# Topologies on the multiplicative spectrum of an ultrametric banach algebra. A survey and new results.

Alain Escassut, Monique Chicourrat

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Alain Escassut, Monique Chicourrat. Topologies on the multiplicative spectrum of an ultrametric banach algebra. A survey and new results. Asian-European Journal of Mathematics, In press. hal-05031516

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# TOPOLOGIES ON THE MULTIPLICATIVE SPECTRUM
OF AN ULTRAMETRIC BANACH ALGEBRA.

## A SURVEY AND NEW RESULTS.

MONIQUE CHICOURRAT AND ALAIN ESCASSUT

**ABSTRACT.** Let $K$ be a complete ultrametric field and $(A, \| \cdot \|)$ be a $K$-Banach algebra. In the past literature the multiplicative spectrum $Mult(A, \| \cdot \|)$ of $A$ has been provided with different topologies and metrics. The usual topology $T_s$ is the pointwise convergence topology and it is compact. Actually $Mult(A, \| \cdot \|)$ can be provided with some uniform convergence topology. These topologies are uniformizable (in the sense of Bourbaki). Using the uniformity point of view we prove new results about $T_s$. In particular considering the case of a separable $A$ we get metrization results generalizing some Maïnetti's results on Tate $K$-algebras. Concerning the affine space we give an answer to some question of Maïnetti. We also study the case of some Krasner algebra $A = H(D)$ for which a third topology was studied by Boussaf and Maïnetti, this topology is linked to a metric defined on circular filters. Finally we consider the case of some $K$-algebras $A$ of functions from an ultrametric space $E$ to $K$. Some of these algebras (the admissible and compatible algebras) yet have been studied by the authors. We give here more results relative to uniform properties and compactifications.

# 1. INTRODUCTION AND PRELIMINARIES

Let $K$ be a (non trivial) complete ultrametric field and $(A, \| \cdot \|)$ be a Banach $K$-algebra with respect to an ultrametric norm. The multiplicative spectrum of $(A, \| \cdot \|)$ is denoted by $Mult(A, \| \cdot \|)$ and it is the set of continuous multiplicative (ultrametric) semi-norms of $A$, that is the set of the multiplicative semi-norms of $A$ such that for all $x \in A$ we have $\phi(x) \le \|x\|$.

The set $Mult(A, \| \cdot \|)$ has been studied for a long time [1], [3], [6], [10], [12], [13], for different algebras $A$. It is known to be compact (Hausdorff) with respect to the usual topology $T_s$ of pointwise convergence.

In this paper we intend to give new topological properties of this space, based on a comparison of some uniformities (in the sense of [2]) on $Mult(A, \| \cdot \|)$ and we examine different examples of $K$-Banach algebras.

In section 2 we consider $Mult(A, \| \cdot \|)$ as a subset of the real Banach algebra $G(A_0, \mathbb{R})$ of bounded functions from the unit ball $A_0$ of $A$ to $\mathbb{R}$, so it is provided with the usual metric of uniform convergence. Considering the uniformity point of

<footnote n="2020">2020 Mathematics Subject Classification: Primary 30G06, 32P05, 46S10; Secondary 54A10, 54E35.</footnote>

<footnote>Keywords: ultrametric Banach algebras, multiplicative spectrum, circular filters, Krasner algebras, uniformities.</footnote>
view we prove some general results on the multiplicative spectrum. In particular we get metrization results when A is separable.

These metrization results generalize results of [15] concerning Tate K-algebras. Section 3 is about these algebras and the affine n-space A^n which is the space of all multiplicative semi-norms on K[X_1, ..., X_n]. It was proved in [15] that whenever K is separable then the pointwise convergence topology on A^n is metrizable. We will answer some open question of [15] about metrics inducing this topology on A^n. In section 4 we consider the special case of the Krasner Tate algebra A = H(D) of analytic functions on the unit ball D of K which is provided with the uniform convergence norm ||.||_0. It is known that there is a bijection between Mult(H(D), ||.||_0) and the set of the so-called circular filters on D [13], [11]. Topological properties of Mult(H(D), ||.||_0) were studied in [3] using some metric on circular filters. In particular this metric gives rise to a topology which is generally finer than T_s. Then, Mult(H(D), ||.||_0) is provided with three topologies and we make a comparison between them.

Finally section 5 deals with some algebras (A, ||.||) of bounded continuous functions from an ultrametric space E to K. We include here compatible and admissible algebras which were studied by the authors in [6], [7]. We establish some general uniform properties and we obtain some results concerning compactifications of E extending results of [6], [7], [12].

Recall first some vocabulary about uniformities. We refer to [2], [4], [19] for more complete terminology and results.

A uniformity $\mathfrak{S}$ on a set $X$ is a filter on the set of reflexive binary relations on $X$, such that:

(1) for all $R$ in $\mathfrak{S}$ the symmetric relation $R^{-1}$ is also in $\mathfrak{S}$;

(2) for all $R$ in $\mathfrak{S}$ there exists some $S$ in $\mathfrak{S}$ such that $S \circ S \subset R$.

It is called separating when the intersection of the elements of $\mathfrak{S}$ is reduced to the diagonal of $X \times X$.

For every binary relation $R$ on $X$, every $x \in X$ and every $A \subset X$ we write :

$$R(x) = \{y \in X \mid (x, y) \in R\} \quad \text{and} \quad R[A] = \bigcup_{x \in A} R(x)$$

Then any uniformity $\mathfrak{S}$ on $X$ generates a topology $\mathcal{T}$ on $X$ for which, given any $x \in X$ the set $\{R(x) \mid R \in \mathfrak{S}\}$ is a neighborhood basis of $x$. It is said that $\mathcal{T}$ is uniformizable and that $\mathfrak{S}$ is compatible with $\mathcal{T}$ or simply that it induces $\mathcal{T}$. The uniformity $\mathfrak{S}$ is separating if and only if the topology $\mathcal{T}$ is separated (Hausdorff).

If $Y \subset X$, the uniformity $\mathfrak{S}_Y$ induced by $\mathfrak{S}$ on $Y$ consists in taking the intersection of elements of $\mathfrak{S}$ with $Y \times Y$. This uniformity is compatible with the topology induced by $\mathcal{T}$ on $Y$.

A uniformity $\mathfrak{S}$ is coarser than another uniformity $\mathfrak{S}'$ if $\mathfrak{S} \subset \mathfrak{S}'$. Of course in this case the corresponding induced topologies can be compared in the same way.
A map $f$ between uniform spaces $(X, \mathfrak{S})$ and $(Y, \mathfrak{S}')$ is uniformly continuous if for every $R \in \mathfrak{S}'$, the set $(f \times f)^{-1}(R) = \{(x, y) \in X \times X \mid (f(x), f(y)) \in R\}$ is a member of $\mathfrak{S}$.

Whenever this map $f$ is one-to-one and if $f$ and $f^{-1}$ are uniformly continuous then we will say that $(X, \mathfrak{S})$ and $(Y, \mathfrak{S}')$ are uniformly isomorphic [19].

It is known that a topology $\mathcal{T}$ is uniformizable if and only if it is completely regular and that a compact (with the Hausdorff separation property) space admits a unique compatible uniformity.

Any metric $\delta$ on $X$ defines a uniformity $\mathfrak{S}_\delta$ on $X$: a filter basis for $\mathfrak{S}_\delta$ is the set of all $U_{\delta, \epsilon}$ with $\epsilon > 0$ where $U_{\delta, \epsilon} = \{(x, y) \in X \times X \mid \delta(x, y) \le \epsilon\}$.

