E_0-semigroups and q-purity

Abstract

An E 0 -semigroup is called q-pure if it is a CP-flow and its set of flow subordinates is totally ordered by subordination. The range rank of a positive boundary weight map is the dimension of the range of its dual map. Let K be a separable Hilbert space. We describe all q-pure E 0 -semigroups of type II 0 which arise from boundary weight maps with range rank one over K. We also prove that no q-pure E 0 -semigroups of type II 0 arise from boundary weight maps with range rank two over K. In the case when K is finite-dimensional, we provide a criterion to determine if two boundary weight maps of range rank one over K give rise to cocycle conjugate q-pure E 0 -semigroups.

Aligned CP-Semigroups

“…Then by Corollary 3.3 of [Jan10] and the assumptions on g(x), we have that ω gives rise to an E 0 -semigroup of type II 0 . Once one applies Theorem 4.4 of [JMP11] to reconcile the earlier definition of q-purity with the one in this paper, we obtain from Lemma 4.3 and Proposition 5.2 of [Jan10] that ω gives rise to a q-pure unital CP-flow. Therefore by Theorem 4 it gives to a aligned E 0semigroup α λ .…”