Prime Poisson suspensions

TL;DR

This paper characterizes when a Poisson suspension is prime, providing explicit examples including non-singular compact group rotations, and compares these with known prime transformations.

Contribution

It establishes a necessary and sufficient condition for prime Poisson suspensions and introduces new explicit examples, expanding the understanding of prime systems.

Findings

Characterization of prime Poisson suspensions

Explicit examples including non-singular compact group rotations

Comparison with known prime transformations

Abstract

We establish a necessary and sufficient condition for a Poisson suspension to be prime. The proof is based on the Fock space structure of the $L^{2}$-space of the Poisson suspension. We give examples of explicit infinite measure preserving systems, in particular non-singular compact group rotations that give rise to prime Poisson suspensions. We also compare some properties of so far known prime transformations with those of our examples, showing that these examples are new.