Markov shift in Non-commutative Probability-II
We study asymptotic behavior of a Markov semigroup on a von-Neumann algebra by exploring a maximal von-Neumann subalgebra where the Markov semigroup is an automorphism. This enables us to prove that strong mixing is equivalent to ergodic property for continuous time Markov semigroup on a type-I von-Neumann algebra with center completely atomic. For discrete time dynamics we prove that an aperiodic ergodic Markov semigroup on a type-I von-Neumann algebra with center completely atomic is strong mixing. There exists a tower of isomorphic von-Neumann algebras generated by the weak Markov process and a unique up to isomorphism minimal dilated quantum dynamics of endomorphisms associated with the Markov semigroup. The dilated endomorphism is pure in the sense of Powers if and only if the adjoint Markov semigroup satisfies Kolmogorov property. As an application of our general results we find a necessary and sufficient condition for a translation invariant state on a quantum spin chain to be pure. We also find a tower of type-II$_1$ factors canonically associated with the canonical conditional expectation on a sub-factor of a type-II$_1$ factor. This tower of factors unlike Jones's tower do not preserve index. This gives a sequence of Jones's numbers as an invariance for the inclusion of a finite sub-factor of a type-II$_1$ factor.