Proposition 1. Let $(\pi, \widehat{H}, V)$ be the minimal Stinespring dilation of a contractive completely positive map $\phi: A \rightarrow B$. Then, there exists a *-homomorphism $$\rho: \phi(A)^{'} \rightarrow \pi(A)^{'}\subset B(\widehat{H})$$ (the $\phi(A)^{'}$ denotes the commutant of $\phi(A)^{'}$) such that $$\phi(a)x=V^{\ast}\pi(a)\rho(x)V$$ for every $a\in A$ and $x\in \phi(A)^{'}$.
Proof. For $x\in \phi(A)^{'}$, we define a linear operator $\rho(x)$ on the span of $\pi(A)VH$ by $$\rho(x)(\sum_{i}\pi(a_{i})V\xi_{i})=\sum_{i}\pi(a_{i})Vx\xi_{i}.$$ Once we prove that $\rho(x)$ is well-defined and bounded for every $x\in \phi(A)^{'}$, it is not hard to check that $\rho$ gives rise to a $*$-representation of $\phi(A)^{'}$ on $\widehat{H}$ such that $\rho(\phi(A)^{'}) \subset\pi(A)^{'}$ and $\rho(a)x=V^{*}\pi(a)\rho(x)V$ for every $a\in A$ and $x\in \rho(A)^{'}$. ...................
I have two question about the quotation above:
How to explain $\rho$ is a *-representation of $\phi(A)^{'}$?
How to compute the equation $\rho(a)x=V^{*}\pi(a)\rho(x)V$?
We have $$ \rho(x)\rho(y)\pi(a)V\xi=\rho(x)\pi(a)Vy\xi=\pi(a)Vxy\xi=\rho(xy)\pi(a)V\xi, $$ so $\rho(xy)=\rho(x)\rho(y)$. The computation for linearity is similar. And $$ \langle \rho(x)^*\pi(a)V\xi,\pi(b)V\eta\rangle=\langle \pi(a)V\xi,\rho(x)\pi(b)V\eta\rangle =\langle \pi(a)V\xi,\pi(b)Vx\eta\rangle=\langle V^*\pi(b)^*\pi(a)V\xi,x\eta\rangle=\langle\phi(b^*a)\xi,x\eta\rangle=\langle x^*\phi(b^*a)\xi,\eta\rangle=\langle \phi(b^*a)x^*\xi,\eta\rangle =\langle V\pi(b)^*\pi(a)Vx^*\xi,\eta\rangle=\langle \pi(a)Vx^*\xi,\pi(b)V\eta\rangle=\langle \rho(x^*)\pi(a)V\xi,\pi(b)V\eta\rangle, $$ so $\rho(x^*)=\rho(x)^*$. All this works because, being well-defined and bounded, we can extend $\rho(x)$ to all of $\widehat H$.
For $\xi\in H$, $$ \phi(a)x\xi=V^*\pi(a)Vx\xi=V^*\rho(x)\pi(a)V\xi=V^*\pi(a)\rho(x)V\xi. $$