Consider an ultraweakly continuous, completely positive generator $L$ of a Quantum Markov Semigroup that is defined on the trace class operators $\mathcal{L}_1$ on a countable infinite dimensional Hilbert space $\mathcal{H}$. Thus $L$ generates a unital completely positive set of maps $\Phi_t = e^{tL}$, which is norm continuous and satisfies $\Phi_t\Phi_s=\Phi_{t+s}$.
I.e. $L: \mathcal{L}_1\rightarrow \mathcal{L}_1$. I want to show that complete positivity is preserved if we add the identity artifically to the space. Here is what I came up with.
To generate a norm continuous semigroup $\Phi_t=e^{tL}$ we require $L$ to be ultraweakly continuous. Thus let $(I_n)_{n=1}^{\infty}$ be the sequence that has $n$ ones on the diagonal. Notice that $$Tr(I_nY) = \sum_{k=1}^n Y_{kk}\rightarrow \sum_{k=1}^{\infty}Y_{kk}=Tr(Y) = Tr(IY)\quad \forall Y\in\mathcal{L}_1 .$$ Hence, $I_n$ is a sequence that converges in the ultraweak topology. Since $L$ is ultraweakly continuous, we may conclude that $$Tr(L(I_n)Y)\rightarrow Tr(L(I)Y)=Tr(0Y)=0\quad \forall Y\in\mathcal{L}_1.$$ Where the last equality follows from the fact that $LI=0$ if $L$ is a generator of a QMS. We may now conclude that $\lim_{n\rightarrow \infty}L(I_n)=0$, because that is the only way this is true for all $Y\in\mathcal{L}_1$. Now that means that if $(X_{i,j})\in M_k(\mathcal{L}_1)$ with the identity artifically added to $\mathcal{L}_1$, so $(X_{i,j})$ is a matrix of operators, is positive, then $L((X_{i,j}))$ is positive. Suppose one of the operators in the $k$ by $k$ matrix is to be the identity. Then by the fact that if we used $I_n$ the result would be positive and the limit of $n$ to infinity yields the identity. Hence, by ultraweak continuity of $L$ we have that if we replaced $I_n$ by $I$ we would still have a positive matrix after applying $L$ by continuity of the inner product on $\mathcal{H}$, which allows us to pull the limit into the inner product, thus the operator $L$ defined on the trace class operators with the identity artifically added on is still completely positive.