Banach $L^p$ lattices with an automorphism

TL;DR

This paper develops a model-theoretic framework for Banach $L^p$ lattices with automorphisms, establishing stability, quantifier elimination, and implications for the structure of automorphism groups.

Contribution

It introduces a model companion for Banach $L^p$ lattices with automorphisms, proving stability, quantifier elimination, and analyzing non-isolated types.

Findings

The theory admits a model companion that is stable and has quantifier elimination.

Non-trivial types in this theory cannot be isolated.

There are no comeagre conjugacy classes in the automorphism group of non-singular transformations.

Abstract

We study the theory of Banach $L^p$ lattices with a distinguished automorphism, in the framework of continuous logic. Using a functional version of the Rokhlin lemma, we prove that it admits a model companion, which is stable and has quantifier elimination. We show that the types of this theory that are not trivial cannot be isolated. We then use this result to obtain a proof of the absence of comeagre conjugacy classes in $\operatorname{Aut}^*({\mu})$, the Polish group of non-singular transformations of a standard probability space.