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.

CONSTRUCTION OF DIRICHLET FORMS ON STANDARD FORMS OF VON NEUMANN ALGEBRAS

2000 ◽  
Vol 03 (01) ◽  
pp. 1-14 ◽  
Author(s):  
YONG MOON PARK
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Admissible Function ◽  
Standard Form ◽  
Dirichlet Forms ◽  
Quantum Spin ◽  
Quantum Spin Systems ◽  
Translation Invariant ◽  
Von Neumann ◽  
Markovian Semigroups

For a von Neumann algebra ℳ acting on a Hilbert space ℋ with a cyclic and separating vector ξ0, we give an explicit expression of Dirichlet forms on the natural standard form [Formula: see text] associated with the pair (ℳ, ξ0). For any self-adjoint analytic element x of ℳ and an admissible function f, we construct a (bounded) Dirichlet form which generates a symmetric Markovian semigroup on ℋ. We then apply our result to construct translation invariant, symmetric, Markovian semigroups for quantum spin systems with finite range interactions.

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.

scholarly journals Non-symmetric Dirichlet Forms on Semifinite von Neumann Algebras

1996 ◽  
Vol 135 (1) ◽  
pp. 50-75 ◽  
Author(s):  
Daniele Guido ◽  
Tommaso Isola ◽  
Sergio Scarlatti
Keyword(s):  
Von Neumann Algebras ◽  
Dirichlet Forms ◽  
Von Neumann

scholarly journals REMARKS ON THE STRUCTURE OF DIRICHLET FORMS ON STANDARD FORMS OF VON NEUMANN ALGEBRAS

2005 ◽  
Vol 08 (02) ◽  
pp. 179-197 ◽  
Author(s):  
YONG MOON PARK
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Bounded Operator ◽  
Standard Form ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Dirichlet Forms ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
Dynamical Semigroup

For a von Neumann algebra ࡕ acting on a Hilbert space ℋ with a cyclic and separating vector ξ0, we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (ࡕ, ξ0). For a general bounded Lindblad type generator L of a conservative quantum dynamical semigroup on ࡕ, we give sufficient conditions so that the bounded operator H induced by L via the symmetric embedding of ࡕ into ℋ to be self-adjoint. It turns out that the self-adjoint operator H can be written in the form of a Dirichlet operator associated to a Dirichlet form given in Ref. 23. In order to make the connection possible, we also extend the range of applications of the formula in Ref. 23.

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