E0-semigroups of type II0and q-purity: Boundary weight maps of range rank one and two Academic Article uri icon

abstract

  • A CP-flow over a separable Hilbert space K is a continuous one-parameter semigroup of completely positive maps on B ( K ⊗ L 2 ( 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 E 0 -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 E 0 -semigroups of type II 0 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 E 0 -semigroups when dim K ∞ . We also show that boundary weight maps of range rank two do not give rise to q -pure E 0 -semigroups of type II 0 .

publication date

  • January 1, 2012