Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$.
Consider an invariant state: $\omega\circ\tau^t\equiv\omega$
Then the dynamics is unitarily implementable: $$\pi_\omega\left[\tau^t(A)\right]=e^{itL}\pi_\omega[A]e^{-itL}$$
Moreover the Liouvillean is determined by: $L\Omega=0$
I'm wondering what other generators can occur: $L\Omega\neq0$
Does somebody have an explicit example on Fock space?