Failure of the Ryll-Nardzewski theorem on the CAR algebra

Spreadability of a sequence of random variables is a distri-butional symmetry that is implemented by suitable actions of J_Z, the unital semigroup of strictly increasing maps on Z with cofinite range. We show that J_Zis left amenable but not right amenable, although it does admit a right Følner sequence. This enables us to prove that on the CAR algebra CAR(Z) there exist spreadable states that fail to be exchangeable. Moreover, we also show that on CAR(Z) there exist stationary states that fail to be spreadable.