Jones Index of a Quantum Dynamical Semigroup

Abstract

In this paper we consider a semigroup of completely positive maps τ=(τ t ,t≥0) with a faithful normal invariant state φ on a type-II 1 factor $\mathcal{A}_{0}$ and propose an index theory. We achieve this via a Kolmogorov’s type of construction for stationary Markov processes which naturally associate a nested family of isomorphic von-Neumann algebras. In particular this construction generalises well known Jones construction associated with a sub-factor of a type-II1 factor.