E 0-semigroups of type II 0 and q-purity: Boundary weight maps of range rank one and two

Christopher Jankowski; Daniel Markiewicz; Robert T. Powers

doi:10.1016/j.jfa.2011.12.022

E ₀-semigroups of type II ₀ and q-purity: Boundary weight maps of range rank one and two

Christopher Jankowski, Daniel Markiewicz, Robert T. Powers

Department of Mathematics

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	American English
Pages (from-to)	3006-3061
Number of pages	56
Journal	Journal of Functional Analysis
Volume	262
Issue number	7
DOIs	https://doi.org/10.1016/j.jfa.2011.12.022
State	Published - 1 Apr 2012

Keywords

CP-flows
CP-semigroups
E -semigroups
Q-Corners
Q-Pure

All Science Journal Classification (ASJC) codes

Analysis

Access to Document

10.1016/j.jfa.2011.12.022

Cite this

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title = "E 0-semigroups of type II 0 and q-purity: Boundary weight maps of range rank one and two",

abstract = "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.",

keywords = "CP-flows, CP-semigroups, E -semigroups, Q-Corners, Q-Pure",

author = "Christopher Jankowski and Daniel Markiewicz and Powers, {Robert T.}",

note = "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.",

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doi = "10.1016/j.jfa.2011.12.022",

language = "American English",

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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.

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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.

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