This is also true for any pseudometric on $X$: following [4], [19], a pseudometric (écart in [2]) on $X$ is a map $\delta$ on $X \times X$ which satisfies all the properties of a metric but the “separating” one, i.e. $\delta(x, y) = 0$ does not imply necessarily that $x = y$. We denote again by $U_{\delta, \epsilon}$ the set of all $(x, y)$ such that $\delta(x, y) \le \epsilon$.

As a matter of fact, any uniformity $\mathfrak{S}$ on $X$ can be defined by some family $(\delta_i)_{i \in I}$ of pseudometrics on $X$ in the sense that the family of all binary relations $U_{\delta_i, \epsilon}$ with $i \in I$ and $\epsilon > 0$ is a subbasis for the filter $\mathfrak{S}$, i.e. the set of all finite intersections is a basis for $\mathfrak{S}$.

The uniformity $\mathfrak{S}$ is said to be pseudometrizable (respectively metrizable) when it can be defined by a family containing only one pseudometric (respectively only one metric). Of course different pseudometrics may give rise to the same uniformity, for example a pseudometric $\delta$ and the pseudometric $2\delta$ give rise to the same uniformity.

**Theorem 1.1:** *The following properties are equivalent :*

(1) $\mathfrak{S}$ is pseudometrizable

(2) $\mathfrak{S}$ admits a countable basis

(3) $\mathfrak{S}$ is defined by a countable family $(\delta_n)_n$ of pseudometrics. In this case the following are pseudometrics defining $\mathfrak{S}$:

$$ \delta_0(x, y) = \sup_n \frac{\min(\delta_n(x, y), 1)}{2^n} \qquad \delta_1(x, y) = \sum_n \frac{1}{2^n} \min(\delta_n(x, y), 1) $$

(Replacing $\frac{1}{2}$ in $\delta_1$ by any $\rho \in ]0, 1[$ one gets another pseudometric defining $\mathfrak{S}$.)

**Corollary 1.2:** $\mathfrak{S}$ is metrizable iff it is separating and has a countable basis.

Finally recall that a uniformity $\mathfrak{S}$ with associated topology $\mathcal{T}$ on $X$ is totally bounded or precompact if for every $R \in \mathfrak{S}$ there exists some finite open cover $(O_i)_{i=1,...,q}$ of $X$ such that $\bigcup_{i=1}^q O_i \times O_i \subset R$. When $\mathfrak{S}$ is the uniformity induced by

a metric $\delta$, this property is equivalent to the precompactness of $(X, \delta)$, that is to say that for every $\epsilon > 0$ there exists finitely many $(x_i)_{i=1,...,q}$ such that the family of (open or closed) balls $(B(x_i, \epsilon))_{i=1,...,q}$ is a cover of $X$.
## 2. GENERAL RESULTS ON $Mult(A, \| \cdot \|)$

**Definitions and notations.** The absolute value on the field $K$ is denoted by $| \cdot |$. The Archimedean value of $\mathbb{R}$ is denoted by $| \cdot |_{\infty}$.

Let $(A, \| \cdot \|)$ be an ultrametric normed $K$-algebra with identity and let $A_0$ be the unit ball of $A$. Let $\mathcal{G}(A_0, \mathbb{R})$ be the set of bounded functions from $A_0$ to $\mathbb{R}$. Then $\mathcal{G}(A_0, \mathbb{R})$ is a real Banach algebra with respect to the usual uniform convergence norm defined as $\|\phi\| = \sup \{|\phi(x)|_{\infty} | x \in A_0\}$.

Let $\Theta$ be the associated metric. Given $\phi, \psi \in \mathcal{G}(A_0, \mathbb{R})$,

$$ \Theta(\phi, \psi) = \sup \left\{ \left| \phi(x) - \psi(x) \right|_{\infty} \mid x \in A_0 \right\}. $$

Given subsets $X, Y$ of $\mathcal{G}(A_0, \mathbb{R})$, we put

$$ \Theta(X, Y) = \inf\{\Theta(\phi, \psi) \mid \phi \in X, \psi \in Y\}. $$

We denote by $\mathcal{T}_u$ the uniform convergence topology induced by $\Theta$. Furthermore we denote by $\mathcal{S}_\Theta$ the *metric uniformity* associated to $\Theta$ which admits as a filter basis the family of all binary relations $V_\epsilon$ where we put

$$ V_{\epsilon} = \{(\psi, \phi) \in \mathcal{G}(A_0, \mathbb{R})^2 \mid \Theta(\psi, \phi) \le \epsilon\} $$

for each $\epsilon > 0$. Of course this uniformity induces the topology $\mathcal{T}_u$ on $\mathcal{G}(A_0, \mathbb{R})$.

On another hand we have the topology $\mathcal{T}_s$ of pointwise convergence on $\mathcal{G}(A_0, \mathbb{R})$ which is obviously coarser than $\mathcal{T}_u$.

Given $\phi \in \mathcal{G}(A_0, \mathbb{R})$, $x_1, ..., x_q \in A_0$ and $\epsilon > 0$ we denote by $\mathcal{V}(\phi, x_1, ..., x_q, \epsilon)$ the set of $\psi \in \mathcal{G}(A_0, \mathbb{R})$ such that $|\phi(x_j) - \psi(x_j)|_{\infty} \le \epsilon$ for all $j = 1, ..., q$.

Then a neighborhood basis of $\phi$ with respect to the topology $\mathcal{T}_s$ is given by the family of all sets $\mathcal{V}(\phi, x_1, ..., x_q, \epsilon)$ where $x_1, ..., x_q \in A_0$ and $\epsilon > 0$.

The topology $\mathcal{T}_s$ is also induced by a uniformity on $\mathcal{G}(A_0, \mathbb{R})$, this one will be denoted by $\mathcal{S}_s$ and admits as a filter basis the family of all binary relations $U_{x_1, ..., x_q, \epsilon}$ where we put

$$ U_{x_1, ..., x_q, \epsilon} = \left\{ (\psi, \phi) \in \mathcal{G}(A_0, \mathbb{R})^2 \mid \forall j = 1, ..., q, \left| \phi(x_j) - \psi(x_j) \right|_{\infty} \le \epsilon \right\} $$

for every $\epsilon > 0$, $x_1, ..., x_q \in A_0$. In other words, $\mathcal{S}_s$ is defined by the family of pseudometrics $(\delta_x)_{x \in A_0}$ where we write $\delta_x(\phi, \psi) = |\phi(x) - \psi(x)|_{\infty}$.

The topology $\mathcal{T}_s$ is induced by $\mathcal{S}_s$ since $\mathcal{V}(\phi, x_1, ..., x_q, \epsilon) = U_{x_1, ..., x_q, \epsilon}(\phi)$.

Of course these uniformities are separating and $\mathcal{S}_s$ is coarser than $\mathcal{S}_\Theta$, we always have: $V_\epsilon \subset U_{x_1, ..., x_q, \epsilon}$.

**Notations:** The set $Mult(A, \| \cdot \|)$ is known to be compact with respect to the usual topology of pointwise convergence [1], [10], [13]. We denote by $\mathcal{T}$ this topology.
Given $\phi \in \operatorname{Mult}(A, \| \cdot \|)$, $x_1, \dots, x_q \in A$ and $\epsilon > 0$ we denote by $\mathcal{W}(\phi, x_1, \dots, x_q, \epsilon)$
the set of $\psi \in \operatorname{Mult}(A, \| \cdot \|)$ such that $|\phi(x_j) - \psi(x_j)|_{\infty} \le \epsilon$ for all $j = 1, \dots, q$.
Then a neighborhood basis of $\phi$ with respect to the topology $\mathcal{T}$ is given by the
family of all sets $\mathcal{W}(\phi, x_1, \dots, x_q, \epsilon)$ where $x_1, \dots, x_q \in A$ and $\epsilon > 0$.

