Une nouvelle définition de l'entropie dynamique des systèmes non commutatifs. (A new definition of dynamical entropy of non-commutative systems)
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 system with singular spectrum has zero entropy. The paper is an essential contribution to the theory of dynamical systems.
conjugacy invariant for automorphisms of \(C^*\)-algebras
dynamical system with singular spectrum has zero entropy
scientific article; zbMATH DE number 61981