Histories Distorted by Partial Isometries

Abstract

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 $\gamma = (\gamma_t)_{t\in \Bbb{R}}$, 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.