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

Yoshito Ishiki Photonics Control Technology Team RIKEN Center for Advanced Photonics 2-1 Hirasawa, Wako, Saitama 351-0198, Japan [email protected]

Abstract.

In 1959, Arens and Eells proved that every metric space can be isometrically embedded into a real linear space as a closed subset. In later years, Michael pointed out that every metric space can be isometrically embedded into a real linear space as a linear independent subset and provided a short proof of Arens–Eells theorem as an application. 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 $K$ . Using Levi-Civita fields, we also show that every Urysohn universal ultrametric sapce has a valued field structure.

Key words and phrases:

Ultrametrics, Isometric embeddings, and Non-Archimedean valued fields

2020 Mathematics Subject Classification:

Primary 54E35, Secondary 54E40, 51F99

1. Introduction

In 1956, Arens and Eells [1] established that for every metric space $(X,d)$ there exist a real normed linear space $(V,\|*\|)$ and an isometric embedding $I\colon X\to V$ such that $I(X)$ is closed in $V$ . Michael [16] pointed out that for every metric space $(X,d)$ , we can take a real normed linear space $(V,\lVert*\rVert)$ and an isometric embedding $I\colon X\to V$ such that $I(X)$ is linearly independent over $V$ . Using this observation, Michael provided a short proof of the Arens–Eells theorem.

A metric $d$ on $X$ is said to be an ultrametric or a non-Archimedean metric if $d(x,y)\leq d(x,z)\lor d(z,y)$ for all $x,y,z\in X$ , where $\lor$ stands for the maximal operator on $\mathbb{R}$ . A set $R$ is a range set if $0\in R$ and $R\subseteq[0,\infty)$ . An ultrametric $d$ on $X$ is said to be R-valued if $d(x,y)\in R$ for all $x,y\in X$ . Some authors try to investigate non-Archimedean analogue of theorems on metric spaces such as the Arens–Eells theorem. Megrelishvili and Shlossberg [13] proved a non-Archimedean Arens–Eells theorem, which embeds ultrametric spaces into linear spaces over $\mathbb{F}_{2}$ . In [14, Theorem 4.3], as a improvement of their theorem, they prove a non-Archimedean Arens–Eells theorem on linear spaces over arbitrary non-Archimedean valued fields. In [7, Theorem 1.1], as a non-Archimedean analogue of the Arens–Eells theorem, the author showed that for every range set $R$ , for every integral domain $A$ with the trivial absolute value $\lvert*\rvert$ (i.e., $\lvert x\rvert=1$ for all $x\neq 0$ ), for every $R$ -valued ultrametric space $(X,d)$ , there exist an $R$ -valued ultra-normed module $(V,\lVert*\rVert)$ over $(A,\lvert*\rvert)$ and an isometric embedding $I\colon X\to V$ such that $I(X)$ is closed and linearly independent over $(A,\rvert*\rvert)$ . Using this embedding theorem, the author proved a ( $R$ -valued) non-Archimedean analogue of the Hausdorff extension theorem of metrics.

There are other attempts to construct an isometric embedding from an ultrametric spaces into a space with a non-Archimedean algebraic structure. Schikhof [21] established that every ultrametric spaces can be isometrically embedded into a non-Archimedean valued field using the Hahn fields, which is a generalization of a field of formal power series. In [2, Conjecture 5.34], Baroughan raise the following conjecture.

Conjecture 1.1.

Let $p$ be an odd prime and $(X,d)$ be an $H_{p}$ -valued metric space, where $H_{p}=\{0\}\cup\{\,p^{n}\mid n\in\mathbb{Z}\,\}$ . Then there exist a non-Archimedean Banach algebra $(B,\lVert*\rVert)$ over $\mathbb{Q}_{p}$ and an isometry $i\colon X\to B$ .

In this paper, as a generalization of the Schikhof theorem ([21]) and known non-Archimedean analogues ([7, Theorem 1.1] and [14, Theorem 4.3]) of the Arens–Eells theorem, we prove that for every non-Archimedean valued field $(K,\rVert*\lVert_{K})$ , for every metric space $(X,d)$ , there exist a valud field $(L,\rVert*\lVert_{L})$ and an isometry embedding $I\colon X\to K$ such that $L$ is a valued field extension of $K$ , the set $I(X)$ is closed in $L$ , and $I(X)$ is algebraically independent over $K$ (Theorem 4.6). A key point of the proof of Theorem 4.6 is to use the notion of $p$ -adic Hahn fields ( $p$ -adic Mal’cev–Neumann fields), which was first introduced by Poonen [20] as $p$ -adic analogues of ordinary Hahn fields.

As an application, we also gives an affirmative solution of Conjecture 1.1 (see Theorem 4.8).

The theory of $p$ -adic Hahn fields have an application to Urysohn universal ultrametric spaces, which is defined as ultrametric spaces possessing high homogeneity. We introduce the concept of $p$ -adic Levi-Civita fields as subspaces of $p$ -adic Hahn fields, which is a $p$ -adic analogue of ordinary Levi-Civita fields (see, for instance, [3]). We also show that if $p$ is $0$ or a prime, then every Urysohn universal ultrametric space has a field structure that is an extension of $\mathbb{Q}_{p}$ , where we consider that $\mathbb{Q}_{0}=\mathbb{Q}$ and it is equipped with the trivial valuation in the case of $p=0$ (see Theorem 5.5).

The paper is organized as follows. Section 2 presents notions and notations of metric spaces and valued fields. We introduce Hahn fields and $p$ -adic Hahn fields, which plays an important role in the proofs of our main results. We also prepare some basic statements on valued fields and Hahn fields. In Section 3, we show statements on algebraic independence in ( $p$ -adic) Hahn fields. Section 4 is devoted to proving Theorem 4.6. We also provide an affirmative solution of Conjecture 1.1. In Section 5, we show that ( $p$ -adic) Levi-Civita fields become Urysohn universal ultrametric spaces. Our arguments in this section are baed on the author’s paper [8].

Acknowledgements.

The author would like to thank to Tomoki Yuji for helpful advices on algebraic arguments.

2. Preliminaries

2.1. Generalities

In this paper, we use the set-theoretic notations of ordinals. For example, for an ordinal $\alpha$ , we have $\beta<\alpha$ if and only if $\beta\in\alpha$ .

2.1.1. Metric spaces

For a metric space $(X,d)$ , and for a subset of $A$ of $X$ , and for $x\in X$ , we define $d(A,x)=\inf\{\,d(a,x)\mid a\in A\,\}$ . For $x\in X$ and $r\in(0,\infty)$ , we denote by $B(a,r;d)$ the closed ball centered at $x$ with radius $r$ . We often simply represent it as $B(a,r)$ when no confusion can arise. Similarly, we define the open ball $U(a,r)$ .

The proofs of the next two lemmas are presented in Propositions 18.2, and 18.4, in [22], respectively.

Lemma 2.1.

Let $X$ be a set, and $d$ be a pseudo-metric on $X$ . Then $d$ satisfies the strong triangle inequality if and only if for all $x,y,z\in X$ , the inequality $d(x,z)<d(z,y)$ implies $d(z,y)=d(x,y)$ .

Lemma 2.2.

Let $(X,d)$ be a pseudo-ultrametric space, $a\in X$ and $r\in[0,\infty)$ . Then for every $q\in B(p,r)$ , we have $B(p,r)=B(q,r)$ .

2.1.2. Valued rings

Let $A$ be a commutative ring. We say that a function $v\colon A\to\mathbb{R}\sqcup\{\infty\}$ is a (additive) valuation if the following conditions are satisfied:

(1)

for every $x\in A$ , we have $v(x)=\infty$ if and only if $x=0$ ;
(2)

for every pair $x,y\in A$ , we have $v(xy)=v(x)+v(y)$
(3)

for every pair $x,y\in A$ , we have $v(x+y)\geq v(x)\land v(y)$ , where $\land$ stands for the minimum operator on $\mathbb{R}$ .

If $A$ is a field, then it is called a valued field. Note that for every valued ring $(A,v)$ , we can extend the valuation $v$ to the fractional field $K$ of $A$ . Namely, for $x=b/a\in K$ , where $a,b\in A$ and $a\neq 0$ , we define $v(x)=v(b)-v(a)$ . In this case, $v$ is well-defined and the pair $(K,v)$ naturally becomes a valued field. For example, for a prime $p$ , we define the $p$ -adic valuation $v_{p}$ on $\mathbb{Z}$ by declaring that $v_{p}$ is the number of the factor $p$ in the prime factorization of $x$ . The completion $\mathbb{Z}_{p}$ of $\mathbb{Z}$ with respect to $v_{p}$ is called the ring of $p$ -adic integers. The fractional field of $\mathbb{Z}_{p}$ is called the field of $p$ -adic numbers. For more discussion on $p$ -adic numbers, we refer the readers to [19] and [22].

We say that a function $\lVert*\rVert\colon A\to[0,\infty)$ is a (non-Archimedean) absolute value or multiplicative valuation if the following conditions are satisfied:

(1)

for every $x\in A$ , we have $\lVert x\rVert=0$ if and only if $x=0$ ;
(2)

for every pair $x,y\in A$ , we have $\lVert xy\rVert=\lVert x\rVert\cdot\lVert y\rVert$
(3)

for every pair $x,y\in A$ , we have $\lVert x+y\rVert\leq\lVert x\rVert\lor\lVert y\rVert$ , where $\lor$ stands for the maximum operator on $\mathbb{R}$ .

In what follows, for every $\eta\in(1,\infty)$ , we consider that $\eta^{-\infty}=0$ and $-\log_{\eta}(0)=\infty$ . For a valuation $v$ on a ring $A$ , and for a real number $\eta\in(1,\infty)$ , we define $\lVert x\rVert_{v,\eta}=\eta^{-v(x)}$ and $v_{\rVert*\rVert,\eta}(x)=-\log_{\eta}(x)$ .

Fix $\eta\in(1,\infty)$ , valuations and absolute values on a ring $A$ are essentially equivalent. Namely, they are corresponding to each other as follows.

Proposition 2.3.

Let $A$ be a commutative ring. Then the following statements are true:

(1)

For every $\eta\in(1,\infty)$ , and for every valuation $v$ on $A$ , the function $\lVert*\rVert_{v,\eta}\colon A\to[0,\infty)$ is an absolute value on $A$ ;
(2)

For $\eta\in(1,\infty)$ , and for every absolute value $\lVert*\rVert$ on $A$ , the function $v_{\rVert*\rVert,\eta}\colon A\to\mathbb{R}$ is a valuation on $A$ .

In this paper, we use both of valuation and absolute values on a ring due to Proposition 2.3. Since we represent a valuation (resp. an absolute value) as a symbol like $v$ , or $w$ (resp. $\rvert*\lvert$ or $\rVert*\lVert$ ), the readers will be able to distinguish them.

For a valued ring $(K,v)$ , the set $\mathfrak{A}(K,v)=\{\,x\in K\mid 0\leq v(x)\,\}$ becomes a ring and $\mathfrak{o}(K,v)=\{\,x\in K\mid 0<v(x)\,\}$ is a maximal ideal of $\mathfrak{A}(K,v)$ . We denote by $\mathfrak{K}(K,v)$ the field $\mathfrak{A}(K,v)/\mathfrak{o}(K,v)$ , and we call it the residue class field of $(K,v)$ . We also denote by $\zeta_{K}\colon\mathfrak{A}(K,v)\to\mathfrak{K}(K,v)$ the canonical projection. We simply represent it as $\zeta$ when no confusions can arise. We say that a subset $J$ of $K$ is a complete system of representative of the residue class field $\mathfrak{K}(A,v)$ if $J\subseteq\mathfrak{A}(A,v)$ , $0\in J$ , and $\zeta|_{J}\colon J\to\mathfrak{K}(A,v)$ is bijective.

2.2. Constructions of valued fields

In this section, we review some constructions of valued fields such as Hahn fields. For more discussion, we refer the readers to [3] and [4]. Most of the proofs in this section is refered to [20] and [23].

2.2.1. Hahn rings and fields

A non-empty subset $S$ of $\mathbb{R}$ is said to be well-ordered if every non-empty subset of $S$ has a minimum.

We denote by $\mathcal{G}$ the set of all subgroups of $\mathbb{R}$ containing $1\in\mathbb{R}$ (equivalently, $\mathbb{Z}\subseteq G$ ). For the sake of convenience, we only consider the setting where $G\in\mathcal{G}$ in this paper.

Now we review the construction of the Hahn fields in [20]. Let $G\in\mathcal{G}$ and $A$ be a commutative ring. For a map $a\colon G\to A$ , we define the support $\text{{\rm supp}}(a)$ of $a$ by the set $\{\,x\in G\mid a(x)\neq 0\,\}$ . We denote by $\mathbb{H}(G,A)$ the set of all $a\colon G\to A$ such that $\text{{\rm supp}}(a)$ is well-ordered. We often symbolically represent $a\in\mathbb{H}(G,A)$ as $a=\sum_{g\in G}a(g)t^{g}$ , where $t$ is an indeterminate. For every pair $a,b\in\mathbb{H}(G,A)$ , we define $a+b$ by

(a+b)(x)=a(x)+b(x).

We also define $ab\colon G\to A$ by

ab=\sum_{g\in G}\left(\sum_{i,j\in G,i+j=g}a_{i}b_{j}\right)t^{g}.

Define a valuation $v_{G,A}$ on $\mathbb{H}(G,A)$ by $v_{G,A}(a)=\min\text{{\rm supp}}(a)$ . Since $\text{{\rm supp}}(a)$ is well-ordered, the minimum $\min\text{{\rm supp}}(a)$ actually exists. Note that $A$ becomes a subring of $\mathbb{H}(G,A)$ .

Proposition 2.4.

Let $G\in\mathcal{G}$ , and $\boldsymbol{k}$ be a field. The pair $(\mathbb{H}(G,\boldsymbol{k}),v_{G,\boldsymbol{k}})$ becomes a valued field and it satisfies that $\mathfrak{K}(\mathbb{H}(G,\boldsymbol{k}),v_{G,\boldsymbol{k}})=\boldsymbol{k}$ .

