Markov shift in non-commutative probability

Abstract

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.(H, S i , P, V i , Ω) is a Popescu system [BJKW], where P is the support projection of the state ψ Ω (X) =< Ω, XΩ > on the von-Neumann algebra π(O d ) ′′ , S i = π(s i ) and V i = P S i P . Let Q be the support projection of the state ψ Ω on the von-Neumann algebra {S I S * J : |I| = |J| < ∞} ′′ and A 0 = QS I S * J Q : |I| = |J| < ∞}. We also set l k = QS k Q for all 1 ≤ k ≤ d and define Markov semigroup τ on A 0 by τ (x) = i l i xl * i . The normal state ψ 0 , defined

Jones Index of a Quantum Dynamical Semigroup

“…Thus the result follows by von-Neumann density theorem. We also note that P is a sub-harmonic projection [10] for (α t : t ≥ 0) i.e. α t (P ) ≥ P for all t ≥ 0 and α t (P ) ↑ [A [0 ] as t ↑ ∞.…”

“…We have studied extensively asymptotic behavior of the dynamics (A 0 , τ t , φ 0 ) in [1] and Kolmogorov's property of the Markov semigroup introduced in [10] was explored to asymptotic behavior of the dynamics (A [0 , α t , φ). In particular this yields a criteria for the inductive limit state canonically associated with (A [0 , α t , φ) to be pure.…”

“…the projection on the fiber at 0 repeatedly. A simple proof follows once we use explicit formulas for F 0] and F [0 given in [10].…”

Pure inductive limit state and Kolmogorov's property

“…Thus asymptotic properties (ergodic, mixing) of the dynamics (A 0 , τ t , φ 0 ) is well determined by the asymptotic properties (ergodic, mixing, respectively) of the reduced dynamics (A p 0 , τ p t , φ p 0 ) provided y = 1. For more details we refer to [13]. In case φ 0 is faithful, normal and invariant for (τ t ), we recall [13] that G = {x ∈ A 0 :τ t τ t (x) = x, t 0} is von-Neumann sub-algebra of F = {x ∈ A 0 : τ t (x * )τ t (x) = τ t (x * x), τ t (x)τ t (x * ) = τ t (xx * ), ∀t 0} and the equality G = C is a sufficient condition for φ 0 to be strong mixing for (τ t ).…”

“…For more details we refer to [13]. In case φ 0 is faithful, normal and invariant for (τ t ), we recall [13] that G = {x ∈ A 0 :τ t τ t (x) = x, t 0} is von-Neumann sub-algebra of F = {x ∈ A 0 : τ t (x * )τ t (x) = τ t (x * x), τ t (x)τ t (x * ) = τ t (xx * ), ∀t 0} and the equality G = C is a sufficient condition for φ 0 to be strong mixing for (τ t ). Since the backward process [1] is related with the forward process via an anti-unitary operator we note that φ 0 is strongly mixing for (τ t ) if and only if same hold for (τ t ).…”