Similarly to $\mathcal{G}(A_0, \mathbb{R})$ this topology is induced by the uniformity $\mathcal{S}$ defined by
all pseudometrics $(\delta_x)_{x \in A_0}$. Actually Proposition 2.2 below proves that they are identical.
For every $\phi \in \operatorname{Mult}(A, \| \cdot \|)$, notice that the restriction $\phi_{A_0}$ of $\phi$ to $A_0$ is in
$\mathcal{G}(A_0, \mathbb{R})$ since $\phi(x) \le \|x\|$ for any $x \in A$. The following lemma is easy to prove.

The uniformity $\mathcal{S}$ is clearly finer than the uniformity defined by the pseudo-
metrics $(\delta_x)_{x \in A_0}$. Actually Proposition 2.2 below proves that they are identical.
For every $\phi \in \operatorname{Mult}(A, \| \cdot \|)$, notice that the restriction $\phi_{A_0}$ of $\phi$ to $A_0$ is in
$\mathcal{G}(A_0, \mathbb{R})$ since $\phi(x) \le \|x\|$ for any $x \in A$. The following lemma is easy to prove.

**Lemma 2.1:** *The map $\phi \in \operatorname{Mult}(A, \| \cdot \|) \longmapsto \phi_{A_0} \in \mathcal{G}(A_0, \mathbb{R})$ is injective. So we may identify $\operatorname{Mult}(A, \| \cdot \|)$ as a subset of $\mathcal{G}(A_0, \mathbb{R})$.*

**Notation.** Next, the metric $\Theta$ is defined on the set of continuous multiplicative semi-norms of $A$: given $\phi, \psi \in \operatorname{Mult}(A, \| \cdot \|)$, we put $\Theta(\phi, \psi) = \Theta(\phi_{A_0}, \psi_{A_0})$.

**Proposition 2.2:** *The usual topology $\mathcal{T}$ of pointwise convergence on $\operatorname{Mult}(A, \| \cdot \|)$ coincides with the topology induced on $\operatorname{Mult}(A, \| \cdot \|)$ by the topology $\mathcal{T}_s$ defined on $\mathcal{G}(A_0, \mathbb{R})$. Consequently the uniformity $\mathcal{S}$ coincides with the uniformity induced by $\mathcal{S}_s$ on $\operatorname{Mult}(A, \| \cdot \|)$.*

**Proof.** The identity map from $\operatorname{Mult}(A, \| \cdot \|)$ provided with the compact topology $\mathcal{T}$ onto $\operatorname{Mult}(A, \| \cdot \|)$ provided with the topology induced by $\mathcal{T}_s$ is clearly continuous, then it is a homeomorphism and the topologies $\mathcal{T}$ and $\mathcal{T}_s$ coincide on $\operatorname{Mult}(A, \| \cdot \|)$. By uniqueness of the uniformity compatible with the compact topology $\mathcal{T}$, the uniformity $\mathcal{S}$ coincides with the uniformity induced by $\mathcal{S}_s$ on $\operatorname{Mult}(A, \| \cdot \|)$. $\square$

Theorems 2.3 and 2.5 recover some results obtained in [11] with other proofs.

**Theorem 2.3:** $\operatorname{Mult}(A, \| \cdot \|)$ as well as every closed subset of $\operatorname{Mult}(A, \| \cdot \|)$ is a compact subspace of $\mathcal{G}(A_0, \mathbb{R})$ with respect to the topology $\mathcal{T}_s$ and it is complete with respect to the metric $\Theta$.

**Proof.** Since $\mathcal{T}$ and $\mathcal{T}_s$ coincide on $\operatorname{Mult}(A, \| \cdot \|)$, then $\operatorname{Mult}(A, \| \cdot \|)$ is compact with respect to $\mathcal{T}_s$, hence it is closed in $\mathcal{G}(A_0, \mathbb{R})$ with respect to $\mathcal{T}_s$. Therefore it is also closed with respect to the finer topology $\mathcal{T}_u$ induced by the metric $\Theta$ and thus it is complete with respect to $\Theta$ since so is $\mathcal{G}(A_0, \mathbb{R})$. The same is true for every closed subspace of $\operatorname{Mult}(A, \| \cdot \|)$ with respect to the topology $\mathcal{T}$. $\square$

Now recall the following result concerning uniform spaces [2]:

**Proposition 2.4:** Let $E$ be a set equipped with a uniformity $\mathcal{S}$. For every compact subspace $X$ and any closed subset $Y$ disjoint from $X$, there exists an element $R \in \mathcal{S}$ such that $R[X] \cap Y = \emptyset$.
Then we get the following result.

**Theorem 2.5:** Let $E$ be a subset of $\mathcal{G}(A_0, \mathbb{R})$ and $X, Y$ be subsets of $E$. If $X$ is compact and $Y$ is closed in $E$ and disjoint from $X$ with respect to the topology of pointwise convergence, then there exist $x \in A_0$ and $\lambda > 0$ such that
$$ \left|\phi(x) - \psi(x)\right|_{\infty} \geq \lambda \text{ for every } \phi \in X, \psi \in Y. \text{ In particular } \Theta(X, Y) > 0. $$

**Proof:** Consider the uniformity $\mathcal{S}_s$ on $E$ (induced by the uniformity $\mathcal{S}_s$ on $\mathcal{G}(A_0, \mathbb{R})$). Then by Proposition 2.4, there exists some $U_{x_1, \dots, x_q, \lambda} \in \mathcal{S}_s$ such that $U_{x_1, \dots, x_q, \lambda}[X] \cap Y = \emptyset$. Thus for every $\phi \in X, \psi \in Y$, we have for each index $i$:
$$ \left|\phi(x_i) - \psi(x_i)\right|_{\infty} \geq \lambda. \text{ This implies of course that } \Theta(X, Y) > 0. \quad \square $$

**Corollary 2.6:** Let $X, Y$ be two compact subsets of $\text{Mult}(A, \| \cdot \|)$ making a partition of $\text{Mult}(A, \| \cdot \|)$. Then there exist $x \in A_0$ and $\lambda > 0$ such that
$$ \left|\phi(x) - \psi(x)\right|_{\infty} \geq \lambda \text{ for every } \phi \in X, \psi \in Y. $$

We consider now the case of a quotient algebra. This result generalizes a result of [15] given for Tate $\mathbb{K}$-algebras.

**Theorem 2.7:** Let $\mathcal{I}$ be a closed ideal in the algebra $(A, \| \cdot \|)$ and denote by $(B, \| \cdot \|_q)$ the usual quotient normed algebra. Let $\pi$ be the quotient map from $A$ onto $B$. Then the map $\mathfrak{P}: \phi \mapsto \phi \circ \pi$ defines a uniform embedding of $\text{Mult}(B, \| \cdot \|_q)$ into $\text{Mult}(A, \| \cdot \|)$, whose range is the subspace $M' = \{\psi \mid \mathcal{I} \subset \ker(\psi)\}$ of $\text{Mult}(A, \| \cdot \|)$.

**Proof:** For any $\phi \in \text{Mult}(B, \| \cdot \|_q)$, the map $\psi = \phi \circ \pi$ is in $\text{Mult}(A, \| \cdot \|)$ and we have $\mathcal{I} \subset \ker(\psi)$. Conversely if $\psi \in \text{Mult}(A, \| \cdot \|)$ and $\mathcal{I} \subset \ker(\psi)$, one can check that if $x - y \in \mathcal{I}$ then $\psi(x) = \psi(y)$. So defining $\phi(\pi(x)) = \psi(x)$, we check that $\phi$ is an element of $\text{Mult}(B, \| \cdot \|_q)$.

