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

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

“…This result runs counter to intuition in that a map φ(A) = tr(A)I (which is in a sense the least pure completely positive map being the average of all pure maps) yields a q-pure weight map. In [JMP11] we classified all the range rank one q-weight maps over a finite dimensional Hilbert space K up to cocycle conjugacy and in this paper we show that in the case of q-pure q-weight maps over a finite dimensional Hilbert space are all cocycle conjugate to q-pure range rank one q-weight maps.…”

“…In [JMP11] the above theorem was proved in the finite dimensional case. In this paper we will only make use of this theorem in the case where K 1 and K 2 are finite dimensional.…”

“…1 2 is a q-pure q-weight map so that φ t = ω| t Λ. The next result except for the last sentence first appeared in [Jan10] and then in [JMP11] but because of its importance and the shortness of the proof we include it here.…”

“…In this terminology we see that an unbounded q-weight map ω over C is q-pure if and only if it is strictly infinite. In [JMP11] we discussed range rank one q-weight maps over C p and our results are summarized in the following theorems.…”

$E_0$-semigroups: around and beyond Arveson's work

“…See [3], [44], [45], [46], [49] for recent progress in this approach. There are several E 0 -semigroups of type II whose gauge groups are known.…”