Consider $C_0$semigroup $S_t:\mathscr{B(H)} \to \mathscr{B(H)}$ with generator $U$. Now define $P_t:\mathscr{B_1(H)} \to \mathscr{B_1(H)}$ where $P_t(\rho)=S_t\rho S_t^*$. How can I prove $P_t$ to be $C_0$semigroup and the generator of $P_t$ is given by $A$, where $A(\rho)=U\rho+\rho U^*$?

The semigroups you construct is in general only weak* continuous. Are you looking for socalled implemented semigroups? See for example this paper. 


Both the semigroup law and strong continuity follow directly from the semigroup law satisfied by $S_t$ and ${S_t}^*$ and by their strong continuity (here it is fundamental that $\mathcal H$ is a Hilbert space, strong continuity of the adjoint semigroup being false in the case of general Banach spaces). Also determining the generator is just an easy exercise in differentiation of a vectorvalued function of one real variable (applying the chain rule). 

