My question is related to K-Theory of $C(X)$ for $X$ totally disconnected
Question: Does anyone know an example of a compact Hausdorff space $X$ (that is not totally disconnected) where $K_0(C(X))$ is not isomorphic to $C(X,\mathbb{Z})$?
I made the blunder of "proving" injectivity with the following faulty argument: the dimension map $\mathrm{dim}$ on $P_n(C(X))=\{p\in M_n(C(X)):p=p^*=p^2\}$ satisfies $$\mathrm{dim}\thinspace p = 0 \iff p=0_n$$ Therefore injectivity holds. The reason this argument fails is because the kernel being trivial implies injectivity for groups (not monoids)---$P_\infty(C(X))=\bigcup_{n\ge 1}P_n(C(X))$ is a monoid with binary operation $\oplus$ which I forgo defining.
I am new to K-Theory, but please feel free to be candid. There must be some sensitivity to the parameter $x\in X$ since $K_0(C(\{x\}))\cong K_0(\mathbb{C})=\mathbb{Z}\cong C(\{x\},\mathbb{Z})$.