Failure of the Ryll-Nardzewski theorem on the CAR algebra

TL;DR

The paper demonstrates that certain symmetry properties of states on the CAR algebra, specifically spreadability and stationarity, do not imply exchangeability or spreadability respectively, challenging the applicability of the Ryll-Nardzewski theorem.

Contribution

It shows the failure of the Ryll-Nardzewski theorem in the context of the CAR algebra by constructing states with specific symmetry properties that do not imply others.

Findings

Existence of spreadable states that are not exchangeable on CAR algebra

Existence of stationary states that are not spreadable on CAR algebra

J_Z is left amenable but not right amenable, affecting symmetry implications

Abstract

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.