Une nouvelle définition de l'entropie dynamique des systèmes non commutatifs. (A new definition of dynamical entropy of non-commutative systems)
type: Item
instance of: scholarly article
title: Une nouvelle définition de l'entropie dynamique des systèmes non commutatifs. (A new definition of dynamical entropy of non-commutative systems)
zbMATH Open document ID: 0772.46036
DOI: 10.1007/BF02099145
author: Jean-Luc Sauvageot
author: Jean-Paul Thouvenot
published in: Communications in Mathematical Physics
publication date: 1992-09-27T00:00:00Z
review text: Extending the work of Connes-Størmer and Connes, Connes-Narnhofer- Thirring introduced a concept of noncommutative dynamical entropy, which is a conjugacy invariant for automorphisms of \(C^*\)-algebras. Their definition goes through abelian models of the noncommutative dynamical system. In the paper under review another approach is developed, which is based on conditional and mutual entropies. Let \(\gamma\) be an automorphism and \(\varphi\) be a state such that \(\varphi\circ\gamma=\varphi\). A coupling of \(({\mathcal A},\gamma,\varphi)\) to a commutative dynamical system \(({\mathcal B},\sigma,\mu)\) means that: (a) \(({\mathcal B},\sigma,\mu)\) is a commutative dynamical system, that is, \({\mathcal B}\) is a commutative \(C^*\)-algebra (isomorphic to a function space \(C(K))\), \(\sigma\) is an automorphism of \({\mathcal B}\) (induced by a homeomorphism \(T\) of \(K\)) and \(\mu\) is a \(T\)-invariant measure on \(K\) (identified with the corresponding integral functional of \(C(K))\). (b) A state \(\lambda\) of \({\mathcal A}\otimes {\mathcal B}\) is given which is invariant under \(\gamma\otimes \sigma\), \(\lambda\mid{\mathcal A}=\varphi\) and \(\lambda\mid{\mathcal B}=\mu\). If \({\mathcal P}\equiv(P_ 1,P_ 2,\dots,P_ k)\) is a partition of \(K\) then the following quantity is constructed. \[ h({\mathcal P},\lambda)=H_ \mu ({\mathcal P}\mid{\mathcal P}^ -)-\sum_ i \mu(P_ i)S(\varphi_ i,\varphi), \] where \(H_ \mu({\mathcal P}\mid {\mathcal P}^ -)\) is the ordinary conditional entropy of the partition \({\mathcal P}\) with respect to its past \({\mathcal P}^ - \equiv T^{-1} {\mathcal P}\vee T^{-2}{\mathcal P}\vee\dots\) and \(\varphi_ i\) is the normalization of the functional \(a\mapsto\lambda(a\otimes \chi(P_ i))\). (So \(\sum_ i \mu(P_ i)\varphi_ i\) is a decomposition of the given state \(\varphi\).) For a nuclear \(C^*\)-algebra \({\mathcal A}\), \(\sup\{h ({\mathcal P},\lambda)\}\) is the CNT-entropy. The new approach allows to prove that a dynamical…
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 28D20
zbMATH DE Number: 61981
zbMATH Keywords: noncommutative dynamical entropy
zbMATH Keywords: conjugacy invariant for automorphisms of \(C^*\)-algebras
zbMATH Keywords: conditional and mutual entropies
zbMATH Keywords: coupling
zbMATH Keywords: commutative dynamical system
zbMATH Keywords: nuclear \(C^*\)-algebra
zbMATH Keywords: CNT-entropy
zbMATH Keywords: dynamical system with singular spectrum has zero entropy
reviewed by: Dénes Petz
MaRDI profile type: Publication
cites work: Relative entropy of states of von Neumann algebras
cites work: Q3719303
cites work: Dynamical entropy of C *-algebras and von Neumann algebras
cites work: Entropy for automorphisms of \(II_1\) von Neumann algebras
cites work: Entropy of Bogoliubov automorphisms of the canonical anticommutation relations
cites work: Entropy of dynamical systems and perturbations of operators
cites work: Some aspects of quantum information theory and their applications to irreversible processes
full work available at URL: https://doi.org/10.1007/bf02099145
OpenAlex ID: W2093881895
description: scientific article; zbMATH DE number 61981