Time reversibility for modular \(K\)-systems
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.
scientific article; zbMATH DE number 1486490
scientific article