Time reversibility for modular \(K\)-systems
type: Item
instance of: scholarly article
title: Time reversibility for modular \(K\)-systems
zbMATH Open document ID: 0953.46032
DOI: 10.1016/S0034-4877(00)88874-6
published in: Reports on Mathematical Physics
publication date: 2000-08-09T00:00:00Z
review text: A one-sided \(K\)-system \((M,N,\tau)\) consists of a von Neumann algebra \(M\) with a faithful normal state \(\omega\) and a von Neumann subalgebra \(N\) of \(M\) such that the modular automorphism group \(\tau=(\tau_t)_{t\in{\mathbb R}}\) leaves \(N\) invariant for \(t\geq 0\) and such that the von Neumann algebra generated by all \(\tau_t(N)\) is \(M\), and that the intersection of all \(\tau_t(N)\) is \(\mathbb C1\). A two-sided \(K\)-system is a quadruple \((M,N,L,\tau)\) such that \((M,N,\tau)\) and \((M,L,\widehat{\tau})\) are one-sided \(K\)-systems, where \(\widehat{\tau}\) is the time reversed modular group. The author provides a necessary and sufficient condition for that a one-sided \(K\)-system extends to a one-sided \(K\)-system for a certain crossed product of \(M\). Then a sufficient criterion is provided for that this extension turns a two-sided \(K\)-system into a two-sided \(K\)-system. Many further results about the structure of such \(K\)-systems are provided, for instance, the form of the correponding Anosov automorphisms and their generators. In an appendix the construction is applied to a free fermion system.
reviewed by: Michael Skeide
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 37D99
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 46L54
zbMATH DE Number: 1486490
zbMATH Keywords: time reversibility
zbMATH Keywords: modular \(K\)-systems
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: faithful normal state
zbMATH Keywords: modular automorphism group \(\tau\)
zbMATH Keywords: two-sided \(K\)-system
zbMATH Keywords: Anosov automorphisms
zbMATH Keywords: free fermion system
author: Heide Narnhofer
MaRDI profile type: Publication
cites work: Entropy for automorphisms of \(II_1\) von Neumann algebras
cites work: Dynamical entropy of C *-algebras and von Neumann algebras
cites work: Dynamical approximation entropies and topological entropy in operator algebras
cites work: Defining quantum dynamical entropy
cites work: Q4249084
cites work: Q3955608
cites work: Q3931654
cites work: Algebraic K-systems
cites work: The CPT-theorem in two-dimensional theories of local observables
cites work: Theory of operator algebras I.
cites work: Q4239833
cites work: Anosov actions on noncommutative algebras
Wikidata QID: Q127391528
Recommended article: Q3650085
Recommended article: Q3014697
Recommended article: Integrability for a class of time-reversible systems
Recommended article: Isochronicity for a class of reversible systems
Recommended article: Time-reversal homotopical properties of concurrent systems
Recommended article: Melnikov-type theorem for time reversible system
Recommended article: Time-Reversibility in 2-Dimensional Systems
Recommended article: Q5474770
Recommended article: Inverse integrating factors for a class of time-reversible systems
Recommended article: Time-Reversibility and Ivariants of Some 3-dim Systems
description: scientific article; zbMATH DE number 1486490