scholarly journals The local index formula in semifinite von Neumann algebras II: The even case

2006 ◽  
Vol 202 (2) ◽  
pp. 517-554 ◽  
Author(s):  
Alan L. Carey ◽  
John Phillips ◽  
Adam Rennie ◽  
Fyodor A. Sukochev
Keyword(s):  
Von Neumann Algebras ◽  
Index Formula ◽  
Local Index ◽  
Von Neumann

scholarly journals The local index formula in semifinite Von Neumann algebras I: Spectral flow

2006 ◽  
Vol 202 (2) ◽  
pp. 451-516 ◽  
Author(s):  
Alan L. Carey ◽  
John Phillips ◽  
Adam Rennie ◽  
Fyodor A. Sukochev
Keyword(s):  
Von Neumann Algebras ◽  
Index Formula ◽  
Spectral Flow ◽  
Local Index ◽  
Von Neumann

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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