scholarly journals The Haagerup Approximation Property for von Neumann Algebras via Quantum Markov Semigroups and Dirichlet Forms

2015 ◽  
Vol 336 (3) ◽  
pp. 1637-1664 ◽  
Author(s):  
Martijn Caspers ◽  
Adam Skalski
Keyword(s):  
Approximation Property ◽  
Von Neumann Algebras ◽  
Dirichlet Forms ◽  
Markov Semigroups ◽  
Von Neumann

scholarly journals Gradient forms and strong solidity of free quantum groups

2020 ◽  
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Groups ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Dirichlet Forms ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Crucial Part

AbstractConsider the free orthogonal quantum groups $$O_N^+(F)$$ O N + ( F ) and free unitary quantum groups $$U_N^+(F)$$ U N + ( F ) with $$N \ge 3$$ N ≥ 3 . In the case $$F = \text {id}_N$$ F = id N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $$L_\infty (O_N^+)$$ L ∞ ( O N + ) is strongly solid. Moreover, Isono obtains strong solidity also for $$L_\infty (U_N^+)$$ L ∞ ( U N + ) . In this paper we prove for general $$F \in GL_N(\mathbb {C})$$ F ∈ G L N ( C ) that the von Neumann algebras $$L_\infty (O_N^+(F))$$ L ∞ ( O N + ( F ) ) and $$L_\infty (U_N^+(F))$$ L ∞ ( U N + ( F ) ) are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.

DIRICHLET FORMS AND SYMMETRIC MARKOVIAN SEMIGROUPS ON ℤ2-GRADED VON NEUMANN ALGEBRAS

2003 ◽  
Vol 15 (08) ◽  
pp. 823-845 ◽  
Author(s):  
CHANGSOO BAHN ◽  
CHUL KI KO ◽  
YONG MOON PARK
Keyword(s):  
Von Neumann Algebras ◽  
Dirichlet Forms ◽  
Gauge Invariant ◽  
Von Neumann ◽  
Markovian Semigroups

We extend the construction of Dirichlet forms and symmetric Markovian semigroups on standard forms of von Neumann algebras given in [1] to the case of ℤ2-graded von Neumann algebras. As an application of the extension, we construct symmetric Markovian semigroups on CAR algebras with respect to gauge invariant quasi-free states and also investigate detailed properties such as ergodicity of the semigroups.

scholarly journals Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras

2014 ◽  
Vol 352 (6) ◽  
pp. 507-510 ◽  
Author(s):  
Martijn Caspers ◽  
Rui Okayasu ◽  
Adam Skalski ◽  
Reiji Tomatsu
Keyword(s):  
Approximation Property ◽  
Von Neumann Algebras ◽  
Von Neumann

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.

scholarly journals Spectral properties of Markov semigroups in von Neumann algebras

2017 ◽  
Vol 453 (2) ◽  
pp. 821-840
Author(s):  
Katarzyna Kielanowicz ◽  
Andrzej Łuczak
Keyword(s):  
Spectral Properties ◽  
Von Neumann Algebras ◽  
Markov Semigroups ◽  
Von Neumann
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