Proof.

See [20, Corollary 1]. ∎

We call $(\mathbb{H}(G,A),v_{G,A})$ the Hahn ring associated with $G$ and $A$ and call it the Hahn field if $A$ is a field. Note that in general, we can define the Hahn fields even if $G$ is a linearly ordered Abelian group (see [20]).

2.2.2. The $p$ -adic Hahn fields

A $p$ -adic analogue of the Hahn fields was first introduced in [20]. Let us review a construction. A field $\boldsymbol{k}$ of characteristic $p$ is said to be perfect if $p=0$ , or $p>0$ and every element of $\boldsymbol{k}$ has a $p$ -th root in $\boldsymbol{k}$ . The following proposition states the existence of rings of Witt vectors. The proof is presented in [23].

Proposition 2.5.

Let $\boldsymbol{k}$ be a perfect field of characteristic $p>0$ . The there exists a unique valued ring $(A,v)$ of characteristic $0$ equipped with a valuation $v$ such that $v(A)=\mathbb{Z}_{\geq 0}$ , $v$ is complete, and $\mathfrak{K}(A,v)=\boldsymbol{k}$ .

For each perfect field $\boldsymbol{k}$ , we denote by $(\mathbb{W}(\boldsymbol{k}),w_{\boldsymbol{k}})$ the valuation field stated in Proposition 2.5 and denote by $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ the fractional field of $\mathbb{W}(\boldsymbol{k})$ . We use the same symbol $w_{\boldsymbol{k}}$ as the valuation on $\mathbb{W}(\boldsymbol{k})$ induced by $w_{\boldsymbol{k}}$ in the canonical way. The ring $(\mathbb{W}(\boldsymbol{k}),w_{\boldsymbol{k}})$ is called the ring of Witt vectors associated with $\boldsymbol{k}$ . Notice that for every prime $p$ , we have $\mathbb{W}(\mathbb{F}_{p})=\mathbb{Z}_{p}$ and $\mathrm{Fr}\mathbb{W}(\mathbb{F}_{p})=\mathbb{Q}_{p}$ , and the valuation $w_{\mathbb{F}_{p}}$ coincides with the $p$ -adic valuation $v_{p}$ .

The next proposition explains the concrete representation of an elements of a ring of Witt vectors.

Proposition 2.6.

Let $\boldsymbol{k}$ be a perfect field of characteristic $p>0$ . Then there uniquely exists a map $f_{\boldsymbol{k}}\colon\boldsymbol{k}\to\mathbb{W}(\boldsymbol{k})$ such that $\{\,f_{\boldsymbol{k}}(a)\mid a\in\boldsymbol{k}\,\}$ is a complete system of representatives, and $f_{\boldsymbol{k}}(ab)=f_{\boldsymbol{k}}(a)f_{\boldsymbol{k}}(b)$ for all $a,b\in\boldsymbol{k}$ . In this case, for every $x\in\mathbb{W}(\boldsymbol{k})$ , there uniquely exists a sequence $\{a_{i}\}_{i\in\mathbb{Z}}$ in $\boldsymbol{k}$ such that $x=\sum_{n\in\mathbb{Z}}f_{\boldsymbol{k}}(a_{n})p^{n}$ and for a sufficient large $m\in\mathbb{Z}_{\geq 0}$ , we have $a_{i}=0$ for all $i<-m$ .

Proof.

See Proposition 8 in the page 35 and and the argument in the page 37 in [23]. ∎

Proposition 2.7.

Let $\boldsymbol{k}$ and $\boldsymbol{l}$ be perfect fields of characteristic $p>0$ . For every homomorphism $\phi\colon\boldsymbol{k}\to\boldsymbol{l}$ , there uniquely exists a homomorphism $\mathbb{W}(\phi)\colon\mathbb{W}(\boldsymbol{k})\to\mathbb{W}(\boldsymbol{l})$ such that $\zeta_{\mathbb{W}(\boldsymbol{l})}\circ\mathbb{W}(\phi)=\phi\circ\zeta_{\mathbb{W}(\boldsymbol{k})}$ . Moreover, if $x=\sum_{n\in\mathbb{Z}}f_{\boldsymbol{k}}(a_{n})p^{n}\in\mathbb{W}(\phi)$ , where $a_{n}\in\boldsymbol{k}$ , then $\mathbb{W}(\phi)(x)=\sum_{n\in\mathbb{Z}}f_{\boldsymbol{l}}(\phi(a_{n}))p^{n}$ . In particular, we have $w_{\boldsymbol{l}}(\mathbb{W}(\boldsymbol{k})(x))=w_{\boldsymbol{k}}(x)$ for all $x\in\mathbb{W}(\boldsymbol{k})$ .

Proof.

See [23, Proposition 10 in the page 39]. ∎

Remark 2.1.

It is known that the construction of rings of Witt vectors is a functor (see [23, Page 39]).

Now we discuss a $p$ -adic analogue of Hahn fields, which is defined by a quotient field of a Hahn ring. For $G\in\mathcal{G}$ , and for a perfect field $\boldsymbol{k}$ of characteristic $p>0$ , we define a subset $\mathbb{N}_{G,\boldsymbol{k}}$ of $\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ by the set of all $\alpha=\sum_{g\in G}\alpha_{g}t^{g}\in\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ such that $\sum_{n\in\mathbb{Z}}\alpha_{g+n}p^{n}=0$ in $\mathbb{W}(\boldsymbol{k})$ for every $g\in G$ .

Proposition 2.8.

For every $G\in\mathcal{G}$ , and for every perfect field $\boldsymbol{k}$ with characteristic $p>0$ , the set $\mathbb{N}_{G,\boldsymbol{k}}$ is an ideal of the ring $\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ , and $\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))/\mathbb{N}_{G,\boldsymbol{k}}$ is a filed.

Proof.

See [20, Proposition 3 ] and [20, Corollary 3]. ∎

Lemma 2.9.

Let $G\in\mathcal{G}$ , $p$ be a prime, and $\boldsymbol{k}$ be a field of characteristic $p$ . Let $J\subseteq\mathbb{W}(\boldsymbol{k})$ be a complete system of representatives of the residue class field of $\boldsymbol{k}$ . Then every element $\alpha=\sum_{g\in G}\alpha_{g}t^{g}\in\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ is equivalent to an element $\beta=\sum_{g\in G}\beta_{g}t^{g}$ modulo $\mathbb{N}_{G,K}$ , where $\beta_{g}$ is in $J$ . In addition, for every $x\in\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ , the family $\{\beta_{g}\}_{g\in G}$ is unique and $\text{{\rm supp}}(\beta)\subseteq\text{{\rm supp}}(\alpha)+\mathbb{Z}_{\geq 0}$ .

Proof.

See [20, Proposition 4]. ∎

Definition 2.1.

Based on Lemma 2.9, for a fixed complete system $J$ of representatives, and for every $a\in\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ , we denote by $\mathrm{St}_{G,\boldsymbol{k},J}(a)$ the standard representation of $a$ with respect to $J$ stated in Lemma 2.9. In this case, we have $\text{{\rm supp}}(\mathrm{St}_{G,\boldsymbol{k},J}(a))\subseteq\text{{\rm supp}}(a)+\mathbb{Z}_{\geq 0}$ .

Let $\mathbb{P}_{p}(G,\boldsymbol{k})$ denote the quotient field $\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))/\mathbb{N}_{G,\boldsymbol{k}}$ , and let $\mathrm{Pr}\colon\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))\to\mathbb{P}_{p}(G,\boldsymbol{k})$ denote the canonical projection. We define $V_{G,\boldsymbol{k},p}\colon\mathbb{P}_{p}(G,\boldsymbol{k})\to G$ by $V_{G,\boldsymbol{k},p}(x)=\min\text{{\rm supp}}(\mathrm{St}_{G,\boldsymbol{k},J}(x))$ .

Proposition 2.10.

Let $G\in\mathcal{G}$ , and $\boldsymbol{k}$ be a perfect field of characteristic $p>0$ . Take a complete system $J\subseteq\mathbb{W}(\boldsymbol{k})$ of representatives of the residue class field $\boldsymbol{k}$ . Then the following statements are true:

(1)

For every $x\in\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ , the value $\min\text{{\rm supp}}(\mathrm{St}_{G,\boldsymbol{k},J}(x))$ is independent from the choice of $J$ .
(2)

The map $V_{G,\boldsymbol{k},p}$ is a valuation on $\mathbb{P}_{p}(G,\boldsymbol{k})$ .

Proof.

See [20, Proposition 5]. ∎

We call the valued field $(\mathbb{P}_{p}(G,\boldsymbol{k}),V_{G,\boldsymbol{k},p})$ the $p$ -adic Mal’cev–Neumann field or $p$ -adic Hahn field. Notice that $(\mathbb{P}_{p}(\mathbb{Z},\mathbb{F}_{p}),V_{\mathbb{Z},\mathbb{F}_{p},p})$ is nothing but the $(\mathbb{Q}_{p},v_{p})$ field of $p$ -adic numbers.

To consider characteristics of a valued field and its residue class field, we supplementally define $\mathcal{CH}$ by the set of all pairs $(q,p)$ such that $q$ and $p$ are $0$ or a prime satisfying the either of the following conditions:

(Q1)

$q=p$ ,
(Q2)

$q=0$ and $0<p$ .

Note that $(q,p)\in\mathcal{CH}$ satisfies (Q2) if and only if $q\neq p$ .

In order to discuss $p$ -adic and ordinary Hahn fields in unified manner, we make a notation as follows.

Definition 2.2.

Let $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , and let $\boldsymbol{k}$ be a perfect field of characteristic $p$ . We define a field $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ by

\mathbb{A}_{q,p}(G,\boldsymbol{k})=\begin{cases}\mathbb{H}(G,\boldsymbol{k})&\text{if $q=p$;}\\ \mathbb{P}_{p}(G,\boldsymbol{k})&\text{if $q\neq p$.}\end{cases}

We also define a valuation $U_{G,\boldsymbol{k},q,p}$ on $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ by

U_{G,\boldsymbol{k},q,p}=\begin{cases}v_{G,\boldsymbol{k}}&\text{if $q=p$;}\\ V_{G,\boldsymbol{k},p}&\text{if $q\neq p$.}\end{cases}

A metric space $(X,d)$ is said to be spherically complete if for every sequence of (closed or open) balls $\{B_{i}\}_{i\in\mathbb{Z}_{\geq 0}}$ with $B_{i+1}\subseteq B_{i+1}$ for all $n\in\mathbb{Z}_{\geq 0}$ , we have $\bigcap_{i\in\mathbb{Z}_{\geq 0}}B_{i}\neq\emptyset$ .

Proposition 2.11.

Let $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , and $\boldsymbol{k}$ be a field of characteristic $p$ . Then the following statements are true:

(1)

$U_{G,\boldsymbol{k},q,p}(\mathbb{A}_{q,p}(G,\boldsymbol{k}))=G$ ;
(2)

$\mathfrak{K}(\mathbb{A}_{q,p}(G,\boldsymbol{k}),U_{G,\boldsymbol{k},q,p})=\boldsymbol{k}$ ;
(3)

$(\mathbb{A}_{q,p}(G,\boldsymbol{k}),U_{G,\boldsymbol{k},q,p})$ is spherically complete. In particular, it is complete.

Proof.

The statements (1) and (2) follows from the construction of $(\mathbb{A}_{q,p}(G,\boldsymbol{k}),U_{G,\boldsymbol{k},q,p})$ . The statement (3) is proven by [20, Theorem 1], and [12, Theorem 4] (see also [3, Theorem 6.11]). ∎

Next we consider homeomorphic embeddings between $p$ -adic or ordinary Hahn fields. We begin with ordinary ones. Let $G,H\in\mathcal{G}$ with $G\subseteq H$ , and $A,B$ be commutative rings. We represent $\iota$ as the inclusion map $G\to H$ . Let $\phi\colon A\to B$ be a ring homomorphism. For $x=\sum_{g\in G}x_{g}t^{g}\in\mathbb{H}(G,A)$ , we define $\mathbb{H}(\iota,\phi)(x)\in\mathbb{H}(H,B)$ by $\mathbb{H}(\iota,\phi)(x)=\sum_{h\in H}y_{h}t^{h}$ , where

y_{h}=\begin{cases}\phi(x_{h})&\text{if $h\in H$;}\\ 0&\text{if $h\not\in H$.}\end{cases}

Then $\mathbb{H}(\iota,\phi)\colon\mathbb{H}(G,A)\to\mathbb{H}(H,B)$ becomes a map. If $G=H$ , we simply write it as $\mathbb{H}(G,\phi)$ . Let us observe properties of $\mathbb{H}(\iota,\phi)$ .

Proposition 2.12.

Let $A,B$ be commutative rings, and $\phi\colon A\to B$ be a ring homomorphism. Let $G,H\in\mathcal{G}$ such that $G\subseteq H$ , and denote by $\iota$ the inclusion map $G\to H$ . Then the map $\mathbb{H}(\iota,\phi)\colon\mathbb{H}(G,A)\to\mathbb{H}(H,B)$ is a ring homomorphism and satisfies $\zeta_{\mathbb{H}(H,B)}\circ\mathbb{H}(\iota,\phi)=\phi\circ\zeta_{\mathbb{H}(G,A)}$ on $\mathfrak{A}(\mathbb{H}(G,A),v_{G,A})$ .

Proof.

The lemma follows from the definitions of $\mathbb{H}(\iota,\phi)$ and Hahn fields. ∎

Next we discuss $p$ -adic Hahn fields, which is defined by quotient fields of Hahn rings.

Proposition 2.13.

