This example/problem is from Valette's notes on the Baum-Connes conjecture (p. 45).
The exercise is to prove that the (trivially equivariant) $K$-homology group $K_1(S^1)$ is $\mathbb{Z}$. For this, given a Fredholm module $\alpha=(\mathcal{H},F,\pi)$ over $C(S^1)$ (i.e. $F^2=\text{id}$ and $\pi$ is a *-representaton of $C(S^1)$ on the Hilbert space $\mathcal{H}$) one sees that for any $f\in C(S^1)$, the operator $\alpha(f)=P\pi(f)P$ (where $P=(1+F)/2$ is a projection) is Fredholm, so that an index map may be defined as $$\text{index}(\alpha)=\text{dim Ker}\ \alpha(z)-\text{dim coKer}\ \alpha(z)$$
where $z$ is the element in $C(S^1)$ given by $z\mapsto z$.
I could see that this is well-defined (i.e. independent of the module chosen within an equivalence class) and surjective. But I can't prove the injectivity.
That is, if $\text{index}(\alpha)=0$ I want to see that $F$ (or actually $\alpha(z)$) are homotopic to a Fredholm operator $F'$ that satisfies $F'^2=\text{id}$ and $F'\pi(f)=\pi(f)F'$.
All I can see is that, after compact perturbation, $\alpha(z)$ becomes an invertible operator $G$ from the closed subspace $P\mathcal{H}$ to itself such that $[\pi(f),G]$ is compact and $G^2-\text{id}$ is compact.