K-theory of $C_0(X)$

Question

Suppose that $X$ is some contractible space. I want to determine the K-theory of $C_0(X)$, i.e. the continuous functions on $X$ which vanish at infinity. But I do not know where to begin.

Answer 1

You need some more assumptions in order to get ahead.

For example, if $X$ is compact, then $C_0(X)=C(X)$. Indeed, $$ C_0(X)\equiv \{f\in C(X)\,|\,\forall\varepsilon\in\mathbb{R}_{>0},\,f^{-1}\left(\mathbb{C}-B_{\varepsilon}\left(0_\mathbb{C}\right)\right)\in Compact\left(X\right)\}$$ However, if $f$ is continuous, since $\mathbb{C}-B_{\varepsilon}\left(0_\mathbb{C}\right)$ is closed, $f^{-1}\left(\mathbb{C}-B_{\varepsilon}\left(0_\mathbb{C}\right)\right)$ is closed. But closed subsets of compact sets are compact, implying that for $X$ compact, $C_0(X)=C(X)$.

Then you are to compute $K_n(C(X))$. By homotopy invariance of the $K_n$ functor the contractibility of $X$ gives you $K_n(\mathbb{C})$ which is $0$ or $\mathbb{Z}$ depending on whether $n$ is odd or even respectively, as is well known.