scholarly journals Free products of finite dimensional and other von Neumann algebras with respect to non-tracial states

1996 ◽  
pp. 41-88 ◽  
Author(s):  
Ken Dykema
Keyword(s):  
Von Neumann Algebras ◽  
Free Products ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Tracial States

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.

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

W*-superrigidity for wreath products with groups having positive first ℓ2-Betti number

2015 ◽  
Vol 26 (01) ◽  
pp. 1550003 ◽  
Author(s):  
Mihaita Berbec
Keyword(s):  
Wreath Product ◽  
Betti Number ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Wreath Products ◽  
Hyperbolic Groups ◽  
Free Products ◽  
Von Neumann ◽  
Product Group ◽  
Group Von Neumann Algebra

In [M. Berbec and S. Vaes, W*-superrigidity for group von Neumann algebras of left–right wreath products, Proc. London Math. Soc.108 (2014) 1116–1152] we have proven that, for all hyperbolic groups and for all nontrivial free products Γ, the left–right wreath product group 𝒢 ≔ (ℤ/2ℤ)(Γ) ⋊ (Γ × Γ) is W*-superrigid, in the sense that its group von Neumann algebra L𝒢 completely remembers the group 𝒢. In this paper, we extend this result to other classes of countable groups. More precisely, we prove that for weakly amenable groups Γ having positive first ℓ2-Betti number, the same wreath product group 𝒢 is W*-superrigid.

scholarly journals Compressions of free products of von Neumann algebras

2000 ◽  
Vol 316 (1) ◽  
pp. 61-82 ◽  
Author(s):  
Kenneth J. Dykema ◽  
Florin Radulescu
Keyword(s):  
Von Neumann Algebras ◽  
Free Products ◽  
Von Neumann

COCYCLE CONJUGACY OF ONE PARAMETER AUTOMORPHISM GROUPS OF AFD FACTORS OF TYPE III

2002 ◽  
Vol 13 (06) ◽  
pp. 579-603 ◽  
Author(s):  
UN KIT HUI
Keyword(s):  
Automorphism Group ◽  
Von Neumann Algebras ◽  
Automorphism Groups ◽  
Crossed Product ◽  
Von Neumann ◽  
Type Iii ◽  
Finite Dimensional ◽  
Cocycle Conjugacy

We classify, up to cocycle conjugacy, one-parameter automorphism groups on an approximately finite dimensional (AFD) factor ℳ of type III with trivial Connes spectrum. Our goal is to find the complete cocycle conjugacy invariants for one-parameter automorphism groups on ℳ. We also study the relations between the flow of weights of ℳ and that of the crossed product ℳ ⋊α ℝ of ℳ by a one-parameter automorphism group α with Γ(α) = {0}. Moreover, we also study model realizations. "Model realizations" means that given certain commutative data, they can be realized as the complete cocycle conjugacy invariants of centrally free and centrally ergodic one-parameter automorphism groups on some properly infinite AFD von Neumann algebras.

scholarly journals Complete Gradient Estimates of Quantum Markov Semigroups

2021 ◽  
Vol 387 (2) ◽  
pp. 761-791
Author(s):  
Melchior Wirth ◽  
Haonan Zhang
Keyword(s):  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Wasserstein Distance ◽  
Gradient Estimate ◽  
Gradient Estimates ◽  
Markov Semigroups ◽  
Free Products ◽  
Von Neumann ◽  
Symmetric Quantum ◽  
Poisson Type

AbstractIn this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.

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