Let $G,H\in\mathcal{G}$ , $\boldsymbol{k}$ and $\boldsymbol{l}$ be perfect field with characteristic $p>0$ and $\phi\colon\boldsymbol{k}\to\boldsymbol{l}$ is a homomorphism. Then the homomorphism $\mathbb{H}(\iota,\mathbb{W}(\phi))\colon\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))\to\mathbb{H}(H,\mathbb{W}(\boldsymbol{l}))$ satisfies

\mathbb{H}(\iota,\mathbb{W}(\phi))(\mathbb{N}_{G,\boldsymbol{k}})\subseteq\mathbb{N}_{H,\boldsymbol{l}}.

in particular, the map $\mathbb{H}(\iota,\mathbb{W}(\phi))$ induces a homomorphism

\mathbb{P}_{p}(\iota,\phi)\colon\mathbb{P}_{p}(G,\boldsymbol{k})\to\mathbb{P}_{p}(H,\boldsymbol{l})

such that $\zeta_{\mathbb{P}_{p}(H,\boldsymbol{l})}\circ\mathbb{P}_{p}(\iota,\phi)=\phi\circ\zeta_{\mathbb{P}_{p}(G,\boldsymbol{k})}$ on $\mathfrak{A}(\mathbb{P}_{p}(G,\boldsymbol{k}),V_{G,\boldsymbol{k},p})$ .

Proof.

Take $x\in\mathbb{N}_{G,\boldsymbol{k}}$ and put $x=\sum_{g\in G}x(g)p^{g}$ . Then for every $g\in G$ we have $\sum_{n\in\mathbb{Z}}x(g+n)p^{n}=0$ in $\mathbb{W}(\boldsymbol{k})$ . Note that for a fixed $g\in G$ and for a sufficient large $m\in\mathbb{Z}_{\geq 0}$ , we have $x(g+n)=0$ for all $n<-m$ since $\{\,g\in G\mid x(g)\neq 0\,\}$ is well-ordered. By the strong triangle inequality, $\sum_{n\in\mathbb{Z}}x(g+n)p^{n}=0$ is equivalent to $x(g+n)p^{n}\to 0$ in $\mathbb{W}(\boldsymbol{k})$ as $n\to\infty$ (see [3, Theorem 2.24]). Since $w_{\boldsymbol{l}}(\mathbb{W}(\phi)(x)=w_{\boldsymbol{k}}(x)$ for all $x\in\mathbb{W}(\boldsymbol{k})$ (see Proposition 2.7), we also have $\mathbb{W}(\phi)(x(g+n))p^{n}\to 0$ in $\mathbb{W}(\boldsymbol{l})$ as $n\to\infty$ . Thus the infinite sum $\sum_{n\in\mathbb{Z}}\mathbb{W}(\phi)(x(g+n))p^{n}$ is convergent and we have

\sum_{n\in\mathbb{Z}}\mathbb{W}(\phi)(x(g+n))p^{n}=\mathbb{W}(\phi)\left(\sum_{n\in\mathbb{Z}}x(g+n)p^{n}\right)=\mathbb{W}(\phi)(0)=0.

This shows that $\mathbb{H}(\iota,\mathbb{W}(\phi))(x)\in\mathbb{N}_{H,\boldsymbol{l}}$ , and hence

\mathbb{H}(\iota,\mathbb{W}(\phi))(\mathbb{N}_{G,\boldsymbol{k}})\subseteq\mathbb{N}_{H,\boldsymbol{l}}.

In particular the map $\mathbb{H}(\iota,\mathbb{W}(\phi))$ induces a map $\mathbb{P}_{p}(\iota,\phi)\colon\mathbb{P}_{p}(G,\boldsymbol{k})\to\mathbb{P}_{p}(G,\boldsymbol{k})$ .

Proposition 2.12 implies that a map $\mathbb{H}(\iota,\mathbb{W}(\phi))\colon\mathbb{H}(H,\mathbb{W}(\boldsymbol{k}))\to\mathbb{H}(G,\mathbb{W}(\boldsymbol{l}))$ satisfies $\zeta\circ\mathbb{H}(\iota,\phi)=\phi\circ\zeta$ . Then $\zeta\circ\mathbb{P}_{p}(\iota,\phi)=\phi\circ\zeta$ . ∎

Let $G,H\in\mathcal{G}$ with $G\subseteq H$ , $(q,p)\in\mathcal{CH}$ , $\boldsymbol{k}$ and $\boldsymbol{l}$ be fields of characteristic $p$ , and $\phi\colon\boldsymbol{k}\to\boldsymbol{l}$ be a homomorphism. Denote by $\iota\colon G\to H$ the inclusion map. We define

\mathbb{A}_{q,p}(\iota,\phi)=\begin{cases}\mathbb{H}(\iota,\phi)&\text{if $q=p$;}\\ \mathbb{P}_{p}(\iota,\phi)&\text{if $q\neq p$.}\end{cases}

Let $(K,v)$ and $(L,w)$ be valued fields. We say that $(L,w)$ is a valued field extension of $(K,v)$ as a valued field if $K\subseteq L$ and $w|_{K}=v$ .

Proposition 2.14.

Let $\boldsymbol{k}$ and $\boldsymbol{l}$ be perfect field with characteristic $p>0$ and $\phi\colon\boldsymbol{k}\to\boldsymbol{l}$ be a homomorphism. The map $\mathbb{A}_{q,p}(\iota,\phi)\colon\mathbb{A}_{q,p}(G,\boldsymbol{k})\to\mathbb{A}_{q,p}(H,\boldsymbol{l})$ is a homomorphism such that $\zeta_{\mathbb{A}_{q,p}(H,\boldsymbol{l})}\circ\mathbb{A}_{q,p}(\iota,\phi)=\phi\circ\zeta_{\mathbb{A}_{q,p}(G,\boldsymbol{k})}$ on $\mathfrak{A}(\mathbb{A}_{q,p}(G,\boldsymbol{k}),U_{G,\boldsymbol{k},q,p})$ . In addition, if $q\neq p$ , then, the following are true:

(1)

$(\mathbb{A}_{q,p}(G,\boldsymbol{k}),U_{G,\boldsymbol{k},q,p})$ is a valued field extension of $(\mathrm{Fr}\mathbb{W}(\boldsymbol{k}),w_{\boldsymbol{k}})$ ;
(2)

In particular, $(\mathbb{A}_{q,p}(G,\boldsymbol{k}),U_{G,\boldsymbol{k},q,p})$ is a valued field extension of $(\mathbb{Q}_{p},v_{p})$ .

Proof.

Propositions 2.12 and 2.13 shows the former part of the statement. By the former part, since $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})=\mathbb{A}_{q,p}(\mathbb{Z},\boldsymbol{k})$ and $\mathbb{Z}\subseteq G$ (see the definition of $\mathcal{G}$ ), we can regard $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ as a subfield of $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ Namely, (1) is true. Similarly, by $\mathbb{Z}\subseteq G$ , $\mathbb{F}_{p}\subseteq\boldsymbol{k}$ , and since $(\mathbb{Q}_{p},v_{p})$ is equal to $(\mathbb{A}_{q,p}(\mathbb{Z},\mathbb{F}_{p}),U_{\mathbb{Z},\mathbb{F}_{p},q,p})$ , the field $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ is a valued field extension of $(\mathbb{Q}_{p},v_{p})$ . This implies (2). ∎

Definition 2.3.

Let $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , and let $\boldsymbol{k}$ be a perfect field of characteristic $p$ . We make the following assumptions and definitions.

(1)

In the rest of the paper, whenever we take a complete system $J\subseteq\mathbb{A}_{q,p}(G,\boldsymbol{k})$ of representatives of the residue class field $\boldsymbol{k}$ , in the case of $q=p$ , we define $J=\boldsymbol{k}$ using the fact that $\boldsymbol{k}\subseteq\mathbb{H}(G,\boldsymbol{k})$ . In the case of $q\neq p$ , we take $J\subseteq\mathbb{P}_{p}(G,\boldsymbol{k})$ such that $J\subseteq\mathbb{W}(\boldsymbol{k})$ based on Proposition 2.14.
(2)

We define an element $\tau\in\mathbb{A}_{q,p}(G,\boldsymbol{k})$ as follows. If $q=p$ , we define $\tau$ by an indeterminate as in the definition of Hahn fields. If $q\neq p$ , we define $\tau=p\in\mathbb{P}_{p}(G,\boldsymbol{k})$ . In this case, every element of $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ can be represented as a power series of $\tau$ with powers in $G$ .
(3)

For $y\in\mathbb{A}_{q,p}(G,\boldsymbol{k})$ , and $g\in G$ , if $q=p$ , then we define $\boldsymbol{C}(y,g)$ the coefficient of $\tau^{g}$ in the power series representation of $y$ . If $q\neq p$ , then we fixed a complete system $J\subseteq\mathbb{W}(\boldsymbol{k})$ of representatives of $\boldsymbol{k}$ , and we define $\boldsymbol{C}(y,g)\in J$ the coefficient of $y$ with respect to $\tau^{g}$ . Of cause, in this case, the value $\boldsymbol{C}(y,g)$ depends on a system $J$ of representatives. Throughout this paper, we will not consider the situation where we change a system of representatives. Thus no confusion can arise even if $J$ does not explicitly appear in the notation of $\boldsymbol{C}(y,g)$ . Notice that we can represent $y=\sum_{g\in G}\boldsymbol{C}(y,g)\tau^{g}$ .

Remark 2.2.

Related to Proposition 2.14, we make the next remarks.

(1)

$\mathbb{A}_{q,p}(\iota,\phi)$ is injective since it is a homomorphism between fields.
(2)

The construction of $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ is a functor.
(3)

In contrast to Proposition 2.7, the author does not know whether $\mathbb{A}_{q,p}(\iota,\phi)$ is a unique homeomorphism such that $\zeta\circ\mathbb{A}_{q,p}(\iota,\phi)=\phi\circ\zeta$ or not.

A group $G$ is said to be divisible if for every $g\in G$ and for every $n\in\mathbb{Z}_{\geq 0}$ there exists $h\in G$ such that $g=n\cdot h$ .

Proposition 2.15.

Let $G\in\mathcal{G}$ be divisible, $(q,p)\in\mathcal{CH}$ , $\boldsymbol{k}$ be an algebraically closed field with characteristic $p>0$ , and $(K,v)$ be a valued field such that $v(K)\subseteq G\sqcup\{\infty\}$ and $\mathfrak{K}(K,v)\subseteq\boldsymbol{k}$ . Then there exists a homomorphic embedding $\phi\colon K\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ such that $v(x)=U_{G,\boldsymbol{k},q,p}(\phi(x))$ fot all $x\in K$ . Namely, the field $(K,v)$ can be regarded as a valued subfield of $(\mathbb{A}_{q,p}(G,\boldsymbol{k}),U_{G,\boldsymbol{k},q,p})$ .

Proof.

See [20, Corollary 5]. ∎

2.2.3. Levi–Civita fields

We next discuss Levi–Civita fields and $p$ -adic Levi–Civita fields, which are will be used in Section 5.

Let $G\in\mathcal{G}$ , and $\boldsymbol{k}$ be a field. For $f\in\mathbb{H}(G,\boldsymbol{k})$ , in this subsection, we mainly consider the following condition.

(Fin)

For every $n\in\mathbb{Z}$ , the set $\text{{\rm supp}}(f)\cap(-\infty,n]$ is finite.

We denote by $\mathbb{L}[G,\boldsymbol{k}]$ the set of all $f\in\mathbb{H}(G,\boldsymbol{k})$ satisfying the condition (Fin). For the next lemma, we refer the readers to [3, Theorem 3.18].

Lemma 2.16.

Let $G\in\mathcal{G}$ , $\boldsymbol{k}$ be a field. Then the set $\mathbb{L}[G,\boldsymbol{k}]$ is a subfield of $\mathbb{H}(G,\boldsymbol{k})$ .

We call $\mathbb{L}[G,\boldsymbol{k}]$ the Levi–Civita field associated with $G$ and $\boldsymbol{k}$

Fix $(q,p)\in\mathcal{CH}$ with $q\neq p$ and assume that $\boldsymbol{k}$ has characteristic $p>0$ . Before defining a $p$ -adic analogue of Levi-Civita fields, we supplementally define a subset $\mathbb{D}[G,\boldsymbol{k}]$ of $\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ $\mathbb{M}_{p}[G,\boldsymbol{k}]$ by the set of all members $f\in\mathbb{M}_{p}[G,\boldsymbol{k}]$ satisfying the condition (Fin). We define $\mathbb{M}_{p}[G,\boldsymbol{k}]$ by $\mathbb{M}_{p}[G,\boldsymbol{k}]=\mathrm{Pr}(\mathbb{D}[G,\boldsymbol{k}])$ , where $\mathrm{Pr}\colon\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))\to\mathbb{P}_{p}(G,\boldsymbol{k})$ is the canonical projection.

Lemma 2.17.

Let $G\in\mathcal{G}$ , $p$ be a prime, and $\boldsymbol{k}$ be a field of characteristic $p>0$ . Then the following statements are true:

(1)

For every complete system $J\subseteq\mathbb{P}_{p}(G,\boldsymbol{k})$ of representatives of $\boldsymbol{k}$ , and for every $a\in\mathbb{D}[G,\boldsymbol{k}]$ , the member $f=\mathrm{St}_{G,\boldsymbol{k},J}\in\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))(x)$ satisfies the condition (Fin) and $f(g)\in J$ for all $g\in G$ .
(2)

We have $\mathbb{D}[G,\boldsymbol{k}]$ is a subring of $\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ ;
(3)

If $a\in\mathbb{D}[G,\boldsymbol{k}]$ satisfies $U_{G,\boldsymbol{k},q,p}(a)>0$ , then $(1-a)^{-1}\in\mathbb{D}[G,\boldsymbol{k}]$ .
(4)

For every $a\in\mathbb{D}[G,\boldsymbol{k}]$ , there exists $b\in\mathbb{D}[G,\boldsymbol{k}]$ such that $ab$ is equivalent to $1$ modulo $\mathbb{N}_{G,\boldsymbol{k}}$ .

Proof.

