### Abstract

Let M be a von Neumann algebra with a von Neumann subalgebra M_{0}. If E is a conditional expectation (i.e., projection of norm one) from M into M_{0}, then any faithful normal state φ_{0} admits a natural extension φ_{0} {ring-operator} E with respect to E in the sense that E = E_{φ0·E•} If E_{ω} is only an ω-conditional expectation, then φ_{0} {ring-operator} E_{ω} is not always an extension of φ_{0}. This paper is devoted to the construction of an extension φ_{0} of φ_{0} generalizing the above situation for ω-conditional expectations, which leads also to a Radon-Nikodym theorem for ω-conditional expectation under suitable majorization condition.

Original language | English |
---|---|

Pages (from-to) | 9-24 |

Number of pages | 16 |

Journal | Pacific Journal of Mathematics |

Volume | 138 |

Issue number | 1 |

Publication status | Published - 1989 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*138*(1), 9-24.

**State extensions and a Radon-Nikodym theorem for conditional expectations on Von Neumann algebras.** / Cecchini, Carlo; Petz, D.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 138, no. 1, pp. 9-24.

}

TY - JOUR

T1 - State extensions and a Radon-Nikodym theorem for conditional expectations on Von Neumann algebras

AU - Cecchini, Carlo

AU - Petz, D.

PY - 1989

Y1 - 1989

N2 - Let M be a von Neumann algebra with a von Neumann subalgebra M0. If E is a conditional expectation (i.e., projection of norm one) from M into M0, then any faithful normal state φ0 admits a natural extension φ0 {ring-operator} E with respect to E in the sense that E = Eφ0·E• If Eω is only an ω-conditional expectation, then φ0 {ring-operator} Eω is not always an extension of φ0. This paper is devoted to the construction of an extension φ0 of φ0 generalizing the above situation for ω-conditional expectations, which leads also to a Radon-Nikodym theorem for ω-conditional expectation under suitable majorization condition.

AB - Let M be a von Neumann algebra with a von Neumann subalgebra M0. If E is a conditional expectation (i.e., projection of norm one) from M into M0, then any faithful normal state φ0 admits a natural extension φ0 {ring-operator} E with respect to E in the sense that E = Eφ0·E• If Eω is only an ω-conditional expectation, then φ0 {ring-operator} Eω is not always an extension of φ0. This paper is devoted to the construction of an extension φ0 of φ0 generalizing the above situation for ω-conditional expectations, which leads also to a Radon-Nikodym theorem for ω-conditional expectation under suitable majorization condition.

UR - http://www.scopus.com/inward/record.url?scp=84973999585&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84973999585&partnerID=8YFLogxK

M3 - Article

VL - 138

SP - 9

EP - 24

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -