1
$\begingroup$

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.

Consider the covariant GNS representation $(H_\phi,\pi_\phi, V_{\phi,\alpha},\xi_\phi)$, together with the selfadjoint projection $E_{\phi,\alpha}$ onto the invariant vectors of $H_\phi$ under the unitary $V_{\phi,\alpha}$.

Are there concrete examples for which ${\rm dim}(E_{\phi,\alpha}H_\phi)>1$?

Improve this question
$\endgroup$
5
  • $\begingroup$ Doesn't the fact that $\dim(E[H]) \geq 2$ contradict the ergodicity of $\phi$? It does in the commutative case. $\endgroup$
    – Adrián González Pérez
    Commented May 26, 2020 at 9:30
  • $\begingroup$ No Adrian, ergodicity means extremality among invariant states. ${\rm dim}(E[H])=1$ implies ergodicity, but the converse is true under the additional assumption of $G$-abelianess (in our situation $Z$-abelianess), see Prop. 3.1.12 in Sakai's book. The question is that, at my best knowledge, conterexamples don't exist in literature $\endgroup$
    – francesco fidaleo
    Commented May 26, 2020 at 9:45
  • $\begingroup$ In commutative case, it does hold true: ${\rm dim}(E[H])=1\iff \phi$ is ergodic. Maybe, it is true also if the support of $\phi$ in the bidual is central. $\endgroup$
    – francesco fidaleo
    Commented May 26, 2020 at 10:35
  • $\begingroup$ What is the relation of the title to your question? I.e., would an example of the sort you want be called a G-abelian (or is it $G$-abelian?) system? $\endgroup$
    – LSpice
    Commented May 26, 2020 at 15:38
  • $\begingroup$ The relation is explained in one of the comments above. $\endgroup$
    – francesco fidaleo
    Commented May 26, 2020 at 19:50

0

Reset to default

You must log in to answer this question.