Consequently the map $\mathfrak{P}: \phi \mapsto \phi \circ \pi$ defines a one-to-one correspondence between $\text{Mult}(B, \| \cdot \|_q)$ and the subspace $M' = \{\psi \mid \mathcal{I} \subset \ker(\psi)\}$ of $\text{Mult}(A, \| \cdot \|)$.

Now $\mathfrak{P}$ is easily seen to be a uniform isomorphism. Actually if we denote any $\pi(x)$ by $\bar{x}$, if $\phi, \phi' \in \text{Mult}(B, \| \cdot \|_q)$ and $\psi = \mathfrak{P}(\phi)$, $\psi' = \mathfrak{P}(\phi')$, we have:
$$ \left|\phi(\bar{x}) - \phi'(\bar{x})\right|_{\infty} = \left|\psi(x) - \psi'(x)\right|_{\infty} \quad \text{therefore } \mathfrak{P} \times \mathfrak{P}: (\phi, \phi') \mapsto (\mathfrak{P}(\phi), \mathfrak{P}(\phi')) \text{ makes a one-to-one correspondence between the pointwise convergence uniformity on } (\text{Mult}(B, \| \cdot \|_q)) \text{ and the uniformity induced by } \mathcal{S}_s \text{ on } M'. \text{ Consequently } \mathfrak{P} \text{ and its inverse map are uniformly continuous. } \quad \square $$

We conclude this section with metrization and sequential compactness results, generalizing results of [15].

**Theorem 2.8:** If $(A, \| \cdot \|)$ is a separable normed algebra, then the uniformity $\mathcal{S}$ of pointwise convergence is defined by a countable family $(d_n)_n$ of pseudometrics on $\text{Mult}(A, \| \cdot \|)$ and so it is metrizable. Consequently the compact pointwise convergence topology on $\text{Mult}(A, \| \cdot \|)$ is metrizable and sequentially compact.
If $\Omega = \{\omega_n | n \in \mathbb{N}^*\}$ is a dense countable subset of A then putting

$$ \mathcal{D}_0 = \sup_{n \ge 1} \frac{1}{2^n} \min(\delta_{\omega_n}, 1) \qquad \mathcal{D}_1 = \sum_{n=1}^{+\infty} \frac{1}{2^n} \min(\delta_{\omega_n}, 1) $$

we get metrics defining $\mathcal{S}$ and inducing $\mathcal{T}$.

**Proof:** Let $\Omega = \{\omega_n | n \in \mathbb{N}^*\}$ be a dense countable subset of A.

The uniformity $\mathcal{S}$ admits as a filter subbasis the family of all $U_{x,\epsilon}$ for $x \in A$, $\epsilon > 0$, where $U_{x,\epsilon} = \{(\psi, \phi) \in \text{Mult}(A, \|\cdot\|)^2 \mid |\phi(x) - \psi(x)|_{\infty} \le \epsilon\}$.

For any $x \in A$ and $\epsilon > 0$ consider any $n \in \mathbb{N}^*$ such that $\|x - \omega_n\| \le \epsilon/3$. Then for every $\phi \in \text{Mult}(A, \|\cdot\|)$ we have:

$$ |\phi(\omega_n) - \phi(x)|_{\infty} \le \phi(\omega_n - x) \le \|\omega_n - x\| \le \epsilon/3 $$

So the triangle inequality gives $U_{\omega_n, \epsilon/3} \subset U_{x, \epsilon}$. We conclude that $\mathcal{S}$ is also defined by the countable family $(\delta_{\omega_n})_n$ and then we apply Theorem 1.1. $\square$

Finally using Theorem 2.7 we get with the above notations:

**Corollary 2.9:** If A is separable then $\text{Mult}(B, \|\cdot\|_q)$ is metrizable and sequentially compact. The metrics defined by $\mathcal{D}_{i,\pi}(\phi, \phi') = \mathcal{D}_i(\phi \circ \pi, \phi' \circ \pi)$ are compatible metrics.

**Remarks:**

1. Notice that the essential argument for the metrizability of the uniformity $\mathcal{S}$ (and then the metrizability of the topology $\mathcal{T}$) on $\text{Mult}(A, \|\cdot\|)$ is the equicontinuity of the multiplicative semi-norms. As a matter of fact the preceding theorem is a consequence of the second Ascoli Theorem [16]. Actually the set $\text{Mult}(A, \|\cdot\|)$ is an equicontinuous set of functions from A to $\mathbb{R}$ then Ascoli theorem asserts that the uniformity defined by the pseudometrics $(\delta_x)_{x \in A}$ is also the uniformity defined by the pseudometrics $(\delta_\omega)_{\omega \in \Omega}$ if $\Omega$ is dense in A. Ascoli Theorem is used in [15] for the particular case of the balls $E(0, R)$ in $\mathbb{A}^n$ (defined in next section).

2. It is not clear that the metrics $\mathcal{D}_0$ and $\mathcal{D}_1$ given in Theorem 2.8 are comparable to the metric $\Theta$ coming from $\mathcal{G}(A_0, \mathbb{R})$. But we obtain such a comparison with a slight modification of these metrics.

Actually recall that the uniformity $\mathcal{S}$ on $\text{Mult}(A, \|\cdot\|)$ is also defined by the family of pseudometrics $(\delta_x)_{x \in A_0}$ (Proposition 2.2). This is to say that a filter basis for $\mathcal{S}$ is given by the family of all binary relations $U_{x_1, \dots, x_q, \epsilon}$ where $\epsilon > 0$, $x_1, \dots, x_q \in A_0$.

Whenever the normed space A is separable, the subspace $A_0$ is also separable. Then taking $\Omega_0 = \{\omega_n | n \in \mathbb{N}^*\}$ as a dense countable subset of $A_0$, the uniformity $\mathcal{S}$ is also defined by the restricted family $(\delta_{\omega_n})_{n \in \mathbb{N}^*}$ of pseudometrics.
Next we notice that $\delta_x \le 1$ for every $x \in A_0$ because if $\psi, \phi \in \text{Mult}(A, \| \cdot \|)$ then $\psi(x)$ and $\phi(x)$ are in $[0, \|x\|]$. So according to Theorem 1.1, the metrics

$$ \mathcal{D}_0 = \sup_{n \ge 1} \frac{1}{2^n} \delta_{\omega_n} \qquad \mathcal{D}_1 = \sum_{n=1}^{+\infty} \frac{1}{2^n} \delta_{\omega_n} $$

induce $\mathfrak{S}$ and $\mathcal{T}$ on $\text{Mult}(A, \| \cdot \|)$ and these are lower than the metric $\Theta$.

### 3. THE CASE OF SOME AFFINE n-SPACE AND TATE ALGEBRAS

Following [15], given a $\mathbb{K}$-algebra $A$, we denote by $M(A)$ the set of all multiplicative semi-norms on $A$.

The set $M(A)$ is also provided with the (Hausdorff) pointwise convergence topology $\mathcal{T}_s$, which is induced by the uniformity $\mathfrak{S}_s$ defined by all pseudometrics $(\delta_x)_{x \in A}$ on $M(A)$ where, as in section 2, $\delta_x(\phi, \psi) = |\phi(x) - \psi(x)|_\infty$.

Of course any algebra norm $\| \cdot \|$ on $A$ is in $M(A)$ and $\text{Mult}(A, \| \cdot \|)$ is a topological subspace of $M(A)$.

**Definition.** ([15], [1]) The space $M(\mathbb{K}[X_1, \dots, X_n])$ is called the analytic affine $n$-space and it is denoted by $\mathbb{A}^n$.

