In section 8.2.3 of Nielsen and Chuang, there is a derivation of the operator-sum representation as follows:
$\xi (\rho) = tr_{env} \left[ U (\rho \otimes \rho_{env}) U^{\dagger}\right] \tag{1}$
And $\rho_{env}=\left|e_{0}\right>\left<e_{0}\right|$ which is the initial state of the environment and $\left|e_{k}\right>$ is an orthonormal basis for the (finite dimensional) state space of the environment. According to the textbook, the equation (1) becomes
$\xi (\rho) = \sum_{k} \left< e_{k}\right|U \left[ \rho \otimes \left| e_{0} \right> \left< e_{0} \right| \right] U^{\dagger} \left| e_{k} \right> \tag{2}$
$ = \sum_{k} E_{k}\rho E_{k}^{\dagger} \tag{3}$
where $E_{k}=\left<e_{k}\right|U\left|e_{0}\right>$
However, I do not understand how to get (2) from (1), and also (3) from (2). Could anyone kindly write the missing steps? Thanks.