E0-semigroups of type II0 and q-purity: Boundary weight maps of range rank one and two

A CP-flow over a separable Hilbert space K is a continuous one-parameter semigroup of completely positive maps on B(K⊗L2(0,∞)) which is intertwined by the right shift semigroup, and CP-flows are obtained from boundary weight maps. In this paper we generalize concepts of q-purity considered previously, by defining an E0-semigroup to be q-pure if it is a CP-flow and its set of CP-flow subordinates is totally ordered by subordination. We provide a complete description of all q-pure E0-semigroups of type II0 arising from boundary weight maps with range rank one, and we provide a criterion to determine if two such boundary weight maps give rise to cocycle conjugate q-pure E0-semigroups when . We also show that boundary weight maps of range rank two do not give rise to q-pure E0-semigroups of type II0.