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Amenable group actions on $L_p$ lattices

Antonio M. Scielzo

Abstract

A result by Ornstein and Weiss states that a free and measure-preserving action of an amenable group on a probability space yields a decomposition of the space in disjoint images, up to a small error, analogous to the one given by the Rokhlin lemma in the case of a single transformation. We generalise this result to non-singular actions, and use it to prove that the theory of an action by automorphisms of an amenable group on a Banach $L_p$ lattice admits a model companion, which is stable and has quantifier elimination.

Amenable group actions on $L_p$ lattices

Abstract

A result by Ornstein and Weiss states that a free and measure-preserving action of an amenable group on a probability space yields a decomposition of the space in disjoint images, up to a small error, analogous to the one given by the Rokhlin lemma in the case of a single transformation. We generalise this result to non-singular actions, and use it to prove that the theory of an action by automorphisms of an amenable group on a Banach lattice admits a model companion, which is stable and has quantifier elimination.
Paper Structure (8 sections, 22 theorems, 70 equations)

This paper contains 8 sections, 22 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Non-singular actions of amenable groups
  3. Approximate multicovers
  4. Non-singular free actions of amenable groups
  5. Representation of group actions on $L_p$ lattices
  6. Free $G$-$L_p$ lattices and their model theory
  7. Decomposition of free $G$-$L_p$ lattices
  8. Model theory of $G$-$L_p$ lattices

Key Result

Theorem 1.1

Let $G$ be a countable amenable group that acts freely and preserving the measure on a standard probability space $(X,£F,μ)$, and suppose that $T_1,…,T_\ell ⊆G$ tile $G$. Then there exist measurable sets $W_1,…,W_\ell∈ £F$ such that

Theorems & Definitions (54)

  • Theorem 1.1: o-w
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Definition 2.5
  • proof
  • ...and 44 more