Failure of the Ryll-Nardzewski theorem on the CAR algebra

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