First we prove (1). By Lemma 2.9, we see that $f(g)\in J$ for all $g\in G$ . Put $A=\text{{\rm supp}}(a)$ and $F=\text{{\rm supp}}(f)$ . Lemma 2.9 also shows that $F\subseteq A+\mathbb{Z}_{\geq 0}$ . Due to this relation, since $A$ satisfies the condition (Fin), so does $W$ . Hence the statement (1) is true.

Next we prove (2). By the definitions of $\mathbb{D}[G,\boldsymbol{k}]$ and $\mathbb{L}[G,\mathrm{Fr}\mathbb{W}(\boldsymbol{k})]$ , we have $\mathbb{D}[G,\boldsymbol{k}]=\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))\cap\mathbb{L}[G,\mathrm{Fr}\mathbb{W}(\boldsymbol{k})]$ (pay an attention to the difference between $\mathbb{W}(\boldsymbol{k})$ and $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ appearing in $\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ and $\mathbb{L}[G,\mathrm{Fr}\mathbb{W}(\boldsymbol{k})]$ , respectively). Thus $\mathbb{D}[G,\boldsymbol{k}]$ is a subring of $\mathbb{H}(G,\mathbb{W}(\boldsymbol{k}))$ .

Now we show (3). Put $m=U_{G,\boldsymbol{k},q,p}(a)$ . As in [20], we have $(1-a)^{-1}=1+a^{2}+a^{3}+\cdots$ in $\mathbb{H}(G,\boldsymbol{k})$ . For every $i\in\mathbb{Z}_{\geq 0}$ , we have $A_{i}=\text{{\rm supp}}(\mathrm{St}_{G,\boldsymbol{k},J}(a^{i}))$ . In this case, we have $i\cdot m\leq\min A_{i}$ for all $i\in\mathbb{Z}_{\geq 1}$ . Put $B=\bigcup_{i\in\mathbb{Z}_{\geq 1}}A_{i}$ . Since $A_{i}$ satisfies $i\cdot m\leq\min A_{i}$ and $(-\infty,n]\cap A_{i}$ is finite for all $i,n\in\mathbb{Z}_{\geq 0}$ , we observe that $B$ satisfies that $(-\infty,n]\cap B$ is finite for all $n\in\mathbb{Z}_{\geq 0}$ . Put $E=\text{{\rm supp}}(\mathrm{St}_{G,\boldsymbol{k},J}(1-a))$ . Then $E\subseteq B+\mathbb{Z}_{\geq 0}$ , and hence $E$ satisfies that $(-\infty,n]\cap E$ is finite for all $n\in\mathbb{Z}_{\geq 0}$ . This implies that $(1-a)^{-1}\in\mathbb{D}[G,\boldsymbol{k}]$ .

We shall prove (4) Put $f=\mathrm{St}_{G,\boldsymbol{k},J}(a)$ . We only need to show that $f$ is invertible in $\mathbb{D}[G,\boldsymbol{k}]$ . We represent $f$ as $f=\sum_{g\in G}f_{g}t^{g}$ . Take $m=\min\text{{\rm supp}}(f)$ and put $y=\sum_{m<g}f_{g}t^{g-m}$ . Since $w_{\boldsymbol{k}}(f_{m})=0$ , we see that $f_{m}$ is invertible in $\mathbb{W}(\boldsymbol{k})$ . Then $f=f_{m}t^{m}(1-(-f_{m}^{-1}y))$ . $U_{G,\boldsymbol{k},q,p}(f_{m}^{-1}y)>0$ . Thus, by $f=p^{m}f_{m}(1-(-f_{m}^{-1}y))$ , and by (3), we see that $x$ is invertible in $\mathbb{D}[G,\boldsymbol{k}]$ . ∎

As a consequence of Lemma 2.17, we obtain the next corollary.

Corollary 2.18.

Let $G\in\mathcal{G}$ , $p$ be a prime, and $\boldsymbol{k}$ be a field of characteristic $p>0$ . Then the following statements are true:

(1)

For every complete system $J\subseteq\mathbb{P}_{p}(G,\boldsymbol{k})$ of representatives of $\boldsymbol{k}$ , the set $\mathbb{M}_{p}[G,\boldsymbol{k}]$ is equal to the set $\mathrm{Pr}(\mathrm{St}_{G,\boldsymbol{k},J}(\mathbb{D}[G,\boldsymbol{k}]))$ . Moreover, for every $a\in\mathbb{M}_{p}[G,\boldsymbol{k}]$ , there uniquely exists $f\in\mathrm{St}_{G,\boldsymbol{k},J}(\mathbb{D}[G,\boldsymbol{k}])$ such that $a=\mathrm{Pr}(f)$ .
(2)

The set $\mathbb{M}_{p}[G,\boldsymbol{k}]$ is a subfield of $\mathbb{P}_{p}(G,\boldsymbol{k})$ .

We call $\mathbb{M}_{p}[G,\boldsymbol{k}]$ the $p$ -adic Levi–Civita field associated with $G$ and $\boldsymbol{k}$ . We simply represent the restriction $U_{G,\boldsymbol{k},q,p}|_{\mathbb{B}_{q,p}(G,\boldsymbol{k})}$ as the same symbol $U_{G,\boldsymbol{k},q,p}$ . In this setting, the field $(\mathbb{B}_{q,p}(G,\boldsymbol{k}),U_{G,\boldsymbol{k},q,p})$ becomes a valued subfield of $(\mathbb{A}_{q,p}(G,\boldsymbol{k}),U_{G,\boldsymbol{k},q,p})$ .

To use $p$ -adic and ordinary Levi-Civita fields in a unified way, we make the next definition.

Definition 2.4.

Let $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , and $\boldsymbol{k}$ be a field of characteristic $p$ . We define $\mathbb{B}_{q,p}(G,\boldsymbol{k})$ by

\mathbb{B}_{q,p}(G,\boldsymbol{k})=\begin{cases}\mathbb{L}[G,\boldsymbol{k}]&\text{if $q=p$;}\\ \mathbb{M}_{p}[G,\boldsymbol{k}]&\text{if $q\neq p$.}\end{cases}

Proposition 2.19.

Let $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , and $\boldsymbol{k}$ be a field of characteristic $p$ . Then the set $\mathbb{B}_{q,p}(G,\boldsymbol{k})$ is a subfield of $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ .

Proof.

The case of $q=p$ is presented in Lemma 2.16. The case of $q\neq p$ is proven in Corollary 2.18. ∎

3. Algebraic independence over valued fields

First we remark that, for valued fields $(K,v)$ and $(L,w)$ such that $(L,w)$ is a valued field extension of $(K,v)$ , there exists a canonical injective embedding from $\mathfrak{K}(K,v)$ into $\mathfrak{K}(L,w)$ . Namely, we can regard $\mathfrak{K}(K,v)$ as a subset of $\mathfrak{K}(L,w)$ since the inclusion map $\iota\colon K\to L$ satisfies $\iota(\mathfrak{A}(K,v))\subseteq\mathfrak{A}(L,w))$ and $\iota(\mathfrak{o}(K,v))\subseteq\mathfrak{o}(L,w)$ . Thus it naturally induce an homomorphic embedding $\mathfrak{K}(K,v)\to\mathfrak{K}(L,w)$ .

Let $K$ and $L$ be fields with $K\subseteq L$ . A member $x$ of $L$ is said to be transcendental over $K$ if $x$ is not a root of any non-trivial polynomial with coefficients in $K$ . A subset $S$ of $L$ is said to be algebraically independent if any finite collection $x_{1},\dots,x_{n}$ in $S$ does not satisfy any non-trivial polynomial equation with coefficients in $K$ . Note that a singleton $\{x\}$ of $L$ is algebraically independent over $K$ if and only if $x$ is transcendental over $K$ .

Lemma 3.1.

Let $(K_{0},v_{0})$ and $(K_{1},v_{1})$ be valued fields. Assume that $(K_{1},v_{1})$ is a valued field extension of $(K_{0},v_{0})$ . If $x\in\mathfrak{A}(K_{1},v_{1})$ such that $\zeta(x)\in\mathfrak{K}(K_{1},v_{1})$ is transcendental over $\mathfrak{K}(K_{0},v_{0})$ , then $x$ is transcendental over $K_{0}$ .

Proof.

The lemma follows from [5, Theorem 3.4.2]. ∎

Lemma 3.2.

Let $(K_{0},v_{0})$ and $(K_{1},v_{1})$ be valued fields such that $(K_{1},v_{1})$ is a valued field extension of $(K_{0},v_{0})$ . If $x_{1},\dots,x_{n},y\in K_{1}$ satisfy that:

(1)

the set $\{x_{1},\dots,x_{n}\}$ is algebraically independent over $K_{0}$ ;
(2)

there exists a filed $L$ containing $x_{1},\dots,x_{n}$ and satisfying $K_{0}\subseteq L\subseteq K_{1}$ for which there exist $z\in L$ and $c\in K_{0}$ satisfying that $c(y-z)\in\mathfrak{A}(K_{1},v_{1})$ and $\zeta(c(y-z))\in\mathfrak{K}(K_{1},v_{1})$ is transcendental over $\mathfrak{K}(L,v_{1}|_{L})$ ,

then the set $\{x_{1},\dots,x_{n},y\}$ is algebraically independent over $K_{0}$ .

Proof.

For the sake of contradiction, suppose that the set $\{x_{1},\dots,x_{n},y\}$ is not algebraically independent over $K_{0}$ . From (1) and the fact that $L$ contains $x_{1},\dots,x_{n}$ , it follows that $y$ is algebraic over $L$ , and hence so is $c(y-z)$ . Using Lemma 3.1 together with (2), we see that $c(y-z)$ is transcendental over $L$ . This is a contradiction. Therefore, the set $\{x_{1},\dots,x_{n},y\}$ is algebraically independent over $K_{0}$ . ∎

Lemma 3.3.

Let $\eta\in(1,\infty)$ , $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , and let $\boldsymbol{k}$ be a perfect field with characteristic $p$ . Fix a cardinal $\theta$ and a complete system $J\subseteq\mathbb{A}_{q,p}(G,\boldsymbol{k})$ of representatives of the residue class field $\boldsymbol{k}$ . If a set $S$ of non-zero elements of $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ satisfies that:

(N1)

for every pair $x,y\in S$ with $x\neq y$ , and for every $g\in G$ satisfying that $g\in[v(x-y),\infty)$ , if either of $\boldsymbol{C}(x,g)$ and $\boldsymbol{C}(y,g)$ is non-zero, then $\boldsymbol{C}(x,g)\neq\boldsymbol{C}(y,g)$ ,

then for every finite subset $A=\{z_{1},\dots,z_{n}\}$ of $S$ , there exist $i\in\{1,\dots,n\}$ and $u\in G$ such that $\boldsymbol{C}(z_{i},u)\neq 0$ and $\boldsymbol{C}(z_{i},u)\neq\boldsymbol{C}(z_{j},u)$ for all $j\in\{1,\dots,n\}$ .

Proof.

Put $u=\min\{\,U_{G,\boldsymbol{k},q,p}(z_{i}-z_{j})\mid i\neq j\,\}$ and take a pair $\{i,n\}$ such that $u=U_{G,\boldsymbol{k},q,p}(z_{i}-z_{n})$ . Then either $\boldsymbol{C}(z_{i},u)$ or $\boldsymbol{C}(z_{n},u)$ is non-zero. We my assume that $\boldsymbol{C}(z_{i},u)\neq 0$ . Using the condition (N1) and the minimality of $u$ , we have $\boldsymbol{C}(y_{i},u)\neq\boldsymbol{C}(y_{j},u)$ for all $j\in\{1,\dots,n\}$ . ∎

Let $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , $\boldsymbol{k}$ be a field of characteristic $p$ , and $K$ be a subfield of $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ . Fix a complete system $J\subseteq\mathbb{A}_{q,p}(G,\boldsymbol{k})$ of representatives if $q\neq p$ . We denote by $\mathbf{AH}(K)$ the subfield of $\boldsymbol{k}$ generated by $\{\,\zeta(\boldsymbol{C}(x,g))\mid x\in K,g\in G\,\}$ over $\mathfrak{K}(K,v)$ . The definition of $\mathbf{AH}(K)$ is “ad-hoc”, which means that it depends on not only information of $K$ , but also the inclusion map $K\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ . Namely, even if $K,L\subseteq\mathbb{A}_{q,p}(G,\boldsymbol{k})$ are isomorphic each other as fields, it can happen that $\mathbf{AH}(K)\neq\mathbf{AH}(L)$ .

Proposition 3.4.

Let $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , and $\boldsymbol{k},\boldsymbol{l}$ be perfect fields with characteristic $p$ such that $\boldsymbol{k}\subseteq\boldsymbol{l}$ , Take a subfield $K$ of $\mathbb{A}_{q,p}(G,\boldsymbol{l})$ such that $\mathfrak{K}(K,v)\subseteq\boldsymbol{k}$ . Fix a system $J\subseteq\mathbb{A}_{q,p}(G,\boldsymbol{l})$ of representatives of $\boldsymbol{l}$ . If a set of $S$ of non-zero elements of $\mathbb{A}_{q,p}(G,\boldsymbol{l})$ satisfies the condition (N1) in Lemma 3.3 and the following:

(T1)

for every pair $x,y\in S$ , and for every distinct pair $g,g^{\prime}\in G$ , if $\boldsymbol{C}(x,g)\neq 0$ , then we have $\boldsymbol{C}(x,g)\neq\boldsymbol{C}(y,g^{\prime})$ ;
(T2)

the set $\{\,\zeta(\boldsymbol{C}(x,g))\mid x\in S,g\in G,\boldsymbol{C}(x,g)\neq 0\,\}$ is algebraically independent over $\mathbf{AH}(K)$ ,

then the set $S$ is algebraically independent over $K$ .

Proof.

Let $\tau$ be the same element of $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ as in Definition 2.4. Take $n$ -many distinct members $z_{1},\dots,z_{n}$ in $S$ . Now we prove that $\{z_{1},\dots,z_{n}\}$ is algebraically independent over $K$ by induction on $n$ .

