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 Quantum dynamical semigroup and its asymptotic behaviors Bull. Korean Math. Soc. 2004 Vol. 41, No. 1, 189-198 Published online March 1, 2004 Veni Choi Yonsei University Abstract : 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 Full-Text :