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Lattices of slowly oscillating functions

Yutaka Iwamoto

TL;DR

The paper addresses how the lattice structure of slowly oscillating function spaces on unbounded chain-connected proper metric spaces encodes coarse and topological geometry. By extending prior work on lattices of function spaces, it proves that lattice isomorphisms induce unique, coarse-compatible homeomorphisms and preserves Higson compactifications, yielding a Banach-Stone-like theorem for these lattices. It then provides a complete description of linear lattice isomorphisms as weighted composition operators, with a representation theorem that pairs coarse homeomorphisms with admissible weight functions to classify all such isomorphisms. These results illuminate the interplay between lattice-theoretic structure and coarse geometric invariants, advancing understanding of how functional lattices reflect underlying space geometry.

Abstract

We show that lattice isomorphisms between lattices of slowly oscillating functions on chain-connected proper metric spaces induce coarsely equivalent homeomorphisms. This result leads to a Banach-Stone-like theorem for these lattices. Furthermore, we provide a representation theorem that characterizes linear lattice isomorphisms among these lattices.

Lattices of slowly oscillating functions

TL;DR

The paper addresses how the lattice structure of slowly oscillating function spaces on unbounded chain-connected proper metric spaces encodes coarse and topological geometry. By extending prior work on lattices of function spaces, it proves that lattice isomorphisms induce unique, coarse-compatible homeomorphisms and preserves Higson compactifications, yielding a Banach-Stone-like theorem for these lattices. It then provides a complete description of linear lattice isomorphisms as weighted composition operators, with a representation theorem that pairs coarse homeomorphisms with admissible weight functions to classify all such isomorphisms. These results illuminate the interplay between lattice-theoretic structure and coarse geometric invariants, advancing understanding of how functional lattices reflect underlying space geometry.

Abstract

We show that lattice isomorphisms between lattices of slowly oscillating functions on chain-connected proper metric spaces induce coarsely equivalent homeomorphisms. This result leads to a Banach-Stone-like theorem for these lattices. Furthermore, we provide a representation theorem that characterizes linear lattice isomorphisms among these lattices.
Paper Structure (6 sections, 18 theorems, 34 equations)

This paper contains 6 sections, 18 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homeomorphisms Induced by Lattice Isomorphisms
  4. Isomorphisms on Lattices of Slowly Oscillating Functions
  5. Nonlinear lattice isomorphisms
  6. Linear lattice isomorphisms

Key Result

Proposition 2.1

If $\mathcal{L}^{\ast}\subset C^{\ast}(X)$ is a complete ring of functions on a Tychonoff space $X$, then $\mathcal{L}^{\ast}$ coincides with the family of all continuous functions on $X$ that are continuously extendable over the compactification $K(\mathcal{L}^{\ast})$ of $X$. $\square$

Theorems & Definitions (36)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • ...and 26 more