A Note on Eigenvalues of Liouvilleans

Abstract

Let L be the Liouvillean of an ergodic quantum dynamical system (\(\mathfrak{M}\), τ, ω). We give a new proof of the theorem of Jadczyk that eigenvalues of L are simple and form a subgroup of \(\mathbb{R}\). If ω is a (τ, β)-KMS state for some β≠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.