Several topological properties of $\mathbb{A}^n$ are given in [1] and [15]. In particular it is locally compact, connected, locally connected [1] and whenever $\mathbb{K}$ is separable it is metrizable [15]. In this section we recover some results of [15] and we answer some open question of this paper. Let us recall some preliminary notations and definitions.

**Notations and definitions.** We use the multi-index notation to describe polynomials $P$ of $A = \mathbb{K}[X_1, \dots, X_n]$, that is: $P = \sum_{i \in I} a_i X^i$ where $I$ is a finite subset of $\mathbb{N}^n$ and for any $i = (i_1, \dots, i_n) \in \mathbb{N}^n$, $X^i = X_1^{i_1} \cdots X_n^{i_n}$.

For such an $i$ we write also $|i| = i_1 + \dots + i_n$.

Let $\| \cdot \|_1$ be the usual Gauss norm on $A$ defined by $\|P\|_1 = \max\{|a_i| \mid i \in I\}$.

More generally, for any real $R > 0$ we denote by $\| \cdot \|_R$ the norm on $A$ defined by $\|P\|_R = \max\{|a_i|R^{|i|} \mid i \in I\}$. It is easy to see that $(A, \| \cdot \|_R)$ is isometric to $(A, \| \cdot \|_1)$.

Notice that if $\phi \in M(A)$ then for every $P \in A$ we have $\phi(P) \le \|P\|_{R_\phi}$ where $R_\phi = \max\{\phi(X_k) \mid k = 1, \dots, n\}$.

It is clear that for any $R > 0$ we have $\phi \in \text{Mult}(A, \| \cdot \|_R)$, i.e. $\phi \le \| \cdot \|_R$ if and only if $R_\phi \le R$.

Then we put $E(0, R) = \text{Mult}(A, \| \cdot \|_R)$ for any $R > 0$. Each of them is uniformly isomorphic to $E(0, 1)$. The family $(E(0, R))_{R>0}$ is increasing and

$$ \mathbb{A}^n = M(A) = \bigcup_{R>0} E(0, R) = \bigcup_{N \in \mathbb{N}^*} E(0, N) $$
A power serie is *restricted* whenever for every $\epsilon > 0$, the absolute values of its coefficients but a finite number are lower than $\epsilon$. We denote by $T_n$ the algebra $\mathbb{K}\{X_1, ..., X_n\}$ of restricted power series in $X_1, ..., X_n$. It is also equipped with the Gauss norm which is denoted again by $\| \cdot \|_1$: the Gauss norm of a restricted power serie is the maximum of the absolute values of its coefficients.

The Banach algebra $T_n$ is a completion of the algebra of polynomials $(A, \| \cdot \|_1)$.
The compact spaces $E(0,1) = \text{Mult}(A, \| \cdot \|_1)$ and $\text{Mult}(T_n, \| \cdot \|_1)$ are homeomorphic (and then uniformly isomorphic.)

One can also consider the completion $T_n(R)$ of $(A, \| \cdot \|_R)$, it is isometric to $T_n$
and we will not deal with these spaces.

Recall that any ideal of $T_n$ is closed with respect to $\| \cdot \|_1$ [8].

**Definition.** A *Tate K-algebra* or *K-affinoid algebra* is the quotient algebra of some $T_n$ by an ideal $\mathcal{I}$ of $T_n$. The quotient norm will be denoted by $\| \cdot \|_q$.

According to the notations and results of Theorem 2.7, if $(B, \| \cdot \|_q)$ is a Tate
$\mathbb{K}$-algebra the space $\text{Mult}(B, \| \cdot \|_q)$ is uniformly isomorphic to the subspace
$\{\psi \mid \mathcal{I} \subset \ker(\psi)\}$ of $\text{Mult}(T_n, \| \cdot \|_1)$. This recovers a result given in [15].

**We assume now that the field K is separable** ($K = \mathbb{C}_p$ for example). In this case it is proved in [15] that each $E(0, R)$ is metrizable and then that the topology $\mathcal{T}_s$ on $\mathbb{A}^n$ is metrizable because it is a locally metrizable, paracompact (Hausdorff) space.

A question is: which metric induces $\mathcal{T}_s$ on $\mathbb{A}^n$? A natural one was constructed in [15]: it was a metric $\Delta$ on $\mathbb{A}^n$ whose restriction on each $E(0, R)$ induces the usual pointwise convergence topology on $E(0, R)$. But the question whether $\Delta$ induces the topology $\mathcal{T}_s$ on the whole space $\mathbb{A}^n$ was remaining open.

The answer is not obvious: see [15] for the counter-example on $\mathbb{Q}$ with the usual
metrizable topology and the discrete topology which induce the same (discrete)
topology on any finite subset of $\mathbb{Q}$. However with a slight modification of $\Delta$ we
give here a positive answer to the question.

As a matter of fact the metrizability of each $E(0, R)$ follows from our Theorem
2.8 since the algebra $A = \mathbb{K}[X_1, \dots, X_n]$ is separable: for any dense countable
subset $D$ of $\mathbb{K}$, the subset $\Omega = D[X_1, \dots, X_n]$ is countable and dense in $A$ provided
with any norm $\| \cdot \|_R$. Notice that $\Omega$ does not depend on $R$.

This countable set $\Omega$ is also dense in $(T_n, \| \cdot \|_1)$. Then Theorem 2.8 and Corollary 2.9 assert that for any Tate $\mathbb{K}$-algebra $(B, \| \cdot \|_q)$, the compact space $\text{Mult}(B, \| \cdot \|_q)$ is metrizable and sequentially compact. This result is given in [15].

**Notations.** The structures induced by the topology $\mathcal{T}_s$ and the uniformity $\mathcal{S}_s$
on the subset $E(0, R)$ of $\mathbb{A}^n$ will be denoted by $\mathcal{T}_{s,R}$ and $\mathcal{S}_{s,R}$.
We write $\Omega = \{P_k \mid k \in \mathbb{N}^*\}$ and if $d_k = \min(\delta_{P_k}, 1)$, using the notations of Theorem 2.8, the metrics $\mathcal{D}_0 = \sup_{k \ge 1} \frac{d_k}{2^k}$ and $\mathcal{D}_1 = \sum_{k=1}^{+\infty} \frac{d_k}{2^k}$ define the uniformity $\mathcal{S}_{s,R}$ and induce the topology $\mathcal{T}_{s,R}$ on $E(0, R)$.

Since $\Omega$ is also dense in $T_n$, the metrics $\mathcal{D}_i$ are also defined on $\text{Mult}(T_n, \|\cdot\|_1)$ and they induce the pointwise convergence topology on this space. Applying Corollary 2.9, for any Tate $\mathbb{K}$-algebra $(B, \|\cdot\|_q)$ the metrics $\mathcal{D}_{i,\pi}$ induce the topology on $\text{Mult}(B, \|\cdot\|_q)$.

Since $\mathbb{A}^n$ is the union of the increasing family $(E(0, R))_{R>0}$ then the metrics $\mathcal{D}_i$ are defined on $\mathbb{A}^n$. The metric $\mathcal{D}_0$ is the metric $\Delta$ in [15]. Now the question is: do the metrics $\mathcal{D}_i$ induce the topology $\mathcal{T}_s$ on the whole space $\mathbb{A}^n$?

With a slight modification of the set $\Omega$ we prove that the answer is “yes”.

**Notations.** We denote by $\mathcal{S}_0$, $\mathcal{S}_1$ and $\mathcal{T}_0$, $\mathcal{T}_1$ the uniformities and topologies on $\mathbb{A}^n$ defined by the metrics $\mathcal{D}_0$ and $\mathcal{D}_1$.

