Histories distorted by partial isometries. (English) Zbl 1264.81207
Summary: In quantum dynamical systems, a history is defined by a pair (\(M, \gamma\)), consisting of a type \(I\) factor \(M\), acting on a Hilbert space \(H\), and an \(E_{0}\)-group \(\gamma=(\gamma_t)_{t\in \mathbb R}\), satisfying certain additional conditions. In this paper, we distort a given history (\(M, \gamma\)), by a finite family \(\mathcal G\) of partial isometries on \(H\). In particular, such a distortion is dictated by the combinatorial relation on the family \(\mathcal G\). Two main purposes of this paper are (i) to show the existence of distortions on histories, and (ii) to consider how distortions work. We can understand Sections 3, 4 and 5 as the proof of the existence of distortions (i), and the properties of distortions (ii) are shown in Section 6.
MSC:
81Q12 | Nonselfadjoint operator theory in quantum theory including creation and destruction operators |
05C21 | Flows in graphs |
05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
16T20 | Ring-theoretic aspects of quantum groups |
22A22 | Topological groupoids (including differentiable and Lie groupoids) |
46N50 | Applications of functional analysis in quantum physics |
47L15 | Operator algebras with symbol structure |
47L75 | Other nonselfadjoint operator algebras |
47L90 | Applications of operator algebras to the sciences |