I am studying the following theorem (see image below) by Davidson. I just want to know where did we use the fact that the representation $\sigma$ is non-degenerate? It seems like the only time it was used was to show that there exists a minimal projection $E$ such that $\sigma(E)\neq 0.$ This was provided by Lemma I.10.1 in the text, which says that if $\mathcal{A}$ and $\sigma$ are both nonzero, then such a minimal projection exists. We know that $\sigma$ is nonzero because it is non-degenerate, but couldn't we just state the theorem using the weaker hypothesis that $\sigma$ is nonzero instead of non-degenerate?
I think you are right. Every time $\sigma$ is used, it is applied to the vector $f$. And you can always "un-degenerate" a representation $\sigma:A\to B(H)$ by changing its codomain to $\overline{\sigma(A)H}$.