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 $\mathrm{C}\mathrm{A}\mathrm{R}(\mathbb{Z})$ there exist spreadable states that fail to be exchangeable. Moreover, we also show that on $\mathrm{C}\mathrm{A}\mathrm{R}(\mathbb{Z})$there exist stationary states that fail to be spreadable.