scholarly journals Measure theory over boolean toposes

2016 ◽  
Vol 163 (1) ◽  
pp. 1-21
Author(s):  
SIMON HENRY
Keyword(s):  
Time Evolution ◽  
Noncommutative Geometry ◽  
Principal Bundle ◽  
Von Neumann Algebras ◽  
Measure Theory ◽  
Topos Theory ◽  
Von Neumann ◽  
Theoretic Version

AbstractIn this paper we develop a notion of measure theory over boolean toposes reminiscent of the theory of von Neumann algebras. This is part of a larger project to study relations between topos theory and noncommutative geometry. The main result is a topos theoretic version of the modular time evolution of von Neumann algebras which take the form of a canonical $\mathbb{R}^{>0}$-principal bundle over any integrable locally separated boolean topos.

The Approximate Jordan-Hahn Decomposition

1989 ◽  
Vol 41 (6) ◽  
pp. 1124-1146 ◽  
Author(s):  
Gottfried T. Rüttimann
Keyword(s):  
Banach Space ◽  
Von Neumann Algebras ◽  
Convex Set ◽  
Measure Theory ◽  
Extreme Points ◽  
Foundations Of Quantum Mechanics ◽  
The Other ◽  
Orthomodular Poset ◽  
Geometrical Aspect ◽  
Von Neumann

Non-commutative measure theory embraces measure theory on cr-fields of subsets of a set, on projection lattices of von Neumann algebras or JBW-algebras and on hypergraphs alike [20], [27], [33], [37], [39], [40], [41]. Due to the unifying structure of an orthoalgebra concepts can easily be transferred from one branch to the other. Additional conceptual inpetus is obtained from the logico-probabilistic foundations of quantum mechanics (see [6], [19], [21]).In the late seventies the author studied the Jordan-Hahn decomposition of measures on orthomodular posets and certain graphs. These investigations revealed an interesting geometrical aspect of this decomposition in that the Jordan-Hahn property of the convex set of probability charges on a finite orthomodular poset can be characterized in terms of the extreme points of the unit ball of the Banach space dual of the base normed space of Jordan charges.

ON THE JLO COCYCLE AND ITS TRANSGRESSION IN ENTIRE CYCLIC COHOMOLOGY

2013 ◽  
Vol 10 (07) ◽  
pp. 1350037
Author(s):  
ALAN LAI
Keyword(s):  
Noncommutative Geometry ◽  
Von Neumann Algebras ◽  
Chern Character ◽  
Cyclic Cohomology ◽  
Character Formula ◽  
Type Ii ◽  
Von Neumann ◽  
Definition Of ◽  
Dixmier Trace ◽  
Math Phys

The JLO character formula due to Jaffe–Lesniewski–Osterwalder [Quantum K-theory: the Chern character, Commun. Math. Phys.112 (1988) 75–88] assigns to each Fredholm module a cocycle in entire cyclic cohomology. It descends to define a cohomological Chern character on K-homology. This paper extends the definition of the JLO character formula for Breuer–Fredholm modules, the modules that represent type II noncommutative geometry; and shows that the JLO character formula coincides with the Connes character formula [see M. Benameur and T. Fack, Type II noncommutative geometry. I. Dixmier trace in von Neumann algebras, Adv. Math.199 (2006) 29–87] at the level of entire cyclic cohomology.

Lectures on von Neumann Algebras

2019 ◽  
Author(s):  
Serban-Valentin Stratila ◽  
Laszlo Zsido
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann

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.

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