Convolution product in an étale groupoid

Question

I am going through Sims - Étale groupoids and their $C^*$ algebras and at Lemma 3.1.4. the author says that $f^**f\in C_c(G^{(0)})$ is supported on $s(supp(f))$ and $(f^**f)(s(\gamma))=|f(\gamma)|^2$ for all $\gamma\in supp(f)$. He says that it follows from the convolution formula $$(f*g)(\gamma)=f(\alpha)g(\beta)$$

I couldn't see how. If $\gamma=\alpha\beta\in UV$ then $(f^**f)(\gamma)=f^*(\alpha)f(\beta)=\overline{f(\alpha^{-1})}f(\beta)$

I tried to think on the examples provided but they don't seem to be related. For example, Example 3.1.5 has a product that is not convolution.

Is there any reference that helps understand these computations?

Answer 1

In the lemma he is supposing that $f$ is supported on a bisection $U$. Then by definition $f^*$ is supported on $U^{-1}$. If $\gamma$ is in the support of $f^**f$ then $\gamma=\alpha\beta$ with $\alpha\in U^{-1}$ and $\beta\in U$. Since $U$ is a bisection the source and range maps are injective on $U$ and you must have $\alpha=\beta^{-1}$. So $\gamma =s(\beta)$ and $f(s(\beta))=\overline{f(\beta)}f(\beta)$. Hope this is clear.