Failure of the Ryll-Nardzewski theorem on the CAR algebra

Vitonofrio Crismale; Stefano Rossi

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arXiv · 2209.02325

Failure of the Ryll-Nardzewski theorem on the CAR algebra

Vitonofrio Crismale·Stefano Rossi

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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.

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Vitonofrio Crismale, Stefano Rossi. 2022-09-06. Failure of the Ryll-Nardzewski theorem on the CAR algebra. https://arxiv.org/abs/2209.02325

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