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≠0
, the crossed product induced by the time
t
automorphism
θ
t
is not a factor if and only if there exist a rational number
r
and an eigenvalue
s
of the restriction of
θ
to the center of
N
, such that
rst=2π
. 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≠0
, the crossed product induced by the time
t
automorphism
α
t
is not simple if and only if there exist a rational number
r
and an eigenvalue
s
of the restriction of
α
to the center of
A
, such that
rst=2π
.