In section 8 of this paper http://arxiv.org/abs/math/0611833v3 the author proves the following: If $E$ is a reflexive Banach space, $G$ a locally compact group and $\pi:L_1(G)\to\mathcal{B}(E)$ a norm-decreasing algebra morphism, then there exists a one-complemented subspace $F$ of $E$ and a group representation $\sigma:G\to\mathcal{B}(F)$ such that $\pi(f)$ restricted to $F$ is given by $\int f(s)\sigma(s)\,ds$.

I have two questions concerning the proof:

1. He takes an approximate identity $(e_{\alpha})$ in $L_1(G)$ of bound 1 and defines $\sigma:G\to\mathcal{B}(E)$ by \begin{equation} \langle\mu,\sigma(s)x\rangle=\lim_{\alpha}\langle\mu,\pi(s\cdot e_{\alpha})x\rangle\qquad (x\in E, \mu\in E^*) \end{equation} where $(s\cdot f)(t)=f(s^{-1}t)$.

Why does that limit exist for every $x\in E$?

2. Then there is a subspace $F$ such that, by restriction, $\sigma:G\to\mathcal{B}(F)$ becomes a group representation. Moreover, there exists a contractive projection $P:E\to F$ such that $P\pi(f)P=\pi(f)$ for all $f\in L_1(G)$.

Why does this contractive projection exist?

(He later concludes that $F=\{\pi(f)x\ :\ f\in L_1(G), x\in E\}$ thanks to the Cohen factorisation theorem)