Without loss of generality we may assume that the unity is in the countable dense subset $D$ of $\mathbb{K}$ and so that each variable $X_i$ is in the countable dense subset $\Omega = D[X_1, \dots, X_n]$ of $\mathbb{K}[X_1, \dots, X_n]$. We write then $\Omega = \{P_k \mid k \in \mathbb{N}^*\}$ where $P_i = X_i$ for each $i \in \{1, \dots, n\}$.

Then we introduce the notation $\delta_{\bar{X}}$ for the pseudometric $\max_{i=1,\dots,n} \delta_{X_i}$ defined on $\mathbb{A}^n$ by: $\delta_{\bar{X}}(\phi, \psi) = \max_{i=1,\dots,n} |\phi(X_i) - \psi(X_i)|_\infty$. The associated uniformity on $\mathbb{A}^n$ and its induced topology (which is not necessarily Hausdorff) are denoted by $\mathcal{S}_{\bar{X}}$ and $\mathcal{T}_{\bar{X}}$ respectively.

Notice that $R_\phi = \delta_{\bar{X}}(0, \phi)$ for any $\phi \in \mathbb{A}^n$. Hence $E(0, R)$ is the closed ball centered at 0 and of radius $R$ with respect to $\delta_{\bar{X}}$ and so it is closed with respect to $\mathcal{T}_{\bar{X}}$.

In the same way, putting $E'(0, R) = \{\phi \in \mathbb{A}^n \mid R_\phi < R\}$, this set is an open ball centered at 0 so it is open with respect to $\mathcal{T}_{\bar{X}}$ and we have also:

$$ \mathbb{A}^n = \bigcup_{R>0} E'(0, R) = \bigcup_{N \in \mathbb{N}^*} E'(0, N) $$

The following Theorem is clear if we notice that:

* the uniformity $\mathcal{S}_{\bar{X}}$ defined by the pseudometric $\delta_{\bar{X}} = \max\{\delta_{X_1}, \dots, \delta_{X_n}\}$ is also defined by the family $(\delta_{X_i})_{i=1,\dots,n}$, i.e. by $(\delta_{P_i})_{i=1,\dots,n}$

* by Theorem 1.1, the uniformities $\mathcal{S}_i$ defined by the metrics $\mathcal{D}_i$ are identical, both are defined by the family of pseudometrics $(\delta_{P_k})_{k \in \mathbb{N}^*}$

* and finally the uniformity $\mathcal{S}_s$ on $\mathbb{A}^n$ is defined by the family of all pseudometrics $(\delta_P)_{P \in \mathcal{A}}$.
**Theorem 3.1:** We have: $\mathcal{S}_{\bar{X}} \subset \mathcal{S}_0 = \mathcal{S}_1 \subset \mathcal{S}_s$. Therefore $\mathcal{T}_{\bar{X}}$ is coarser than $\mathcal{T}_0$ which coincides with $\mathcal{T}_1$ and which in turn is coarser than $\mathcal{T}_s$ on $\mathbb{A}^n$.

**Corollary 3.2:** The subsets $E(0, R)$ are closed with respect to $\mathcal{T}_0$ (and $\mathcal{T}_1$) and $\mathcal{T}_s$, the subsets $E'(0, R)$ are open with respect to these topologies.

**Theorem 3.3:** The topologies $\mathcal{T}_0$ and $\mathcal{T}_1$ coincide with $\mathcal{T}_s$ on $\mathbb{A}^n$ i.e. the metrics $D_0$ and $D_1$ induce the pointwise convergence topology on $\mathbb{A}^n$.

**Proof:** For each $R > 0$, the topologies induced on $E(0, R)$ by the topologies $\mathcal{T}_0$ (which is also $\mathcal{T}_1$) and $\mathcal{T}_s$ on $\mathbb{A}^n$ are identical. Then the same is true for the topologies induced on $E'(0, R)$.

So the identity map between $(\mathbb{A}^n, \mathcal{T}_0)$ and $(\mathbb{A}^n, \mathcal{T}_s)$ gives rise to homeomorphisms when restricted to each $E'(0, R)$. Since the family $(E'(0, R))_{R>0}$ is an open cover of $\mathbb{A}^n$ with respect to $\mathcal{T}_0$ and $\mathcal{T}_s$, we conclude that the identity map from $(\mathbb{A}^n, \mathcal{T}_0)$ to $(\mathbb{A}^n, \mathcal{T}_s)$ is an homeomorphism too. $\square$

**Remark and conjecture.** The question formulated by Maïnetti can be generalized to the uniformity: do the metrics $D_0$ and $D_1$ define the uniformity $\mathcal{S}_s$ on the whole space $\mathbb{A}^n$, since they do so in each subspace $E(0, R)$? This is not obvious but we conjecture that it is generally false: actually it seems that the essential argument of equicontinuity for the elements of $E(0, R)$ cannot be avoided.

## 4. THE PARTICULAR CASE OF $H(D)$

In this section $\mathbb{K}$ is again a separable complete ultrametric field and we assume as well that it is algebraically closed.

We put $|\mathbb{K}| = \{|x| \mid x \in \mathbb{K}\}$. Then $|\mathbb{K}|$ is dense in $[0, +\infty[$. Given $a \in \mathbb{K}$ and $r > 0$ we denote by $d(a, r)$ the disk $\{\xi \in \mathbb{K} \mid |\xi - a| \le r\}$ and $\theta$ is the ultrametric associated to $|\cdot|$ on $\mathbb{K}$.

Given any infinite closed bounded subset $D$ of $\mathbb{K}$, any rational function with coefficients in $\mathbb{K}$, having no pole in $D$, is bounded [9]. Thus we denote by $R(D)$ the $\mathbb{K}$-algebra of such rational functions, provided with the norm of uniform convergence $\|\cdot\|_0$ on $D$. The $\mathbb{K}$-Banach algebra $A = H(D)$ denotes the completion of $R(D)$ with respect to the norm $\|\cdot\|_0$. The elements of $H(D)$ are called *analytic elements in D* [9], [14].

Topological properties of $\text{Mult}(H(D), \|\cdot\|_0)$ were studied in [3]. Using a bijection between $\text{Mult}(H(D), \|\cdot\|_0)$ and the set of all *circular filters* on $D$ some metric was defined on this space.

When $D$ is *affinoid* then the algebra $H(D)$ is a Krasner-Tate algebra [8], hence a Tate $\mathbb{K}$-algebra and results of Section 3 apply to $H(D)$. Therefore $\text{Mult}(H(D), \|\cdot\|_0)$ is equipped with different metrics and topologies:

* the metrics $D_i$ of section 3 inducing the pointwise convergence topology $\mathcal{T}_s$

* the metric $\Theta$ of section 2 inducing the finer uniform convergence topology $\mathcal{T}_{\Theta}$
• and the metric defined in [3] using the set of circular filters. This metric extends the metric of $K$, we denote it again by $\theta$. The induced topology will be denoted by $\mathcal{T}_{\theta}$.

Can we compare these topologies ? It was proved in [3] that $\mathcal{T}_{\theta}$ is always finer than $\mathcal{T}_s$ and that these topologies are identical if and only if the domain $D$ is compact. For example the disk $D = d(0, 1)$ is not compact since $|K|$ is dense in $\mathbb{R}^+$ and therefore in this case $\mathcal{T}_{\theta}$ is strictly finer than $\mathcal{T}_s$.

Henceforth we consider the special case $D = d(0, 1)$. Recall first some definitions and some facts about circular filters on $K$ and $D$ [3], [11].

