Conditional expectations in complete normed complex algebras satisfying the von Neumann inequality

2021 ◽  
pp. 125563
Author(s):  
Ángel Rodríguez Palacios
Keyword(s):  
Von Neumann ◽  
Conditional Expectations

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.

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.

scholarly journals SMOOTH PATHS OF CONDITIONAL EXPECTATIONS

2011 ◽  
Vol 22 (07) ◽  
pp. 1031-1050
Author(s):  
ESTEBAN ANDRUCHOW ◽  
GABRIEL LAROTONDA
Keyword(s):  
Conditional Expectation ◽  
Von Neumann Algebra ◽  
Boundedness Condition ◽  
Derivative Operator ◽  
Von Neumann ◽  
Conditional Expectations ◽  
Finite Trace

Let [Formula: see text] be a von Neumann algebra with a finite trace τ, represented in [Formula: see text], and let [Formula: see text] be sub-algebras, for t in an interval I (0 ∈ I). Let [Formula: see text] be the unique τ-preserving conditional expectation. We say that the path t ↦ Et is smooth if for every [Formula: see text] and [Formula: see text], the map [Formula: see text] is continuously differentiable. This condition implies the existence of the derivative operator [Formula: see text] If this operator satisfies the additional boundedness condition, [Formula: see text] for any closed bounded subinterval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras [Formula: see text] are *-isomorphic. More precisely, there exists a curve [Formula: see text], t ∈ I of unital, *-preserving linear isomorphisms which intertwine the expectations, [Formula: see text] The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps [Formula: see text] onto [Formula: see text]. We show that this restriction is a multiplicative isomorphism.

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