Bulletin of the
Korean Mathematical Society BKMS

In this study we consider quantum dynamical semigroup with a normal faithful invariant state. A quantum dynamical semigroup $\alpha = \{\alpha _t \}_{t \geq 0}$ is a class of linear normal identity-preserving mappings on a von Neumann algebra $\mathcal M $ with semigroup property and some positivity condition. We investigate the asymptotic behaviors of the semigroup such as ergodicity or mixing properties in terms of their eigenvalues under the assumption that the semigroup satisfies positivity. This extends the result of \cite{Wa} which is obtained under the assumption that the semigroup satisfy 2-positivity.

Keywords: quantum dynamical semigroup, positivity, Schwarz inequality, Jordan product, ergodicity, weak mixing

MSC numbers: 46L55, 82C10