scholarly journals The Birman–Schwinger principle in von Neumann algebras of finite type

2007 ◽  
Vol 247 (2) ◽  
pp. 492-508 ◽  
Author(s):  
Vadim Kostrykin ◽  
Konstantin A. Makarov ◽  
Anna Skripka
Keyword(s):  
Finite Type ◽  
Von Neumann Algebras ◽  
Von Neumann

Normal AW*-algebras

1980 ◽  
Vol 85 (1-2) ◽  
pp. 137-141 ◽  
Author(s):  
J. D. Maitland Wright
Keyword(s):  
Finite Type ◽  
Partially Ordered Set ◽  
Von Neumann Algebras ◽  
Ordered Set ◽  
Von Neumann ◽  
Partially Ordered

SynopsisIn recent years it has become clear that AW*-algebras can be much more pathological and unlike von Neumann algebras than was originally expected. When AW*-algebras are monotone complete, then the work of Kadison and Pederson shows that a particularly smooth and elegant theory can be developed. A technically weaker requirement on an AW*-algebra is that it be “normal”. This condition, which says that the lattice of projections is embedded in a well-behaved way in the partially ordered set of all self-adjoint elements, can sometimes be used as a substitute for monotone completeness. In this note we prove that when an AW*-algebra is of finite type (that is x*x = 1 implies xx* = 1) then it is normal.

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.

Sign in / Sign up

Export Citation Format

Share Document