In the case of $n=1$ , put $z_{1}=\sum_{g\in G}\boldsymbol{C}(z_{1},g)\tau^{g}$ . Take $u\in G$ such that $\boldsymbol{C}(z_{1},u)\neq 0$ . Put $A=\{\,\boldsymbol{C}(z_{1},g)\mid g\in G,g\neq u\,\}$ . Then, due to the condition (T1), we have $\boldsymbol{C}(z_{1},u)\not\in A$ . Note that the set $\zeta(A)$ is algebraically independent over $\boldsymbol{k}$ . Let $\boldsymbol{m}$ be the perfect subfield of $\boldsymbol{l}$ generated by $\mathbf{AH}(K)\cup\zeta(A)$ , and put $L=\mathbb{A}_{q,p}(G,\boldsymbol{m})$ . Notice that $\mathbb{A}_{q,p}(G,\boldsymbol{m})$ is a subfield of $\mathbb{A}_{q,p}(G,\boldsymbol{l})$ (see Proposition 2.14). The fact that $\mathbf{AH}(K)\subseteq\boldsymbol{m}$ implies that $K\subseteq L$ . By assumption (T2) and $\boldsymbol{C}(z_{1},u)\not\in A$ , we see that $\boldsymbol{C}(z_{1},u)$ is transcendental over $\boldsymbol{m}$ . Thus, by Lemma 3.1, we conclude that $z_{1}$ is transcendental over $L$ . In particular, $z_{1}$ is transcendental over $K$ .

Next, we fix $k\in\mathbb{Z}_{\geq 0}$ and assume that the case of $n=k$ is true. We consider the case of $n=k+1$ . Since $S$ satisfies the condition (N1), we can take $i\in\{1,\dots,n\}$ and $u\in G$ stated in Lemma 3.3. We may assume that $i=k+1$ . Put

A=\{\,\boldsymbol{C}(z_{i},g)\mid g\in G,i=1,\dots,k\,\}\cup\{\,\boldsymbol{C}(z_{k+1},g)\mid g\in G,g\neq u\,\}.

According to (T1) and the conclusion of Lemma 3.3, we see that $\boldsymbol{C}(z_{k+1},u)\not\in A$ . Let $\boldsymbol{m}$ be a perfect subfield of $\boldsymbol{l}$ generated by $\mathbf{AH}(K)\cup\zeta(A)$ , and put $L=\mathbb{A}_{q,p}(G,\boldsymbol{m})$ . Similarly to the case of $n=1$ , we observe that $K\subseteq L$ . Since $A\subseteq\boldsymbol{m}$ , we have $z_{i}\in L$ for all $i\in\{1,\dots,k\}$ . Define $f=z_{k+1}-\boldsymbol{C}(z_{k+1},u)\tau^{u}$ . Due to the condition (T1), we have $f\in L$ . By the definition, we also have $\tau^{-u}(z_{k+1}-f)=\boldsymbol{C}(z_{k+1},u)$ . Thus the condition (T2) shows that $\zeta(\tau^{-u}(z_{k+1}-f))$ is transcendental over $\boldsymbol{m}$ . Hence Lemma 3.2 shows that the set $\{z_{1},\dots,z_{k},z_{k+1}\}$ is algebraically independent over $K$ . This finishes the proof. ∎

Remark 3.1.

Put $w=\boldsymbol{C}(x,g)$ . The condition (T1) means that $\boldsymbol{C}(x,g)$ is zero or $g$ is a unique member in $G$ such that $\boldsymbol{C}(y,g)=w$ for some $y\in X$ .

4. Isometric embeddings of ultrametric spaces

In this section, we prove our non-Archimedean analogue of the Arens–Eells theorem. As a consequence, we give an affirmative solution of Conjecture 1.1.

4.1. A non-Archimedean Arens–Eells theorem

4.1.1. Preparations

This subsection is devoted to proving the following technical theorem, which plays a central role of our first main theorem. Our proof of the next theorem can be considered as a sophisticated version of the proof of the main theorem of [21].

Theorem 4.1.

Let $\eta\in(1,\infty)$ , $(q,p)\in\mathcal{CH}$ , $G\in\mathcal{G}$ , $\boldsymbol{k}$ be a field, and $\boldsymbol{l}$ be a perfect field of characteristic $p$ Fix a cardinal $\theta$ and a complete system $J\subseteq\mathbb{A}_{q,p}(G,\boldsymbol{l})$ of representatives of the residue class field $\boldsymbol{l}$ . Let $C$ be a subset of $J$ . Put $R=\{0\}\sqcup\{\,\eta^{-g}\mid g\in G\,\}$ . If the following condition are satisfied:

(A1)

$\boldsymbol{l}$ is a field extension of $\boldsymbol{k}$ ;
(A2)

the subset $\zeta(C)$ of $\boldsymbol{l}$ is algebraically independent over $\boldsymbol{k}$ ;
(A3)

$\operatorname{Card}(C)=\theta$ ,

then for every $R$ -valued ultrametric space $(X,d)$ with $\operatorname{Card}(X)\leq\theta$ , there exists a map $I\colon X\to\mathbb{A}_{q,p}(G,\boldsymbol{l})$ such that:

(B1)

each $I(x)$ is non-zero;
(B2)

the map $I$ is an isometric embedding from $(X,d)$ into the ultrametric space $(\mathbb{A}_{q,p}(G,\boldsymbol{k}),\lVert*\rVert_{U_{G,\boldsymbol{k},q,p},\eta})$ ;
(B3)

for every pair $x,y\in X$ , and for every $r\in(0,d(x,y)]$ , if either of $\boldsymbol{C}(I(x),-\log_{\eta}(r))$ or $\boldsymbol{C}(I(y),-\log_{\eta}(r))$ is non-zero, then we have $\boldsymbol{C}(I(x),-\log_{\eta}(r))\neq\boldsymbol{C}(I(y),-\log_{\eta}(r))$ ;
(B4)

for every pair $x,y\in X$ , and for every distinct pair $g,g^{\prime}\in G$ , if $\boldsymbol{C}(I(x),g)\neq 0$ , then $\boldsymbol{C}(I(x),g)\neq\boldsymbol{C}(I(y),g^{\prime})$ .
(B5)

for every $\alpha<\theta$ , the set $\{\,\boldsymbol{C}(I(\xi_{\alpha}),g)\mid g\in G\,\}$ is contained in $C$ .

In this subsection, in what follows, we fix objects in the assumption of Theorem 4.1. We divide the proof of Theorem 4.1 into some lemmas.

First we prepare notations. Take $\varpi\not\in X$ , and put $E=X\sqcup\{\varpi\}$ . Fix $r_{0}\in R\setminus\{0\}$ and $x_{0}\in X$ , and define an $R$ -valued ultrametric $h$ on $E$ by $h|_{X\times X}=d$ and $h(x,\varpi)=d(x,x_{0})\lor d(x_{0},\varpi)$ . Then $h$ is actually an ultrametric (see, for example, [7, Lemma 5.1]). A one-point extension of a metric space is a traditional method to prove analogues of the Arens–Eells theorem.

Put $C=\{\,b_{\alpha}\mid\alpha<\theta\,\}$ and $E=\{\,\xi_{\alpha}\mid\alpha<\theta\,\}$ with $\xi_{0}=\varpi$ . For every $\beta<\kappa$ , we also put $E_{\beta}=\{\,\xi_{\alpha}\mid\alpha<\beta\,\}$ and $C_{\beta}=\{\,b_{\alpha}\mid\alpha<\beta\,\}$ .

Fix $\lambda<\theta$ . We say that a map $f\colon E_{\lambda}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ is well-behaved if the following conditions are true:

(C1)

if $\lambda=0$ , then $H_{0}$ is the empty map and if $\lambda>0$ , then $H_{\lambda}(\varpi)=0$ , where $0$ is the zero element of $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ ;
(C2)

the map $f$ is an isometric embedding from $E_{\lambda}$ into $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ ;
(C3)

for every pair $x,y\in E_{\lambda}$ , and for every $r\in(0,d(x,y)]$ , if either of $\boldsymbol{C}(f(x),-\log_{\eta}(r))$ or $\boldsymbol{C}(f(y),-\log_{\eta}(r))$ is non-zero, then we have $\boldsymbol{C}(f(x),-\log_{\eta}(r))\neq\boldsymbol{C}(f(y),-\log_{\eta}(r))$ ;
(C4)

for every pair $x,y\in E_{\lambda}$ , and for every distinct pair $g,r^{\prime}\in G$ , if $\boldsymbol{C}(f(x),g)\neq 0$ , then $\boldsymbol{C}(f(x),g)\neq\boldsymbol{C}(f(y),g^{\prime})$ ;
(C5)

for every $\alpha<\lambda$ , the set $\{\,\boldsymbol{C}(f(\xi_{\alpha}),g)\mid g\in G\,\}$ is contained in $C_{\alpha+1}$ .

For an ordinal $\lambda<\theta$ , a family $\{H_{\alpha}\colon E_{\alpha}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})\}_{\alpha<\lambda}$ is said to be coherent if the following condition is true:

(Coh)

for every $\beta<\theta$ and for every $\alpha<\beta$ , we have $H_{\beta}|_{E_{\alpha}}=H_{\alpha}$ .

For the sake of simplicity, we define an ultrametric $e$ on $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ by $e(x,y)=\lVert x-y\rVert_{U_{G,\boldsymbol{k},q,p},\eta}$ .

We shall construct a coherent family $\{H_{\alpha}\}_{\alpha<\theta}$ of well-behaved maps using transfinite recursion. We begin with the following convenient criterion.

Lemma 4.2.

Fix $\lambda<\theta$ with $\lambda\neq 0$ . Put $u=d(E,\xi_{\lambda})$ and $m=-\log_{\eta}(u)$ . Let $H_{\lambda}\colon E_{\lambda}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ be a well-behaved map. Assume that an isometric embedding $H_{\lambda+1}\colon E_{\lambda+1}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ satisfies $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$ and the following property:

(P1)

For every $g\in G\cap(m,\infty)$ , we have $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}),g)=0$ ;
(P2)

If $m<\infty$ and $m\in G$ , then $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}),m)\in\{0,b_{\lambda}\}$ ;

Then tha map $H_{\lambda}$ satisfies the conditions (C3)–(C5).

Proof.

First we note that the following claim is true:

(CL)

For every $a\in(u,\infty)$ , there exists $z\in E_{\lambda}$ such that

$e(H_{\lambda+1}(z),H_{\lambda}(\xi_{\lambda}))<a.$

On the other words, for every $g\in(-\infty,m)$ , there exists $z\in E_{\lambda}$ such that

$\boldsymbol{C}(H_{\lambda+1}(z),n)=\boldsymbol{C}(H_{\lambda}(\xi_{\lambda}),n)$

for all $n<g$

Due to (C3) for $H_{\lambda}$ , it suffices to consider the case of $y=\xi_{\lambda}$ . Take $x\in E_{\lambda+1}$ and $r\in(0,d(x,\xi_{\lambda})]$ . Assume that either of $\boldsymbol{C}(f(x),-\log_{\eta}(r))$ or $\boldsymbol{C}(f(\xi_{\lambda}),-\log_{\eta}(r))$ is non-zero. In this case, we have $r\leq d(x,\xi_{\lambda})\leq u$ . Thus $m\leq-\log_{\eta}(r)$ . If $m<-\log_{\eta}(r)$ , then the property (P1) shows that $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}),-\log_{\eta}(r))=0$ . Thus the condition (C3) is valid. If $m=-\log_{\eta}(r)$ , then the property (P2) implies that $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}),m)\not\in C_{\lambda}$ . Thus, using (C5) for $H_{\lambda}$ , the condition (C3) is satisfied. In any case, we conclude that the condition (C3) is true.

Owing to (C4) for $H_{\lambda}$ , it is enough to confirm the case where $x=\xi_{\lambda}$ and $y\in E_{\lambda}$ , or $x\in E_{\lambda}$ and $y=\xi_{\lambda}$ . Take arbitrary distinct pair $g,g^{\prime}\in G$ . First assume that $x=\xi_{\lambda}$ , i.e., $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda}),g)\neq 0$ . By (P1), we have $g\leq m$ . In the case of $g=m$ , using (P2) and (C5) for $H_{\lambda}$ , we have $\boldsymbol{C}(f(x),g)\neq\boldsymbol{C}(f(y),g^{\prime})$ . In the case of $g<m$ . Then $\eta^{-g}\in(u,\infty)$ . Thus the property (CL) enables us to take $z\in E_{\lambda}$ such that $e(y_{i},\xi_{\lambda})<\eta^{-g}$ . Thus we also have $\boldsymbol{C}(H_{\lambda+1}(z),a)=\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda+1}),a)$ for all $a\leq g$ . Applying (C4) for $H_{\lambda}$ to $y$ and $z$ , we obtain that $\boldsymbol{C}(H_{\lambda+1}(z),g)\neq\boldsymbol{C}(H_{\lambda+1}(y),g^{\prime})$ . Hence, we have $\boldsymbol{C}(H_{\lambda+1}(\xi_{\lambda+1}),g)\neq\boldsymbol{C}(H_{\lambda+1}(y),g^{\prime})$ . Second assume that $y=\xi_{\lambda}$ , i.e., $\boldsymbol{C}(H_{\lambda+1}(x),g)\neq 0$ . By (P1), it is sufficient consider that $g\leq m$ . By (P2), if $g^{\prime}=m$ , then we have $\boldsymbol{C}(H_{\lambda+1}(x),g)\neq\boldsymbol{C}(H_{\lambda+1}(y),g^{\prime})$ . If $g^{\prime}<m$ , then the property (CL) enables us to take $z\in E_{\lambda}$ such that $e(y_{i},\xi_{\lambda})<\eta^{-g}$ . The remaining proof is similar to that of the first case.

By (CL), if $g<m$ we have $\boldsymbol{C}(H_{\lambda}(\xi_{\lambda}),g)\in C_{\lambda}$ . If $m<g$ , (P1) implies that $\boldsymbol{C}(H_{\lambda}(\xi_{\lambda}),g)=0$ . Thus By (P2) we have the set $\{\,\boldsymbol{C}(f(\xi_{\alpha}),g)\mid g\in G\,\}$ is contained in $C_{\lambda}\sqcup\{b_{\lambda}\}=C_{\lambda+1}$ . Namely, the condition (C5) is true. ∎

We next see the elementary lemma.

