $K$?

It seems true based on this paper(first sentence in the proof of Theorem $5$).

Can anyone prove it?

Answer 1

Even more is true. Every copy of $c_0$ in $C(K)$ for $K$ compact, metric is complemented by a projection of norm at most 2. Indeed, $C(K)$ is in this case separable (as $K$ is second-countable we may use the Stone–Weierstrass theorem to get the claim) and then we may apply Sobczyk's theorem.

Answer 2

Let $t_n \in K$ be a convergent sequence of distinct points with limit $t_\infty$ also distinct from all $t_n$, consider $C_0(K)=\lbrace f\in C(K): f(t_\infty)=0\rbrace$ and consider $P:C_0(K)\to c_0$, $f\mapsto (f(t_n))_{n\in\mathbb N}$. Conversely, choose peak functions $\varphi_n \in C(K)$ with disjoint supports and $\varphi_n(t_k)=\delta_{n,k}$ and define $I:c_0\to C_0(K)$, $(x_n)_{n\in\mathbb N} \mapsto \sum_{n=1}^\infty x_n \varphi_n$ (this is only formally a series, for each $t\in K$ at most one term of $\sum_{n=1}^\infty x_n \varphi_n(t)$ does not vanish). This should establish $c_0$ as a complemented subspace of $C_0(K)$. It remains to see that $C(K)\to C_0(K)$, $f\mapsto f-f(t_\infty)$ is a projector.