scholarly journals State extensions and a Radon-Nikodým theorem for conditional expectations on von Neumann algebras

1989 ◽  
Vol 138 (1) ◽  
pp. 9-24 ◽  
Author(s):  
Carlo Cecchini ◽  
Dénes Petz
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann ◽  
Conditional Expectations ◽  
Nikodym Theorem

scholarly journals Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

2021 ◽  
Author(s):  
Ivan Bardet ◽  
Ángela Capel ◽  
Cambyse Rouzé
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Spin Systems ◽  
Von Neumann ◽  
Strong Subadditivity ◽  
Finite Dimensional ◽  
Lattice Spin ◽  
Lattice Spin Systems ◽  
Conditional Expectations

AbstractIn this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. This generalisation, referred to as approximate tensorization of the relative entropy, consists in a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

scholarly journals The Radon-Nikodym theorem for von neumann algebras

1973 ◽  
Vol 130 (0) ◽  
pp. 53-87 ◽  
Author(s):  
Gert K. Pedersen ◽  
Masamichi Takesaki
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann ◽  
Nikodym Theorem

A UNIFIED APPROACH TO GENERALIZED CONDITIONAL EXPECTATIONS OPERATOR VALUED WEIGHTS AND RADON–NIKODYM DERIVATIVES ON VON NEUMANN ALGEBRAS

2002 ◽  
Vol 05 (03) ◽  
pp. 409-419
Author(s):  
STEFANO CAVALLARO ◽  
CARLO CECCHINI
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebras ◽  
Unified Approach ◽  
Von Neumann ◽  
Canonical Map ◽  
Conditional Expectations

Given two von Neumann algebras, ℳ and [Formula: see text] with [Formula: see text], and two normal semifinite faithful weights, φ and ψ on ℳ and [Formula: see text] respectively, we define a canonical map from {b ∈ ℳ+ | φ(b)< ∞} to the set of positive forms on the Hilbert space of the GNS representation of [Formula: see text] associated to ψ. We show that generalized conditional expectations, operator valued weights and Radon–Nikodym derivatives on von Neumann algebras can be obtained from particular cases of this canonical map.

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