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

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. ‖β_t(I)‖=1 and α_t−β_t 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 q-pure E₀-semigroups induced by CP-flows over a finite-dimensional Hilbert space K up to cocycle conjugacy.