Failure of the Ryll-Nardzewski theorem on the CAR algebra

Spreadability of a sequence of random variables is a distributional symmetry that is implemented by suitable actions of $\mathbb{J}_\mathbb{Z}$, the unital semigroup of strictly increasing maps on $\mathbb{Z}$ with cofinite range. We show that $\mathbb{J}_\mathbb{Z}$ is left amenable but not right amenable, although it does admit a right Folner sequence. This enables us to prove that on the CAR algebra ${\rm CAR}(\mathbb{Z})$ there exist spreadable states that fail to be exchangeable. Moreover, we also show that on ${\rm CAR}(\mathbb{Z})$there exist stationary states that fail to be spreadable.