$K_0$ of an inner automorphism

Question

An exercise in a book I am studying says

$K_0(\alpha) =id$ for every inner automorphism $\alpha$.

I am not sure why this is true, I suppose it has to do with the fact that inner automorphisms look like $Ad \; u$ for some unitary $u\in A$.

Answer 1

If $\alpha = Ad_u$, then for any projection $p$ we have $\alpha(p) = upu^*$. The projection $upu^*$ and $p$ are unitarily equivalent. Since unitarily equivalent projections define the same classes in $K$-theory $$[p] = [upu^*] = [\alpha(p)] = K_0(\alpha)([p]).$$

It now follows that $K_0(\alpha)$ is just the identity map on $K_0(A)$ since $K_0(\alpha)([p] - [q]) = [p] - [q]$.