scholarly journals A description of amalgamated free products of finite von Neumann algebras over finite-dimensional subalgebras

2010 ◽  
Vol 43 (1) ◽  
pp. 63-74 ◽  
Author(s):  
Ken Dykema
Keyword(s):  
Von Neumann Algebras ◽  
Free Products ◽  
Von Neumann ◽  
Amalgamated Free Products ◽  
Finite Dimensional ◽  
Finite Von Neumann Algebras

scholarly journals Radial multipliers on amalgamated free products of II1-factors

2014 ◽  
Vol 25 (03) ◽  
pp. 1450026
Author(s):  
Sören Möller
Keyword(s):  
Free Product ◽  
Von Neumann Algebras ◽  
Hankel Matrix ◽  
Trace Class ◽  
Free Products ◽  
Von Neumann ◽  
Amalgamated Free Products ◽  
Amalgamated Free Product ◽  
Radial Multipliers ◽  
Completely Bounded Map

Let ℳi be a family of II1-factors, containing a common II1-subfactor 𝒩, such that [ℳi : 𝒩] ∈ ℕ0 for all i. Furthermore, let ϕ: ℕ0 → ℂ. We show that if a Hankel matrix related to ϕ is trace-class, then there exists a unique completely bounded map Mϕ on the amalgamated free product of the ℳi with amalgamation over 𝒩, which acts as a radial multiplier. Hereby, we extend a result of Haagerup and the author for radial multipliers on reduced free products of unital C*- and von Neumann algebras.

scholarly journals Correspondences, von Neumann Algebras and Holomorphic L2 Torsion

2000 ◽  
Vol 52 (4) ◽  
pp. 695-736 ◽  
Author(s):  
A. Carey ◽  
M. Farber ◽  
V. Mathai
Keyword(s):  
Complex Manifold ◽  
Von Neumann Algebras ◽  
Compact Complex Manifold ◽  
Dolbeault Cohomology ◽  
Hermitian Metrics ◽  
Variation Formula ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Finite Von Neumann Algebras

AbstractGiven a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic L2 torsion, which lies in the determinant line of the twisted L2 Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von Neumann algebras as developed in [CFM]. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute ametric variation formula for the holomorphic L2 torsion, which shows that it is not in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic L2 torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles.

Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

2021 ◽  
pp. 1-54 ◽  
Author(s):  
Michael Brannan ◽  
Li Gao ◽  
Marius Junge
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Beltrami Operator ◽  
Free Products ◽  
Von Neumann ◽  
Amalgamated Free Products ◽  
Quantum Tori ◽  
Bounded Below

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

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.

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