Classification of q-pure q-weight maps over finite dimensional Hilbert spaces

Abstract

An E0-semigroup of B(H) is a one parameter strongly continuous semigroup of *endomorphisms of B(H) that preserve the identity. Every E0-semigroup that possesses a strongly continuous intertwining semigroup of isometries is cocycle conjugate to an E0-semigroup induced by the Bhat induction of a CP -flow over a separable Hilbert space K. We say an E0-semigroup α is q-pure if the CP -subordinates β of norm one (i.e. βt(I) = 1 and αt − βt is completely positive for all t ≥ 0) are totally ordered in the sense that if β and γ are two CP -subordinates of α of norm one, then β ≥ γ or γ ≥ β. This paper shows how to construct and classify all qpure E0-semigroups induced by CP -flows over a finite-dimensional Hilbert space K up to cocycle conjugacy.