A Note on Eigenvalues of Liouvilleans
Résumé
Let L be the Liouvillean of an ergodic quantum dynamical system $(\mathfrak{M} ,\tau,\omega)$. We give a new proof of the theorem of Jadczyk that eigenvalues of L are simple and form a subgroup of $\mathbb{R}$ . If $\omega$ is a $(\tau, \beta)$-KMS state for some $\beta>0$ we show that this subgroup is trivial, namely that zero is the only eigenvalue of L. Hence, for KMS states ergodicity is equivalent to weak mixing.