Lemma 4.3.

Let $\tau$ be the same element of $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ as in Definition 2.4. Then, for every $a\in\mathbb{A}_{q,p}(G,\boldsymbol{k})$ and for every $r\in(0,\infty)$ , we can take $\gamma\in B(a,r;e)$ such that $\gamma=\sum_{g\in G\cap(-\infty,-\log_{\eta}(r))}s_{g}\tau^{g}$ , where $s_{g}\in J$ . In this case, we have $B(a,r;e)=B(\gamma,r;e)$ .

Proof.

We put $a=\sum_{g\in G\cap(-\infty,-\log_{\eta}(r))}s_{g}\tau^{g}$ , where $s_{g}\in J$ and we define $\gamma=\sum_{g\in G\cap(-\infty,-\log_{\eta}(r))}s_{g}\tau^{g}$ . By the definition of $e$ , or $U_{G,\boldsymbol{k},q,p}$ , we have $\gamma\in B(a,r;e)$ . Thus Lemma 2.2 implies that $B(a,r;e)=B(\gamma,r;e)$ . ∎

Next we show lemmas corresponding to steps of isolated ordinals in transfinite induction.

Lemma 4.4.

Fix $\lambda<\theta$ with $\lambda\neq 0$ , and let $H_{\lambda}\colon E_{\lambda}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ be a well-behaved map. If $h(E_{\lambda},\xi_{\lambda})>0$ , then we can obtain a well-behaved isometric embedding $H_{\lambda+1}\colon E_{\lambda+1}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ such that $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$ .

Proof.

Put $Y_{\lambda}=H_{\lambda}(E_{\lambda})$ . Put $u=h(E_{\lambda},\xi_{\lambda})>0$ and $m=-\log_{\eta}(u)$ . Let $\tau$ be the same element in $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ as in Lemma 4.3.

Case 1. [There is no $a\in E_{\lambda}$ such that $h(a,\xi_{\lambda})=u$ ]: Take a sequence $\{y_{i}\}_{i\in\mathbb{Z}_{\geq 0}}$ in $E_{\lambda}$ such that $h(y_{i+1},\xi_{\lambda})<h(y_{i},\xi_{\lambda})$ for all $i\in\mathbb{Z}_{\geq 0}$ and $h(y_{i},\xi_{\lambda})\to u$ as $i\to\infty$ . Put $r_{i}=h(y_{i},\xi_{\lambda})$ . Since $r_{i+1}<r_{i}$ and $B(H(y_{i+1}),r_{i+1};e)\subseteq B(H(y_{i}),r_{i};e)$ for all $i\in\mathbb{Z}_{\geq 0}$ (see Lemma 2.2), using the spherical completeness of $\mathbb{A}_{q,p}(G,\boldsymbol{l})$ ((3) in Proposition 2.11), we obtain $\bigcap_{i\in\mathbb{Z}_{\geq 0}}B(H_{\lambda}(y_{i}),r_{i};e)\neq\emptyset$ . In this case, the set $\bigcap_{i\in\mathbb{Z}_{\geq 0}}B(H_{\lambda}(y_{i}),r_{i};e)$ is a closed ball of radius $r$ centered at some point in $\mathbb{P}_{p}(G,\boldsymbol{l})$ . Lemma 4.3 implies that there exists $\gamma=\sum_{g\in G\cap(-\infty,m)}s_{r}\tau^{g}$ in $\mathbb{A}_{q,p}(G,\boldsymbol{k})$ such that $B(\gamma,r;e)=\bigcap_{i\in\mathbb{Z}_{\geq 0}}B(H_{\lambda}(y_{i}),r_{i};e)$ . We define a map $H_{\lambda+1}\colon E_{\lambda+1}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ by $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$ and $H_{\lambda+1}(\xi_{\lambda})=\gamma$ . This definition implies the condition (C1). Next we verify the condition (C2). It suffices to show that $d(x,\xi_{\lambda})=e(H_{\lambda+1}(x),H_{\lambda+1}(\xi_{\lambda}))$ . Take $n\in\mathbb{Z}_{\geq 0}$ such that $d(y_{n},\xi_{\lambda})<d(x,\xi_{\lambda})$ . Then Lemma 2.1 implies $h(x,\xi_{\lambda})=h(y_{n},x)$ , and hence $h(y_{n},x)=e(H_{\lambda+1}(y_{n}),H_{\lambda+1}(x))$ . By $\gamma\in B(H_{\lambda+1}(y_{n}),r_{n};e)$ , we have

e(H_{\lambda+1}(\xi_{\lambda}),H_{\lambda+1}(y_{n}))\leq h(\xi_{\lambda},y_{n})<h(\xi_{\lambda},x)=e(H_{\lambda+1}(y_{n}),H_{\lambda+1}(x)).

Thus, using Lemma 2.1 again, we have

e(H_{\lambda+1}(y_{n}),H_{\lambda+1}(x))=e(H_{\lambda+1}(\xi_{\lambda}),H_{\lambda+1}(x)),

and hence $e(H_{\lambda+1}(\xi_{\lambda}),H_{\lambda+1}(x))=h(\xi_{\lambda},x)$ . By the construction, the mao $H_{\lambda+1}$ satisfies the properties (P1)–(CL). Thus, we see that $H_{\lambda+1}$ satisfies the conditions (C3)–(C5).

Case 2. [There exists $a\in X_{\lambda}$ such that $d(a,\xi_{\lambda})=u$ ]: By Lemma 4.3, there exists $\gamma=\sum_{g\in G\cap(-\infty,m)}s_{g}\tau^{g}\in\mathbb{A}_{q,p}(G,\boldsymbol{k})$ such that $B(\gamma,u;e)=B(H_{\lambda}(a),u;e)$ . We put

\delta=b_{\lambda}\cdot\tau^{m}+\gamma=b_{\lambda}\cdot\tau^{m}+\sum_{g\in G\cap(-\infty,m)}s_{g}\tau^{g}.

Then Lemma 2.2 states that $B(\delta,u;e)=B(H_{\lambda}(a),u;e)$ . We define a map $H_{\lambda+1}\colon E_{\lambda+1}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ by $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$ and $H_{\lambda+1}(\xi_{\lambda})=\delta$ . To verify the condition (C2), we show that $e(H_{\lambda+1}(\xi_{\lambda}),H_{\lambda+1}(x))=e(\xi_{\lambda},x)$ for all $x\in X_{\lambda}$ . If $x\not\in B(a,u;h)$ , then we have $h(x,a)=h(x,\xi_{\lambda})$ . Similarly, we have $e(H_{\lambda+1}(x),H_{\lambda+1}(a))=e(H_{\lambda+1}(x),H_{\lambda+1}(\xi_{\lambda}))$ . Since $e(H_{\lambda}(x),H_{\lambda}(a))=e(H_{\lambda}(x),H_{\lambda}(a))=d(x,a)$ , we have

e(H_{\lambda+1}(x),H_{\lambda+1}(\xi_{\lambda}))=h(x,\xi_{\lambda}).

If $x\in B(a,u;h)$ , then $h(x,\xi_{\lambda})\leq h(x,a)\lor h(a,\xi_{\lambda})\leq u$ . By the definition of $u$ , we have $h(\xi_{\lambda},x)=u$ . We also see that $H_{\lambda+1}(x)\in B(H_{\lambda+1}(a),u;e)$ . Then we can represent $H_{\lambda+1}(x)=\gamma+\sum_{g\in G\cap[m,\infty)}c_{g}\tau^{g}$ , where $c_{g}\in C_{\lambda}$ . Take $\alpha<\lambda$ with $\xi_{\alpha}=x$ . Then the condition (C2) for $H_{\lambda}$ implies that $c_{g}\in C_{\alpha+1}$ for all $g\in G\cap(-\infty,m]$ . From the definition of $U_{G,\boldsymbol{k},q,p}$ and $b_{\lambda}\not\in C_{\alpha+1}$ , it follows that $e(H_{\lambda+1}(x),H_{\lambda+1}(\xi_{\lambda}))=u$ . Hence

e(H_{\lambda+1}(x),H_{\lambda+1}(\xi_{\lambda}))=h(x,\xi_{\lambda}).

Similarly to Case 1, by the construction, we see that $H_{\lambda+1}$ satisfies the properties (P1)–(CL). Thus, we see that $H_{\lambda+1}$ satisfies the conditions (C3)–(C5). ∎

Lemma 4.5.

Fix $\lambda<\theta$ with $\lambda\neq 0$ , and let $H_{\lambda}\colon E_{\lambda}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ be a well-behaved map. If $h(E_{\lambda},\xi_{\lambda})=0$ , then there exists a well-behaved map $H_{\lambda+1}\colon E_{\lambda+1}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ such that $H_{\lambda+1}|_{X_{\lambda}}=H_{\lambda}$ .

Proof.

We now define $H_{\lambda+1}(\xi_{\lambda})$ as follows. Take a sequence $\{x_{i}\}$ in $E_{\lambda}$ such that $x_{i}\to\xi_{\lambda}$ as $i\to\infty$ , and define $H_{\lambda+1}(\xi_{\lambda})=\lim_{i\to\infty}H_{\lambda}(x_{i})$ . The value $H_{\lambda+1}(\xi_{\lambda})$ is independent of the choice of a sequence $\{x_{i}\}_{i\in\mathbb{Z}_{\geq 0}}$ .

First we confirm that $H_{\lambda+1}$ is well-behaved. Since $H_{\lambda}$ satisfies the condition (C1), so does $H_{\lambda+1}$ .

To prove (C2), it is enough to show $d(x,\xi_{\xi_{\lambda}})=e(H_{\lambda+1}(x),H_{\lambda+1}(\xi_{\lambda}))$ for all $x\in E_{\lambda+1}$ . Take a sufficient large $n\in\mathbb{Z}_{\geq 0}$ so that $d(x_{n},\xi_{\lambda})<d(x,\xi_{\lambda})$ and $e(H_{\lambda+1}(x_{n}),H_{\lambda+1}(\xi_{\lambda}))<e(H_{\lambda+1}(x),H_{\lambda+1}(\xi_{\lambda}))$ . Lemma 2.1 implies that

d(x,\xi_{\lambda})=d(x_{n},x)

and

e(H_{\lambda+1}(x),H_{\lambda+1}(\xi_{\lambda}))=e(H_{\lambda+1}(x_{n}),H_{\lambda+1}(x)).

Since $H_{\lambda}$ is an isometry and $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$ , we have

d(x_{n},x)=e(H_{\lambda+1}(x_{n}),H_{\lambda+1}(x)).

Therefore we conclude that the condition (C2) is ture.

By the construction, we see that $H_{\lambda+1}$ satisfies the properties (P1)–(CL). Thus, we see that $H_{\lambda+1}$ satisfies the conditions (C3)–(C5). ∎

Let us prove Theorem 4.1.

Proof of Theorem 4.1.

The proof is based on [21]. Using transfinite recursion, we first construct a coherent family $\{H_{\mu}\colon E_{\mu}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})\}_{\mu<\theta}$ of well-behaved maps. Fix $\mu\leq\theta$ and assume that we have already obtained a coherent family $\{H_{\alpha}\}_{\alpha<\mu}$ of well-behaved maps. Now we construct $H_{\mu}$ as follows. If $\lambda=0$ , then we define $H_{0}$ as the empty map. If $\lambda=1$ , then we have $E_{1}=\{b_{0}\}$ and we define $H_{1}\colon E_{1}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ by $H_{1}(\xi_{0})=0$ . If $\mu=\lambda+1$ for some $\lambda<\mu$ with $\lambda\neq 0$ , then using Lemmas 4.4 and 4.5, we obtain a well-behaved map $H_{\lambda+1}\colon E_{\lambda+1}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ such that $H_{\lambda+1}|_{E_{\lambda}}=H_{\lambda}$ . If $\mu$ is a limit ordinal, then we define $H_{\mu}\colon E_{\mu}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ by $H_{\mu}(x)=H_{\alpha}(x)$ , where $x\in E_{\alpha}$ . In this case, $H_{\mu}$ is well-defined since the family $\{H_{\alpha}\}_{\alpha<\theta}$ is coherent. Therefore, according to transfinite recursion, we obtain a well-behaved isometric embedding $H_{\theta}\colon E_{\theta}\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ . In this case, note that $E_{\theta}=E$ . Now we define $I\colon X\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ by $I=H_{\theta}|_{X}$ . Due to (C1), we have $H_{\theta}(\varpi)=0$ and $\varpi\not\in X$ . Thus the condition (B1) is true. Since $H_{\theta}$ satisfies (C2)–(C5), the map $I$ satisfies the conditions (B2)–(B5). This finishes the proof of Theorem 4.1. ∎

4.1.2. The proof of the first main result

The following is our first main result.

Theorem 4.6.

Let $\eta\in(1,\infty)$ , $(q,p)\in\mathcal{CH}$ , $\theta$ be a cardinal, and $G\in\mathcal{G}$ be dividable. Put $R=\{\,\eta^{-g}\mid g\in G\sqcup\{\infty\}\,\}$ . Take an arbitrary valued field $(K,v)$ of characteristic $q$ such that $v(K)\subseteq G\sqcup\{\infty\}$ and $\mathfrak{K}(K,v)$ has characteristic $p$ . Then there exists a valued field $(L,w)$ such that:

(L1)

the field $(L,w)$ is a valued field extension of $(K,v)$ ;
(L2)

$w(L)=G\sqcup\{\infty\}$ ;
(L3)

for each $R$ -valued ultrametric $(X,d)$ with $\operatorname{Card}(X)\leq\theta$ , there exists an isometric embedding $I\colon(X,d)\to(L,\lVert*\rVert_{w,\eta})$ such that the set $I(X)$ is algebraic independent over $K$ .