**Definitions and notations:** Given $a \in K$, $r > 0$ and $s > r$, we denote by $\Gamma(a, r, s)$ the annulus $\{\xi \in K \mid r < |\xi - a| < s\}$.

Let $a \in K$ and let $R \in ]0, +\infty[$. We call circular filter of center $a$ and diameter $R$ on $K$ the filter $\mathcal{F}$ which admits as a generating system the family of sets $\Gamma(\alpha, r', r'')$ with $\alpha \in d(a, R)$, $r' < R < r''$, i.e. the family of sets of the form $\bigcap_{i=1}^q \Gamma(\alpha_i, r'_i, r''_i)$ with $\alpha_i \in d(a, R)$, $r'_i < R < r''_i$ ($1 \le i \le q$, $q \in \mathbb{N}$) is a basis of the filter $\mathcal{F}$.

Consider now a decreasing sequence of disks $(D_n)_{n \in \mathbb{N}}$ whose intersection is empty, which exists when the field $K$ is not spherically complete, $K = \mathbb{C}_p$ for example. The filter $\mathcal{F}$ admitting as a filter basis this family of disks is called circular filter with no center, of canonical basis $(D_n)_{n \in \mathbb{N}}$. The number $\lim_{n \to +\infty} \mathrm{diam}(D_n)$ is called diameter of $\mathcal{F}$.

Finally the filter of neighborhoods of a point $a \in K$ is called circular filter of the neighborhoods of $a$ on $K$. It is also named circular filter of center $a$ and diameter $0$ or Cauchy circular filter of limit $a$.

Now $D = d(0, 1)$ is closed and a filter $\mathcal{F}_D$ on $D$ is a circular filter on $D$ if $\mathcal{F}_D$ is the trace on $D$ of some circular filter on $K$ which is secant with $D$ (that is to say each element of $\mathcal{F}$ meets $D$): $\mathcal{F}_D = \{F \cap D \mid F \in \mathcal{F}\}$.

We denote by $\Phi(D)$ the set of all these circular filters on $D$. Then $\Phi(D)$ is provided with an order relation defined as follows. Let $\mathcal{F}, \mathcal{G}$ be two circular filters on $D$. Then we put $\mathcal{F} \preceq \mathcal{G}$ if either $\mathcal{F} = \mathcal{G}$ or $\mathcal{G}$ has a center $a$ and a diameter $r$ and $\mathcal{F}$ is secant with $d(a, r)$. Then $\preceq$ is an order relation on $\Phi(D)$ and given circular filters $\mathcal{F}, \mathcal{G}$, they have a least upper bound in $\Phi(D)$ which we denote by $\sup(\mathcal{F}, \mathcal{G})$.

With this order relation a metric is defined on $\Phi(D)$ by:

$$ \theta(\mathcal{F}, \mathcal{G}) = \mathrm{diam}(\sup(\mathcal{F}, \mathcal{G})) - \frac{\mathrm{diam}(\mathcal{F}) + \mathrm{diam}(\mathcal{G})}{2}. $$

If we identify each point $a \in D$ with the circular filter of center $a$ and diameter $0$, then this metric extends the metric associated to the absolute value on $D$ because
if $\mathcal{F}$ and $\mathcal{G}$ are the filters of neighborhoods of $a$ and $b$ respectively then $\sup(\mathcal{F}, \mathcal{G})$ is the circular filter of center $a$ and diameter $|a - b|$.

**Theorem 4.1** [9], [11]: $\Phi(D)$ is complete with respect to the metric $\theta$.

**Theorem 4.2** [9], [13]: For every circular filter $\mathcal{F} \in \Phi(D)$, for every $f \in H(D)$, $|f(x)|$ admits a limit along $\mathcal{F}$, i.e. the image filter $|f|(\mathcal{F})$ has a limit in $\mathbb{R}$. Denoting this limit by $\varphi_{\mathcal{F}}(f)$, the mapping $\Omega$ defined by $\Omega(\mathcal{F}) = \varphi_{\mathcal{F}}$ is a bijection from $\Phi(D)$ onto $\text{Mult}(H(D), \| \cdot \|_0)$.

**Corollary 4.3:** Let $\mathcal{F}$ be the circular filter of center 0 and diameter $r$. Then, given $f \in H(D)$, $\varphi_{\mathcal{F}}(f)$ is the limit of $|f(x)|$ when $|x|$ approaches $r$ while being different from $r$, it is also denoted by $|f|(r)$.

Then according to Theorem 4.2 a metric denoted by $\theta$ again is defined on $\text{Mult}(H(D), \| \cdot \|_0)$ by: $\theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) = \theta(\mathcal{F}, \mathcal{G})$.

We can now make a comparison between the topologies $\mathcal{T}_s, \mathcal{T}_{\theta}, \mathcal{T}_{\Theta}$ defined on $\text{Mult}(H(D), \| \cdot \|_0)$. We yet observed that when $D = d(0, 1)$, a consequence of results in [3] is the following:

**Theorem 4.4:** The topology $\mathcal{T}_{\theta}$ is strictly finer than $\mathcal{T}_s$ on $\text{Mult}(H(D), \| \cdot \|_0)$.

As a matter of fact another argument to prove that these topologies are not identical can be given by the following comparison. Let $\mathcal{G}$ be the circular filter of center 0 and diameter $r \in ]0, 1[$ and $\psi = \varphi_{\mathcal{G}}$. Consider also a sequence $(a_n)_{n \in \mathbb{N}}$ such that $|a_n| < |a_{n+1}|$ and $\lim_{n \to +\infty} |a_n| = r$, let $\phi_n$ be the multiplicative semi-norm defined by $\phi_n(f) = |f(a_n)|$, $n \in \mathbb{N}$, i.e. $\phi_n = \varphi_{\mathcal{F}_n}$ where $\mathcal{F}_n$ is the circular filter of neighborhoods of $a_n$.

Since $a_n \in d(0, r)$ we have $\mathcal{F}_n \subsetneq \mathcal{G}$ for every $n$. Therefore $\theta(\phi_n, \psi) = \frac{r}{2}$ for every $n \in \mathbb{N}$ and the sequence $(\phi_n)_n$ does not converge to $\psi$ with respect to $\theta$. But for every $f \in H(D)$ and every $n \in \mathbb{N}$, we have $\phi_n(f) = |f(a_n)|$, hence $\lim_{n \to +\infty} \phi_n(f) = |f|(r) = \psi(f)$ and then the sequence $(\phi_n)_n$ converges to $\psi$ with respect to the pointwise topology $\mathcal{T}_s$.

On another hand, we noticed in section 2 that the topology $\mathcal{T}_{\Theta}$ is finer than $\mathcal{T}_s$. Unsurprisingly it is strictly finer than $\mathcal{T}_s$. This will be a consequence of Theorems 4.4 and 4.5 below.

**Theorem 4.5:** The topology $\mathcal{T}_{\Theta}$ is strictly finer than $\mathcal{T}_{\theta}$ on $\text{Mult}(H(D), \| \cdot \|_0)$.

Theorem 4.5 is partially proved with the following lemma:

**Lemma 4.6:** The metric $\Theta$ is greater than the metric $\theta$.

**Proof of Lemma 4.6:** We prove that for any distinct circular filters $\mathcal{F}$ and $\mathcal{G}$ we have:

$$ (1) \qquad \Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \geq \theta(\mathcal{F}, \mathcal{G}) $$
The circular filter with center $a \in D$ and diameter $r \ge 0$ will be denoted by $\mathcal{F}_{a,r}$.
The following results can be checked. Notice that for any $a \in D$, the polynomial
$x - a$ is in the unit ball of $H(D)$.

