Jones Index of a Quantum Dynamical Semigroup

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 A₀\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.     

Extremal Unital Completely Positive Maps and Their Symmetries

We consider the set \(P_1({\mathcal A},{\mathcal M})\) (respectively \(CP_1({\mathcal A},{\mathcal M})\) of unital positive (completely) maps from a \(C^*\) algebra \({\mathcal A}\) to a von-Neumann sub-algebra \({\mathcal M}\) of \({\mathcal B}({\mathcal H})\), the algebra of bounded linear operators on a Hilbert space \({\mathcal H}\). We study the extreme points of the convex set \(P_1({\mathcal A},{\mathcal M})\) (\(CP_1({\mathcal A},{\mathcal M})\)) via their canonical lifting to the convex set of (unital) positive (completely) normal maps from \(\hat{{\mathcal A}}\) to \({\mathcal M}\), where \({\mathcal A}^{**}\) is the universal enveloping von-Neumann algebra over \({\mathcal A}\). If \({\mathcal A}={\mathcal M}\) then a (completely) positive map \(\tau \) admits a unique decomposition into a sum of a normal and a singular (completely) positive maps. Furthermore, if \({\mathcal M}\) is a factor then a unital complete positive map is a unique convex combination of unital normal and singular completely positive maps. We also used a duality argument to find a criteria for an element in the convex set of unital completely positive maps with a given faithful normal invariant state on \({\mathcal M}\) to be extremal. In our investigation, gauge symmetry in the minimal Stinespring representation of a completely positive map and Kadison theorem on order isomorphism played an important role.     

Pure inductive limit state and Kolmogorov's property

Let be a C^*-dynamical system where be a semigroup of injective endomorphism and ψ be an (λ_t) invariant state on the C^* subalgebra and is either non-negative integers or real numbers. The central aim of this exposition is to find a useful criteria for the inductive limit state canonically associated with ψ to be pure. We achieve this by exploring the minimal weak forward and backward Markov processes associated with the Markov semigroup on the corner von-Neumann algebra of the support projection of the state ψ to prove that Kolmogorov's property [A. Mohari, Markov shift in non-commutative probability, J. Funct. Anal. 199 (2003) 189–209] of the Markov semigroup is a sufficient condition for the inductive state to be pure. As an application of this criteria we find a sufficient condition for a translation invariant factor state on a one-dimensional quantum spin chain to be pure. This criteria in a sense complements criteria obtained in [O. Bratteli, P.E.T. Jorgensen, A. Kishimoto, R.F. Werner, Pure states on , J. Operator Theory 43 (1) (2000) 97–143; A. Mohari, Markov shift in non-commutative probability, J. Funct. Anal. 199 (2003) 189–209] as we could go beyond lattice symmetric states.     

Markov shift in non-commutative probability

We consider a class of quantum dissipative semigroup on a von-Neumann algebra which admits a normal invariant state. We investigate asymptotic behavior of the dissipative dynamics and their relation to that of the canonical Markov shift. In case the normal invariant state is also faithful, we also extend the notion of ‘quantum detailed balance’ introduced by Frigerio-Gorini and prove that forward weak Markov process and backward weak Markov process are equivalent by an anti-unitary operator.     

Translation Invariant Pure State on \otimes _{{\mathbb {Z}}}\!M_d({\mathbb {C}}) and Haag Duality

We prove Haag duality property of any translation invariant pure state on ${\mathcal B}= \otimes _{{\mathbb {Z}}}\!M_d({\mathbb {C}}), \;d \ge 2$ , where $M_d({\mathbb {C}})$ is the set of $d \times d$ dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on ${\mathcal B}$ .