Jones index of a quantum dynamical semigroup

In this paper we consider a completely positive map τ = (τ t , t ≥ 0) with a faithful normal invariant state φ on a type-II 1 factor A 0 and propose an index theory. We achieve this via a more general Kolmogorov's type of construction for stationary Markov processes which naturally associate a nested isomorphic von-Neumann algebras. In particular this construction generalizes well known Jones construction associated with a sub-factor of type-II 1 factor. 2 1 Introduction: Let τ = (τ t , t ≥ 0) be a semigroup of identity preserving completely positive normal maps [Da,BR] on a von-Neumann algebra A 0 acting on a separable Hilbert space H 0 , where either the parameter t ∈ R + , the set of positive real numbers or Z + , the set of positive integers. In case t ∈ R + , i.e. continuous, we assume that for each x ∈ A 0 the map t → τ t (x) is continuous in the weak * topology. Thus variable t ∈ IT + where IT is either IR or IN. We assume further that (τ t) admits a normal invariant state φ 0 , i.e. φ 0 τ t = φ 0 ∀t ≥ 0. As a first step following well known Kolmogorov's construction of stationary Markov processes, we employ GNS method to construct a Hilbert space H and an increasing tower of isomorphic von-Neumann type−II factors {A [t : t ∈ R or Z} generated by the weak Markov process (H, j t , F t] , t ∈ R or Z, Ω) [BP,AM] where j t : A 0 → A [t is an injective homomorphism from A 0 into A [0 so that the projection F t] = j t (I) is the cyclic space of Ω generated by {j s (x) : −∞ < s ≤ t, x ∈ A 0 }. The tower of increasing isomorphic von-Neumann algebras {A [t , t ∈ R or Z} are indeed a type-II ∞ factor if and only if τ is not an endomorphism. In any case the projection j 0 (I) is a finite projection in A [−t for all t ≤ 0. In particular we also find an increasing tower of type-II 1 factors {M t : t ≥ 0} defined by M t = j 0 (I)A [−t j 0 (I). Thus …