Continuous and discrete flows on operator algebras

Abstract

Let $(N,\R,\theta)$ be a centrally ergodic W* dynamical system. When $N$ is not a factor, we show that, for each $t\not=0$, the crossed product induced by the time $t$ automorphism $\theta_t$ is not a factor if and only if there exist a rational number $r$ and an eigenvalue $s$ of the restriction of $\theta$ to the center of $N$, such that $rst=2\pi$. In the C* setting, minimality seems to be the notion corresponding to central ergodicity. We show that if $(A,\R,\alpha)$ is a minimal unital C* dynamical system and $A$ is either prime or commutative but not simple, then, for each $t\not=0$, the crossed product induced by the time $t$ automorphism $\alpha_t$ is not simple if and only if there exist a rational number $r$ and an eigenvalue $s$ of the restriction of $\alpha$ to the center of $A$, such that $rst=2\pi$.