Moreover, for every $R$ -valued ultrametric space $(X,d)$ , there exist a valued field $(F,u)$ and a map $I\colon X\to F$ such that:

(F1)

the map $I\colon(X,d)\to(F,\lVert*\rVert_{u,\eta})$ is an isometric embedding;
(F2)

the field $(L,u)$ is a valued field extension of $(K,v)$ ;
(F3)

$u(F)\subseteq G\sqcup\{\infty\}$ ;
(F4)

the set $I(X)$ is closed in $F$ ;
(F5)

the set $I(X)$ is algebraically independent over $K$ ;
(F6)

if $(X,d)$ is complete, then $(F,u)$ can be chosen to be complete.

Proof.

Let $\boldsymbol{k}$ be the algebraic closure of $\mathfrak{K}(K,v)$ . Notice that $\boldsymbol{k}$ is perfect. Take a perfect filed $\boldsymbol{l}$ such that the transcendental degree of $\boldsymbol{l}$ over $\boldsymbol{k}$ is $\theta$ , and take a transcendental basis $B$ of $\boldsymbol{l}$ over $\boldsymbol{k}$ . In this case, we have $\operatorname{Card}(B)=\theta$ . Take a system $J\subseteq\mathbb{A}_{q,p}(G,\boldsymbol{l})$ of representatives of $\boldsymbol{l}$ . Notice that $B\subseteq\zeta(J)$ . Put $C=\zeta^{-1}(B)=\{\,b_{\alpha}\mid\alpha<\theta\,\}$ . Applying Theorem 4.1 to $(X,d)$ and $(K,\lVert*\rVert_{v,\eta})$ , we can take an isometric embedding $\psi\colon K\to\mathbb{A}_{q,p}(G,\boldsymbol{k})$ satisfying the conditions (B1)–(B5). Let $\phi\colon\boldsymbol{k}\to\boldsymbol{l}$ be the inclusion map. Using a homomorphic embedding $\mathbb{A}_{q,p}(G,\phi)\circ\psi\colon K\to\mathbb{A}_{q,p}(G,\boldsymbol{l})$ , in what follows, we consider that $K$ is a subfield of $\mathbb{A}_{q,p}(G,\boldsymbol{l})$ . In this case, we see that $\mathbf{AH}(K)\subseteq\boldsymbol{k}$ . Put $L=\mathbb{A}_{q,p}(G,\boldsymbol{l})$ and $w=U_{G,\boldsymbol{l},q,p}$ . Then $(L,w)$ satisfies the conditions (L1) and (L2) (see Proposition 2.11). Now we prove (L3). Take an $R$ -valued ultrametric space $(X,d)$ . Applying Theorem 4.1 to $C$ , $J$ , $\boldsymbol{l}$ , $\boldsymbol{k}$ , and $(X,d)$ of $\operatorname{Card}(X)\leq\theta$ , we can take $I\colon X\to L$ satisfying the conditions (B1)–(B5). The condition (B2) shows that $I$ is isometric. Due to the conditions (B1), (B3), and (B4), and due to the fact that $\mathbf{AH}(K)\subseteq\boldsymbol{k}$ , the set $\{\,I(x)\mid x\in X\,\}$ satisfies the assumptions in Proposition 3.4. Then, according to Proposition 3.4, we see that $I(X)$ is algebraic over $K$ . This means that the condition (L3).

We now prove the latter part. Let $(Y,e)$ be the completion of $(X,d)$ . Then $(Y,e)$ is $R$ -valued (see [3, (12) in Theorem 1.6]). Take an isometric embedding $I\colon Y\to L$ stated in the former part of Theorem 4.6. Since $(Y,e)$ is complete and $I$ is isometric, the set $I(Y)$ is closed in $L$ . Let $F$ be the subfield of $L$ generated by $\{\,I(x)\mid x\in X\,\}$ over $K$ and put $w=U_{G,\boldsymbol{l},q,p}|_{K}$ . Since $I(Y)$ is algebraically independent over $K$ , we have $I(Y)\cap L=I(X)$ . Thus $I(X)$ is closed in $F$ . The condition (F6) follows from the construction. This finishes the proof. ∎

Letting $K=\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ , and using Proposition 2.14, we obtain the next corollary.

Corollary 4.7.

Let $\eta\in(1,\infty)$ , $(q,p)\in\mathcal{CH}$ with $q\neq p$ , $\theta$ be a cardinal, and $G\in\mathcal{G}$ be dividable. Put $R=\{\,\eta^{-g}\mid g\in G\sqcup\{\infty\}\,\}$ . Then there exists a valued field $(L,v)$ such that:

(1)

the field $(L,v)$ is a valued field extension of $(\mathrm{Fr}\mathbb{W}(\boldsymbol{k}),w_{\boldsymbol{k}})$ ;
(2)

the absolute value $\lVert*\rVert_{v,\eta}$ is $R$ -valued;
(3)

for each $R$ -valued ultrametric $(X,d)$ with $\operatorname{Card}(X)\leq\theta$ , there exists an isometric embedding $I\colon(X,d)\to(L,\lVert*\rVert_{w,\eta})$ such that the set $I(X)$ is algebraic independent over $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ .

Moreover, for every $R$ -valued ultrametric space $(X,d)$ , there exist a valued field $(F,u)$ and a map $I\colon X\to F$ such that

(1)

the map $I\colon(X,d)\to(F,\lVert*\rVert_{u,\eta})$ is an isometric embedding;
(2)

the field $(L,u)$ is a valued field extension of $(\mathrm{Fr}\mathbb{W}(\boldsymbol{k}),w_{\boldsymbol{k}})$ ;
(3)

$u(F)\subseteq G\sqcup\{\infty\}$ ;
(4)

$I(X)$ is closed in $F$ ;
(5)

$I(X)$ is algebraically independent over $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ ;
(6)

if $(X,d)$ is complete, then $F$ can be chosen to be complete.

Proof.

The proof is the same to that of Theorem 4.6. Remark that, since $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ is naturally a subset of $\mathbb{A}_{0,p}(G,\boldsymbol{k})$ (see Proposition 2.14), we do not need to use Proposition 2.15. Thus, the assumption of the corollary does not require that $G$ be divisible. ∎

4.2. Broughan’s conjecture

Next we give an affirmative solution of Conjecture 1.1. We begin with the definition of non-Archimedean Banach algebras. Let $(K,\lvert*\rvert)$ be a field equipped with a non-Archimedean absolute value, and $(B,\lVert*\rVert)$ be a normed linear space over a field $K$ . We say that $(B,\lVert*\rVert)$ is a non-Archimedean Banach algebra over $K$ (see [26]) if the following conditions are fullfiled:

(1)

$B$ is a ring (not necessarily commutative and unitary);
(2)

$(B,\lVert*\rVert)$ is a normed linear space over $(K,\lvert*\rvert)$ ;
(3)

for every pair $x,y\in B$ , we have $\lVert x+y\rVert\leq\lVert x\rVert\lor\lVert y\rVert$ and $\lVert xy\lVert\leq\lVert x\rVert\cdot\lVert y\rVert$ ;
(4)

$B$ is complete with respect to $\lVert*\rVert$ ;
(5)

if $B$ has a unit $e$ , then $\lVert e\rVert=1$ .

Some authors assume commutativity and the existence of a unit in the definition of a Banach algebra (see, for example, [24] and [17]). For instance, a valued field extension $(L,\lvert*\rvert_{L})$ of $(K,\lvert*\rvert_{K})$ is a (unitary) Banach algebra over $(K,\lvert*\rvert_{K})$ .

As an application of Corollary 4.7, we obtain the next theorem.

Theorem 4.8.

Let $p$ be a prime, $\theta$ be a cardinal, and $\boldsymbol{k}$ be a perfect field of characteristic $p$ such that the transcendental degree of $\boldsymbol{k}$ over $\mathbb{F}_{p}$ is equal to $\theta$ . Then every $H_{p}$ -valued ultrametric $(X,d)$ with $\operatorname{Card}(X)\leq\theta$ can be isometrically embedded into $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ . In particular, Conjecture 1.1 is true.

Proof.

The theorem follows from Corollary 4.7. Theorem 4.8 is actually an affirmative solution of Conjecture 1.1 since for every cardinal $\theta$ , there exists a perfect filed $\boldsymbol{k}$ of characteristic $p$ such that the transcendental degree of $\boldsymbol{k}$ over $\mathbb{F}_{p}$ is equal to $\theta$ , and since $\mathrm{Fr}\mathbb{W}(\boldsymbol{k})$ is a non-Archimedean (commutative and unitary) Banach algebra over $\mathbb{Q}_{p}$ (see Proposition 2.14). ∎

Remark 4.1.

An solution of Conjecture 1.1 is given as the ring $\mathbb{W}(\boldsymbol{k})$ of Witt vectors.

5. Algebraic structures on Urysohn universal ultrametric spaces

In this section, we show that every Urysohn universal ultrametric space has a valued field structure that are extensions of a given prime valued field. Such an algebraic structure is realized as a $p$ -adic or ordinary Levi–Civita field.

For a class $\mathscr{C}$ of metric spaces, a metric space $(X,d)$ is said to be $\mathscr{C}$ -injective if for every pair $(A,a)$ and $(B,b)$ in $\mathscr{C}$ and for every pair of isometric embeddings $\phi\colon(A,a)\to(B,b)$ and $\psi\colon(A,a)\to(X,d)$ , there exists an isometric embedding $\theta\colon(B,b)\to(X,d)$ such that $\theta\circ\phi=\psi$ . We denote by $\mathscr{F}$ (resp. $\mathscr{N}(R)$ for a range set $R$ ) the class of all finite metric spaces (resp. all finite $R$ -valed ultrametric spaces). There exists a complete $\mathscr{F}$ -injetive separable complete metric space $(\mathbb{U},\rho)$ unique up to isometry, and this space is called the Urysohn universal metric space (see [25] and [15]).

Similarly, if $R$ is countable, there exists a separable complete $R$ -valued $\mathscr{N}(R)$ -injective ultrametric space up to isometry, and it is called the $R$ -Urysohn universal ultrametric space, which is a non-Archimedean analogue of $(\mathbb{U},\rho)$ (see [6] and [18]). Remark that if $R$ is uncountable, then every $R$ -injective ultrametric space is not separable (see [6]). Due to this phenomenon, mathematicians often consider only the case where $R$ is countable for separability.

In [11], the author discovered the concept of petaloid spaces, which can be considered to as $R$ -Urysohn universal ultrametric spaces in the case where $R$ is uncountable.

We begin with preparation of notations. A subset $E$ of $[0,\infty)$ is said to be semi-sporadic if there exists a strictly decreasing sequence $\{a_{i}\}_{i\in\mathbb{Z}_{\geq 0}}$ in $(0,\infty)$ such that $\lim_{i\to\infty}a_{i}=0$ and $E=\{0\}\cup\{\,a_{i}\mid i\in\mathbb{Z}_{\geq 0}\,\}$ . A subset of $[0,\infty)$ is said to be tenuous if it is finite or semi-sporadic (see [10]). For a range set $R$ , we denote by $\mathbf{TEN}(R)$ the set of all tenuous range subsets of $R$ . Let us recall the definition of petaloid spaces ([11]).

Definition 5.1.

Let $R$ be an uncountable range set. We say that a metric space $(X,d)$ is $R$ -petaloid if it is an $R$ -valued ultrametric space and there exists a family $\{\Pi(X,S)\}_{S\in\mathbf{TEN}(R)}$ of subspaces of $X$ satisfying the following properties:

(P1)

For every $S\in\mathbf{TEN}(R)$ , the subspace $(\Pi(X,S),d)$ is isometric to the $S$ -Urysohn universal ultrametric space.
(P2)

We have $\bigcup_{S\in\mathbf{TEN}(R)}\Pi(X,S)=X$ .
(P3)

If $S,T\in\mathbf{TEN}(R)$ , then $\Pi(X,S)\cap\Pi(X,T)=\Pi(X,S\cap T)$ .
(P4)

If $S,T\in\mathbf{TEN}(R)$ and $x\in\Pi(X,T)$ , then $d(x,\Pi(X,S))$ belongs to $(T\setminus S)\cup\{0\}$ .

We call the family $\{\Pi(X,S)\}_{S\in\mathbf{TEN}(R)}$ an $R$ -petal of $X$ , and call $\Pi(X,S)$ the $S$ -piece of the $R$ -petal $\{\Pi(X,S)\}_{S\in\mathbf{TEN}(R)}$ .

Notice that even if $R$ is countable, the $R$ -Urysohn universal ultrametric space has a petal structure satisfying the conditions (P1)–(P4) (see [11]). This means that a petal space is a natural generalization of Urysohn universal ultrametric spaces.

The following is [9, Theorem 2.3] (see the property (P1) and [22, Propositions 20.2 and 21.1]).

Theorem 5.1.

Let $R$ be an uncountable range set. The following statements hold:

(1)

There exists an $R$ -petaloid ultrametric space and it is unique up to isometry.
(2)

The $R$ -petaloid ultrametric space is complete, non-separable, and $\mathscr{F}(R)$ -injective.
(3)

Every separable $R$ -valued ultrametric space can be isometrically embedded into the $R$ -petaloid ultrametric space.

Based on Theorem 5.1, for a range set $R$ , we denote by $(\mathbb{V}_{R},\sigma_{R})$ the $R$ -Urysohn universal ultrametric space if $R$ is countable; otherwise, the $R$ -petaloid ultrametric space. In what follows, by abuse of notation, we call $(\mathbb{V}_{R},\sigma_{R})$ the $R$ -Urysohn universal ultrametric space even if $R$ is uncountable.

We explain an example of a petaloid space. Let $N$ be a countable set. Let $R$ be a range set. We also denote by $\mathrm{G}(R,N)$ the set of all function $f\colon R\to N$ such that $f(0)=0$ and the set $\{0\}\cup\{\,x\in R\mid f(x)\neq 0\,\}$ is tenuous. For $f,g\in\mathrm{G}(R,N)$ , we define an $R$ -ultrametric $\mathord{\vartriangle}$ on $\mathrm{G}(R,N)$ by $\mathord{\vartriangle}(f,g)=\max\{\,r\in R\mid f(r)\neq g(r)\,\}$ if $f\neq g$ ; otherwise, $\mathord{\vartriangle}(f,g)=0$ . For more information of this construction, we refer the readers to [6] and [18].

