On perturbations of Hilbert spaces and probability algebras with a generic automorphism

TL;DR

This paper investigates the stability properties of theories involving infinite dimensional Hilbert spaces and atomless probability algebras with generic automorphisms, showing they are stable up to perturbation but not without it.

Contribution

It establishes that these theories are $eth_0$-stable up to perturbation and have prime models over any set, highlighting the importance of perturbation in their stability analysis.

Findings

$IHS_A$ is $eth_0$-stable up to perturbation.

$APr_A$ is $eth_0$-stable up to perturbation.

Without perturbation, the theories are not superstable.

Abstract

We prove that $IHS_A$, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is $\aleph_0$-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, $APr_A$, the theory of atomless probability algebras equipped with a generic automorphism is $\aleph_0$-stable up to perturbation. However, not allowing perturbation it is not even superstable.