$K_0(C_0(X, A))$ , when X is compact and contractible.

Question

Let A be a $C^{*}$-algebra and $B = C_0(X, A)$ be the set of all continuous functions from a locally compact Hausdorff space $X$ to $A$, vanishing at infinity. Prove that

$K_0(B) \cong K_0(A)$ and there is a deformation retract from $B$ to $A$ when $X$ is compact and contractible.

Can anybody help me to prove that?

Answer 1

If $X$ is compact then $C_0(X,A)=C(X,A)$ and $A$ embeds into $C(X,A)$ via the constant functions. If $X$ is contractible then let $h:[0,1]\times X\to X$ be a contraction and define: $$\varphi_t :C(X,A)\to C(X,A), \qquad f\mapsto[x\mapsto f(h(t,x))]$$ clearly $\varphi_t$ is a $*$-morphism for every $t\in[0,1]$ and for any $f\in C(X,A)$ the map $t\mapsto \varphi_t(f)$ is continuous, hence $\varphi$ is a homotopy to the constant functions, which we are identified with $A$.