Pure E_0-semigroups and absorbing states

TL;DR

This paper classifies pure E_0-semigroups with invariant states, showing they are cocycle perturbations of CAR/CCR flows, and develops methods to construct such perturbations with specific eigenvalue properties.

Contribution

It characterizes pure E_0-semigroups with invariant states and introduces systematic construction methods for cocycle perturbations with prescribed eigenvalues.

Findings

Pure E_0-semigroups with invariant states are cocycle perturbations of CAR/CCR flows.

Convergence to equilibrium occurs in trace norm for these semigroups.

Methods for constructing perturbations with specific eigenvalue lists are developed.

Abstract

An E_0-semigroup acting on B(H) is called pure if the intersection of the ranges $\alpha_t(B(H))$, $t>0$, is the algebra of scalars. We determine all pure E_0-semigroups which have a weakly continuous invariant state $\omega$ and which are minimal in an appropriate sense. In such cases, for every normal state $\rho$ there is convergence to equilibrium in the trace norm $\lim_{t\to\infty}\| \rho\circ\alpha_t - \omega \| = 0.$ A normal state $\omega$ with this property is called an absorbing state. Such E_0-semigroups mest be cocycle perturbations of CAR/CCR flows, and we develop systematic methods for constructing those perturbations which have absorbing states with prescribed finite eigenvalue lists.