Suppose $s_1,s_2,\ldots, s_n$ are generators of Cuntz algebra $\mathcal{O}_n$,let $\phi$ be a unital endomorphism on $\mathcal{O}_n$,show that $\phi=\phi_u$ for some unitary in $\mathcal{O}_n$,where $\phi_u(s_i)=us_i$ for $i=1,2,\ldots,n$.
I tried to let $u=\sum_{i=1}^{n}\phi(s_i)s_i^*$,when $n=2$,it is easy to verify the above element is unitary in $\mathcal{O}_2$ by using the fact: $s_1s_1^*+s_2s_2^*=1=s_1^*s_1=s_2^*s_2$.
I wonder whether $u$ is unitary in $\mathcal{O}_n$?
Yes it's unitary.
Use that $s_i^*s_j = \delta_{ij}$, then for $u = \sum_i \phi(s_i)s_i^*$ and $u^* = \sum_i s_i\phi(s_i^*)$ you can check just as in the case of $n=2$:
\begin{align*} \left(\sum_i s_i\phi(s_i^*)\right)\left(\sum_j \phi(s_j)s_j^*\right) &= \sum_i s_i\phi(s_i^*s_i)s_i^* = \sum_i s_is_i^* = 1,\\ \left(\sum_j \phi(s_j)s_j^*\right)\left(\sum_i s_i\phi(s_i^*)\right) &= \sum_j \phi(s_j)s_j^*s_j\phi(s_j^*) = \sum \phi(s_j)\phi(s_j^*) = 1. \end{align*}