Lemma 5.2.

For every countable set $N$ and every range set $R$ , the ultrametric space $(\mathrm{G}(R,N),\mathord{\vartriangle})$ is isometric to $(\mathbb{V}_{R},\sigma_{R})$ .

Proof.

The lemma follows from [11, Theorem 1.3]. Notice that whereas in [11], it is only shown that $(\mathrm{G}(R,\mathbb{Z}_{\geq 0}),\mathord{\vartriangle})$ is isometric to $(\mathbb{V}_{R},\sigma_{R})$ , the proof in [11] is still valid even if $N$ is a countable set in general. ∎

Lemma 5.3.

Let $A$ be a subset of $\mathbb{R}$ , and $\eta\in(0,\infty)$ . Put $S=\{0\}\sqcup\{\,\eta^{-g}\mid g\in A\,\}$ . Then the following statements are equivalent:

(1)

for every $n\in\mathbb{Z}_{\geq 0}$ , the set $A\cap(-\infty,n]$ is finite;
(2)

the set $R$ is tenuous.

Proof.

The lemma follows from [10, Lemma 2.12] and the fact that a map $f\colon\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\eta^{-x}$ reverses the order on $\mathbb{R}$ . ∎

In the next lemma, in order to treat the cases of $q=p$ and $q\neq p$ , even if $q\neq p$ , we define $\mathrm{St}_{G,\boldsymbol{k},J}$ by $\mathrm{St}_{G,\boldsymbol{k},J}(x)=x$ .

Lemma 5.4.

Let $\eta\in(1,\infty)$ , $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , $\boldsymbol{k}$ be a field of $\operatorname{Card}(\boldsymbol{k})=\aleph_{0}$ and characteristic $p$ . Put $R=\{0\}\cup\{\,\eta^{-g}\mid g\in G\,\}$ . Fix a complete system $J\subseteq\mathbb{B}_{q,p}(G,\boldsymbol{k})$ of representatives of the residue class field $\boldsymbol{k}$ . Take $S\in\mathbf{TEN}(R)$ . Define a subset $\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)$ of $\mathbb{B}_{q,p}(G,\boldsymbol{k})$ by the set of all $f\in\mathbb{B}_{q,p}(G,\boldsymbol{k})$ such that $\{\,\eta^{-g}\mid g\in\text{{\rm supp}}(\mathrm{St}_{G,\boldsymbol{k},J}(f))\,\}\subseteq S$ , and define an ultrametric $d$ on $\mathbb{B}_{q,p}(G,\boldsymbol{k})$ by $d(x,y)=\lVert x-y\rVert_{U_{G,\boldsymbol{k},q,p},\eta}$ . Then $(\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S),d)$ is isometric to $(\mathbb{V}_{S},\sigma_{S})$ .

Proof.

According to $\operatorname{Card}(\boldsymbol{k})=\aleph_{0}$ , we see that $\operatorname{Card}(J)=\aleph_{0}$ . Then, using Lemma 2.9 or (1) in Corollary 2.18, for every $x\in\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)$ , there uniquely exists $s\in\mathbb{H}(G,\boldsymbol{k})$ such that $\mathrm{St}_{G,\boldsymbol{k},J}(x)=s$ and $s(g)\in J$ . Put $s=\sum_{g\in G}s_{x}(g)t^{g}$ , where $s_{x}(g)\in J$ . We define a map $T(x)\colon R\to J$ by $T(x)(0)=0$ and $T(x)(r)=s_{x}(-\log_{\eta}(r))$ if $r\neq 0$ . Then $\{\,r\in R\mid s_{x}(r)\neq 0\,\}\subseteq S$ . Thus $T(x)\in\mathrm{G}(S,J)$ for all $x\in\mathbb{B}_{q,p}(G,\boldsymbol{k})$ . By the definition of $T$ , we see that the map $T\colon\mathbb{B}_{q,p}(G,\boldsymbol{k})\to\mathrm{G}(S,J)$ defined by $x\mapsto T(x)$ is bijective. Using the definitions of $U_{G,\boldsymbol{k},q,p}$ and $\mathord{\vartriangle}$ , the map $T\colon\mathbb{B}_{q,p}(G,\boldsymbol{k})\to\mathrm{G}(S,J)$ becomes an isometric bijection. Thus $(\mathrm{G}(S,J),\mathord{\vartriangle})$ is isometric to $(\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S),d)$ . Due to $\operatorname{Card}(J)=\aleph_{0}$ , Lemma 5.2 finishes the proof. ∎

Theorem 5.5.

Let $\eta\in(1,\infty)$ , $G\in\mathcal{G}$ , $(q,p)\in\mathcal{CH}$ , $\boldsymbol{k}$ be a field of $\operatorname{Card}(\boldsymbol{k})=\aleph_{0}$ and characteristic $p$ . Define an ultrametric $d$ on $\mathbb{B}_{q,p}(G,\boldsymbol{k})$ by $d(x,y)=\lVert x-y\rVert_{U_{G,\boldsymbol{k},q,p},\eta}$ . Then the Levi-Civita field $(\mathbb{B}_{q,p}(G,\boldsymbol{k}),d)$ is isometric to $(\mathbb{V}_{R},\sigma_{R})$ .

Proof.

We define $\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)$ as in Lemma 5.4. Let us prove that $\{\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)\}_{S\in\mathbf{TEN}(R)}$ satisfies the conditions (P1)–(P4).

Lemma 5.4 implies the conditions (P1). From the definition of $\{\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)\}_{S\in\mathbf{TEN}(R)}$ , the conditions (P2) is true. Next we show the condition (P3). It is sufficient to verify

(5.1)

\displaystyle\mathbb{B}_{q,p}(G,\boldsymbol{k})\subseteq\bigcup_{S\in\mathbf{TEN}(R)}\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S).

Take $f\in\mathbb{B}_{q,p}(G,\boldsymbol{k})$ and put $A=\text{{\rm supp}}(\mathrm{St}_{G,\boldsymbol{k},J}(f))$ . We also define $S=\{\,\eta^{-g}\mid g\in A\,\}$ . Then $S\in\mathbf{TEN}(R)$ by Lemma 5.3. Hence $f\in\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)$ . Thus, the inclusion (5.1) is true.

Now we show (P4). We may assume that $x\not\in\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)$ . Put $U=\text{{\rm supp}}(\mathrm{St}_{G,\boldsymbol{k},J}(x))$ . Since $x\not\in\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)$ , we have $U\setminus S\neq\emptyset$ . Put $u=\max(U\setminus S)$ and $m=-\log_{\eta}(u)$ . Notice that $\boldsymbol{C}(x,m)\neq 0$ . Define a point $y\in\mathbb{B}_{q,p}(G,\boldsymbol{k})$ by $y=\sum_{g\in G\cap(-\infty,m)}x(g)\tau^{g}$ . Then $y\in\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)$ and $d(x,y)=u$ . Then $d(x,\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S))\leq u$ . For the sake of contradiction, suppose that $d(x,\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S))<u$ . In this setting, we can take $z\in\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S)$ such that $m<v(x-z)$ . Then $\boldsymbol{C}(z,m)=\boldsymbol{C}(x,m)$ . In particular, $\boldsymbol{C}(z,m)\neq 0$ . Namely, we have $\text{{\rm supp}}(\mathrm{St}_{G,\boldsymbol{k},J}(z))\not\subseteq S$ . This is a contradiction. Thus we have $d(x,\Pi(\mathbb{B}_{q,p}(G,\boldsymbol{k}),S))=u\in T\setminus S$ , which means that the condition (P4) holds.

Therefore, we conclude that the Levi-Civita field $\mathbb{B}_{q,p}(G,\boldsymbol{k})$ is isometric to $(\mathbb{V}_{R},\sigma_{R})$ . ∎

We find another application of the theory of Urysohn universal ultrametric spaces to that of valued field. For a class $\mathscr{C}$ of metric spaces, we say that a metric space $(X,d)$ is $\mathscr{C}$ -universal or universal for $\mathscr{C}$ if for every $(A,e)\in\mathscr{C}$ there exists an isometric embedding $f\colon A\to X$ . An ultrametric space $(X,d)$ is said to be $(R,\aleph_{0})$ -haloed if for every $a\in X$ and for every $r\in R\setminus\{0\}$ , there exists a subset $A$ of $\mathrm{B}(a,r)$ such that $\kappa\leq\operatorname{Card}(A)$ and $d(x,y)=r$ for all distinct $x,y\in A$ (see [9]).

Theorem 5.6.

Let $\eta\in(1,\infty)$ , $G\in\mathcal{G}$ , $p$ be a prime, and let $(K,v)$ be a valued field such that $\mathfrak{K}(K,v)$ is an infinite set and has characteristic $p$ . Put $R=\{\,\eta^{-g}\mid g\in G\sqcup\{\infty\}\,\}$ . Then $(K,\lVert*\rVert_{v,\eta})$ is universal for all separable $R$ -valued ultrametric spaces.

Proof.

According to $\aleph_{0}\leq\operatorname{Card}(\boldsymbol{k})$ , we observe that $(K,\lVert*\rVert_{v,\eta})$ is $(R,\omega_{0})$ -haloed, and hence it is $\mathscr{N}(R,\omega_{0})$ -injective (see [9, Theorem 1.1]). In [9, Theorem 1.5], it is stated that every complete $\mathscr{N}(R,\omega_{0})$ -injective ultrametric space contains an isometric copy of $(\mathbb{V}_{R},\sigma_{R})$ . Thus $(K,\lVert*\rVert_{v,\eta})$ contains a metric subspace isometric to $(\mathbb{V}_{R},\sigma_{R})$ . Hence $(K,\lVert*\rVert_{v,\eta})$ is universal for all separable $R$ -valued ultrametric spaces. ∎

For a prime $p$ , the field $\mathbb{C}_{p}$ of $p$ -adic complex numbers is defined as the completion of the algebraic closure of $\mathbb{Q}_{p}$ . The $p$ -adic valuation $v_{p}$ can be extended on $\mathbb{C}_{p}$ . In this case, we have $v_{p}(\mathbb{C}_{p})=\mathbb{Q}$ (see [22]).

Corollary 5.7.

Let $\eta\in(0,\infty)$ , and $p$ be a prime. Put $R=\{\,\eta^{-g}\mid g\in\mathbb{Q}\,\}$ . The field $(\mathbb{C}_{p},v_{p})$ of $p$ -adic complex numbers is universal for all separable $R$ -valued ultrametric spaces.

6. Questions

As a sophisticated version of a non-Archimedean Arens–Eells theorem for linear spaces, we ask the next question.

Question 6.1.

Let $R,T$ be range sets, and $S$ be a range set such that $S\setminus\{0\}$ is a multiplicative subgroup of $\mathbb{R}_{\geq 0}$ . Take a non-Archimedean valued field $(K,\rvert*\lvert_{K})$ and assume that

(1)

$R\cdot S=\{\,r\cdot s\mid r\in R,s\in S\,\}\subseteq T$ ;
(2)

$\rvert*\lvert_{K}$ is $S$ -valued.

For every $R$ -valued ultrametric space $(X,d)$ , do there exist a $T$ -valued ultra-normed linear space $(V,\rVert*\lVert_{L})$ over $(K,\rvert*\lvert_{K})$ , and an isometric map $I\colon X\to F$ ?

Similarly to Corollary 5.7, as a counterpart of Theorem 4.6, we make the following question.

Question 6.2.

Let $p$ be $0$ or a prime, $\boldsymbol{k}$ be an infinite field of characteristic $p$ , and $R$ be a rang set such that $R\setminus\{0\}$ is a multiplicative subgroup of $\mathbb{R}_{>0}$ . Take an arbitrary complete valued field $(K,\rvert*\lvert_{*})$ with the residue class field $\boldsymbol{k}$ such that $\{\,\rvert x\lvert_{K}\,\mid x\in K\,\}=R$ and $(K,v)$ is a valued field extension of $(\mathbb{Q}_{p},v_{p})$ . For every $R$ -valued ultrametric space $(X,d)$ with $\operatorname{Card}(X)\leq\operatorname{Card}(\boldsymbol{k})$ , does there exist an isometric embedding $I\colon X\to K$ ?

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

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Conjecture 1.1.

Acknowledgements.

2. Preliminaries

2.1. Generalities

2.1.1. Metric spaces

Lemma 2.1.

Lemma 2.2.

2.1.2. Valued rings

Proposition 2.3.

2.2. Constructions of valued fields

2.2.1. Hahn rings and fields

Proposition 2.4.

Proof.

2.2.2. The pp-adic Hahn fields

Proposition 2.5.

Proposition 2.6.

Proof.

Proposition 2.7.

Proof.

Remark 2.1.

Proposition 2.8.

Proof.

Lemma 2.9.

Proof.

Definition 2.1.

Proposition 2.10.

Proof.

Definition 2.2.

Proposition 2.11.

Proof.

Proposition 2.12.

Proof.

Proposition 2.13.

Proof.

Proposition 2.14.

Proof.

Definition 2.3.

Remark 2.2.

Proposition 2.15.

Proof.

2.2.3. Levi–Civita fields

Lemma 2.16.

Lemma 2.17.

Proof.

Corollary 2.18.

Definition 2.4.

Proposition 2.19.

Proof.

3. Algebraic independence over valued fields

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Proposition 3.4.

Proof.

Remark 3.1.

4. Isometric embeddings of ultrametric spaces

4.1. A non-Archimedean Arens–Eells theorem

4.1.1. Preparations

Theorem 4.1.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Proof of Theorem 4.1.

4.1.2. The proof of the first main result

Theorem 4.6.

Proof.

Corollary 4.7.

2.2.2. The $p$ -adic Hahn fields