scholarly journals Type decomposition in posets

2016 ◽  
Vol 66 (5) ◽  
Author(s):  
Tristan Bice
Keyword(s):  
Von Neumann Algebras ◽  
Classical Type ◽  
Von Neumann ◽  
Decomposition Theory ◽  
Type Decomposition

AbstractMotivated by the classical type decomposition of von Neumann algebras, and various more recent extensions to other structures, we develop a type decomposition theory for general posets.

Type Decomposition and the Rectangular AFD Property for W*-TRO’s

2004 ◽  
Vol 56 (4) ◽  
pp. 843-870 ◽  
Author(s):  
Zhong-Jin Ruan
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Space ◽  
Dual Operator ◽  
Von Neumann ◽  
Direct Sum ◽  
Cardinal Numbers ◽  
Finite Dimensional ◽  
Dimensional Property ◽  
Type Decomposition

AbstractWe study the type decomposition and the rectangular AFD property for W*-TRO’s. Like von Neumann algebras, every W*-TRO can be uniquely decomposed into the direct sum of W*- TRO's of type I, type II, and type III. We may further considerW*-TRO's of type Im,n with cardinal numbers m and n, and considerW*-TRO's of type IIλ,μ with λ, μ = 1 or ∞. It is shown that every separable stable W*-TRO (which includes type I∞, ∞, type II∞, ∞ and type III) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for W*-TRO’s. One of our major results is to show that a separable W*-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular space (equivalently, a rectangular space).

The role of the algebraic structure in Wold-type decomposition

2021 ◽  
Vol 33 (4) ◽  
pp. 1033-1049
Author(s):  
G. A. Bagheri Bardi ◽  
Zbigniew Burdak ◽  
Akram Elyaspour
Keyword(s):  
Linear Algebra ◽  
Algebraic Structure ◽  
Von Neumann Algebras ◽  
Algebraic Approach ◽  
Von Neumann ◽  
Weak Operator ◽  
Commuting Pairs ◽  
Decomposition Theorems ◽  
Type Decomposition

Abstract In recent works [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, Wold-type decompositions in Baer ∗ \ast -rings, Linear Algebra Appl. 539 2018, 117–133] and [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, The role of algebraic structure in the invariant subspace theory, Linear Algebra Appl. 583 2019, 102–118], the algebraic analogues of the three major decomposition theorems of Wold, Nagy–Foiaş–Langer and Halmos–Wallen were established in the larger category of Baer * {*} -rings. The results have their versions for commuting pairs in von Neumann algebras. In the corresponding proofs, both norm and weak operator topologies are heavily involved. In this work, ignoring topological structures, we give an algebraic approach to obtain them in Baer * {*} -rings.

Banach–Mazur stability of von Neumann algebras

2020 ◽  
pp. 1-26
Author(s):  
Jean Roydor
Keyword(s):  
Von Neumann Algebra ◽  
Hochschild Cohomology ◽  
Von Neumann Algebras ◽  
Cohomology Groups ◽  
Von Neumann ◽  
Type Decomposition

We initiate the study of perturbation of von Neumann algebras relatively to the Banach–Mazur distance. We first prove that the type decomposition is continuous, i.e. if two von Neumann algebras are close, then their respective summands of each type are close. We then prove that, under some vanishing conditions on its Hochschild cohomology groups, a von Neumann algebra is Banach–Mazur stable, i.e. any von Neumann algebra which is close enough is actually Jordan ∗-isomorphic. These vanishing conditions are possibly empty.

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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