TY - JOUR
T1 - E 0-semigroups of type II 0 and q-purity
T2 - Boundary weight maps of range rank one and two
AU - Jankowski, Christopher
AU - Markiewicz, Daniel
AU - Powers, Robert T.
N1 - Funding Information:
* Corresponding author. E-mail addresses: [email protected] (C. Jankowski), [email protected] (D. Markiewicz), [email protected] (R.T. Powers). 1 Partially supported by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev. 2 Research supported by grant 2008295 from the US–Israel Binational Science Foundation.
PY - 2012/4/1
Y1 - 2012/4/1
N2 - A CP-flow over a separable Hilbert space K is a continuous one-parameter semigroup of completely positive maps on B(KoL2(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 dimK<∞. 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.
AB - A CP-flow over a separable Hilbert space K is a continuous one-parameter semigroup of completely positive maps on B(KoL2(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 dimK<∞. 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.
KW - CP-flows
KW - CP-semigroups
KW - E -semigroups
KW - Q-Corners
KW - Q-Pure
UR - http://www.scopus.com/inward/record.url?scp=84856704834&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2011.12.022
DO - 10.1016/j.jfa.2011.12.022
M3 - Article
AN - SCOPUS:84856704834
SN - 0022-1236
VL - 262
SP - 3006
EP - 3061
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 7
ER -