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A non-Archimedean Arens--Eells isometric embedding theorem on valued fields

Yoshito Ishiki

TL;DR

The paper proves a non-Archimedean Arens–Eells–type embedding: for any non-Archimedean valued field $K$ and any ultrametric space $(X,d)$, there exists a valued-field extension $(L,\lVert\cdot\rVert)$ of $K$ and an isometric embedding $I:X\to L$ with $I(X)$ algebraically independent over $K$. The construction leverages Hahn fields and their $p$-adic analogues, notably $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ built from Witt vectors, and employs a transfinite recursive argument to extend partial embeddings while preserving distances and algebraicity. A key technical theorem (the Preparations) enables the main embedding; the results also solve Broughan's conjecture by embedding Hp-valued spaces into $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$, and are extended to generalized ultrametric spaces valued in linearly ordered groups. Altogether, the work connects non-Archimedean metric embedding with explicit field-theoretic constructions, enabling precise representations of ultrametric structures inside valued fields and offering a framework for further generalizations and applications.

Abstract

In 1959, Arens and Eells proved that every metric space can be isometrically embedded into a normed linear space as a closed subset. In later years, in the paper on a short proof of the Arens--Eells theorem, Michael implicitly pointed out that the Arens--Eells theorem follows from the statement that every metric space can be isometrically embedded into a normed linear space as a linearly independent subset. In this paper, we prove a non-Archimedean analogue of the Arens--Eells isometric embedding theorem, which states that for every non-Archimedean valued field $K$, every ultrametric space can be isometrically embedded into a non-Archimedean valued field that is a valued field extension of $K$ such that the image of the embedding is algebraically independent over $K$.

A non-Archimedean Arens--Eells isometric embedding theorem on valued fields

TL;DR

The paper proves a non-Archimedean Arens–Eells–type embedding: for any non-Archimedean valued field and any ultrametric space , there exists a valued-field extension of and an isometric embedding with algebraically independent over . The construction leverages Hahn fields and their -adic analogues, notably built from Witt vectors, and employs a transfinite recursive argument to extend partial embeddings while preserving distances and algebraicity. A key technical theorem (the Preparations) enables the main embedding; the results also solve Broughan's conjecture by embedding Hp-valued spaces into , and are extended to generalized ultrametric spaces valued in linearly ordered groups. Altogether, the work connects non-Archimedean metric embedding with explicit field-theoretic constructions, enabling precise representations of ultrametric structures inside valued fields and offering a framework for further generalizations and applications.

Abstract

In 1959, Arens and Eells proved that every metric space can be isometrically embedded into a normed linear space as a closed subset. In later years, in the paper on a short proof of the Arens--Eells theorem, Michael implicitly pointed out that the Arens--Eells theorem follows from the statement that every metric space can be isometrically embedded into a normed linear space as a linearly independent subset. In this paper, we prove a non-Archimedean analogue of the Arens--Eells isometric embedding theorem, which states that for every non-Archimedean valued field , every ultrametric space can be isometrically embedded into a non-Archimedean valued field that is a valued field extension of such that the image of the embedding is algebraically independent over .
Paper Structure (16 sections, 29 theorems, 25 equations)

This paper contains 16 sections, 29 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalities
  4. Metric spaces
  5. Valued rings
  6. Constructions of valued fields
  7. Hahn rings and fields
  8. The $p$-adic Hahn fields
  9. Algebraic independence over valued fields
  10. Isometric embeddings of ultrametric spaces
  11. A non-Archimedean Arens--Eells theorem
  12. Preparations
  13. The proof of the first main result
  14. Broughan's conjecture
  15. Isometric embeddings of generalized ultrametric spaces
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $X$ be a set, and $d\colon X\times X\to \mathbb{R}$ be a symmetric function, i.e., $d(x, y)=d(y, x)$ for all $x, y\in X$. Then $d$ satisfies the strong triangle inequality: $d(x, y)\le d(x, z)\lor d(z, y)$ if and only if for every triple $x, y, z\in X$, the inequality $d(x, z)<d(z, y)$ implies $

Theorems & Definitions (63)

  • Conjecture 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Remark 2.1
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • ...and 53 more