FIRST CASE: the case $\mathcal{F} = \mathcal{F}_{a,r}$ and $\mathcal{G} = \mathcal{F}_{b,s}$ where $a, b \in D$ and $r, s \in [0, +\infty[$.

Then we have: $\sup(\mathcal{F}, \mathcal{G}) = \mathcal{F}_{a,u} = \mathcal{F}_{b,u}$ where $u = \max(r, s, |a - b|)$.

• Assume first that $|a - b| \le \max(r, s)$, then we get:

$$
\theta(\mathcal{F}, \mathcal{G}) = \max(r, s) - \frac{r+s}{2} = \frac{|r-s|}{2}
$$

with $r \neq s$ since $\mathcal{F} \neq \mathcal{G}$. Without loss of generality we may assume that $r < s$
and then $\mathcal{F} = \mathcal{F}_{a,r} \subset \mathcal{G} = \mathcal{F}_{b,s} = \mathcal{F}_{a,s}$ and as a matter of fact $\sup(\mathcal{F}, \mathcal{G}) = \mathcal{G}$. We
have: $\varphi_{\mathcal{F}}(x-a) = r$ and $\varphi_{\mathcal{G}}(x-a) = s$ so $\Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \ge |s-r|$ and (1) is true.

• Assume now that $|a - b| > \max(r, s)$, then we get:

$$
\theta(\mathcal{F}, \mathcal{G}) = |a - b| - \frac{r + s}{2}
$$

We have again: $\varphi_{\mathcal{F}}(x-a) = r$.

On another hand: $\varphi_{\mathcal{G}}(x-a) = \varphi_{\mathcal{F}_{b,s}}(x-b+b-a) = |b-a|$ since $|b-a| > s$.

So $\Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \ge |a-b|-r$. Similarly we get that $\Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \ge |a-b|-s$.

Finally $\Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \ge |a-b| - \frac{r+s}{2}$ and (1) is true again.

SECOND CASE: the case $\mathcal{F} = \mathcal{F}_{a,r}$ where $a \in D$ and $r \ge 0$ and $\mathcal{G}$ has no center.
Assume that $\mathcal{G}$ has diameter $s > 0$ and canonical basis the decreasing sequence
$(D_n)_n$ where $D_n = d(b_n, s_n)$ and $s = \lim_{n \to +\infty} s_n$.

For $n$ large enough, $|x-a|$ is constant on each $D_n$. We denote by $\rho_a$ this constant.
Then we have: $\sup(\mathcal{F}, \mathcal{G}) = \mathcal{F}_{a,u} = \mathcal{F}_{b_n, u}$ for large $n$ where $u = \max(r, \rho_a)$.
Notice that for large $n$, we have: $\rho_a > s_n \ge s$.

• Assume first that $\rho_a \le r$, then we have: $\sup(\mathcal{F}, \mathcal{G}) = \mathcal{F}_{a,r} = \mathcal{F}_{b_n,r}$ for large $n$ and $\theta(\mathcal{F}, \mathcal{G}) = r - \frac{r+s}{2} = \frac{r-s}{2}$.

Then for large $n$, we have: $\varphi_{\mathcal{F}}(x-b_n) = \varphi_{\mathcal{F}_{b_n,r}}(x-b_n) = r$. On another hand, $\lim_{n \to +\infty} \varphi_{\mathcal{G}}(x-b_n) = s$. Therefore $\Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \ge r-s > \theta(\mathcal{F}, \mathcal{G})$.

• Assume now that $\rho_a > r$, we have: $\sup(\mathcal{F}, \mathcal{G}) = \mathcal{F}_{a,\rho_a} = \mathcal{F}_{b_n,\rho_a}$ for large $n$ and
$\theta(\mathcal{F}, \mathcal{G}) = \rho_a - \frac{r+s}{2}$.

Then: $\varphi_{\mathcal{F}}(x-a) = r$ and $\varphi_{\mathcal{G}}(x-a) = \rho_a$.

On another hand, $\lim_{n \to +\infty} \varphi_G(x - b_n) = s$ and for large $n$:

$$
\varphi_{\mathcal{F}}(x - b_n) = \varphi_{\mathcal{F}_{a,r}}(x - a + a - b_n) = |a - b_n| = \rho_a \text{ since } \rho_a > r. \text{ Therefore}
$$

$$
\Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \geq \max(\rho_a - s, \rho_a - r) \geq \rho_a - \frac{r+s}{2} = \theta(\mathcal{F}, \mathcal{G})
$$

THIRD CASE: the case $\mathcal{F}$ and $\mathcal{G}$ have no center. Assume that:
$\mathcal{F}$ has diameter $r > 0$ and canonical basis $(d(a_n, r_n))_n$ with $r = \lim_{n \to +\infty} r_n$ and
$\mathcal{G}$ has diameter $s > 0$ and canonical basis $(d(b_n, s_n))_n$ with $s = \lim_{n \to +\infty} s_n$.

Since $\mathcal{F} \neq \mathcal{G}$ they are not secant and we have $d(a_N, r_N) \cap d(b_N, s_N) = \emptyset$ for some $N$. Let $\rho = |a_N - b_N|$. Then $\rho > \max(r, s)$ and $\rho = |a_n - b_n|$ for every $n \ge N$.

We have: $\sup(\mathcal{F}, \mathcal{G}) = \mathcal{F}_{a_N, \rho} = \mathcal{F}_{b_N, \rho}$ and $\theta(\mathcal{F}, \mathcal{G}) = \rho - \frac{r+s}{2}$.

Then $\lim_{n \to +\infty} \varphi_{\mathcal{F}}(x - a_n) = r$ and
$$ \lim_{n \to +\infty} \varphi_{\mathcal{G}}(x - a_n) = \lim_{n \to +\infty} \varphi_{\mathcal{G}}(x - b_n + b_n - a_n) = \lim_{n \to +\infty} |b_n - a_n| = \rho \text{ since } \rho > s. $$

Therefore $\Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \ge \rho - r$. Similarly $\Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \ge \rho - s$.

Finally $\Theta(\varphi_{\mathcal{F}}, \varphi_{\mathcal{G}}) \ge \rho - \frac{r+s}{2} = \theta(\mathcal{F}, \mathcal{G})$. $\square$

**Proof of Theorem 4.5:** The preceding lemma shows that $\mathcal{T}_{\Theta}$ is finer than $\mathcal{T}_{\theta}$.
Now let us consider the filter $\mathcal{G}$ of center 0 and diameter 1.

For any filter $\mathcal{F}$ of center 0 and of diameter $r < 1$, we have: $\mathcal{F} \preceq \mathcal{G}$ and $\theta(\varphi_{\mathcal{G}}, \varphi_{\mathcal{F}}) = \theta(\mathcal{G}, \mathcal{F}) = \frac{1-r}{2}$. On the other hand, the element $x^n$ belongs to the unit ball of $H(D)$ and we have: $\varphi_{\mathcal{F}}(x^n) = r^n$ and $\varphi_{\mathcal{G}}(x^n) = 1$. Then we have for every $n$, $|\varphi_{\mathcal{G}}(x^n) - \varphi_{\mathcal{F}}(x^n)|_{\infty} = 1 - r^n$, hence: $\Theta(\varphi_{\mathcal{G}}, \varphi_{\mathcal{F}}) \ge 1$.

In particular considering the filter $\mathcal{F}_n$ of center 0 and of diameter $1 - \frac{1}{n}$ for every $n \in \mathbb{N}^*$ we have: $\theta(\varphi_{\mathcal{G}}, \varphi_{\mathcal{F}_n}) = \frac{1}{2n}$ and $\Theta(\varphi_{\mathcal{G}}, \varphi_{\mathcal{F}_n}) \ge 1$ for every $n$, so the sequence $(\varphi_{\mathcal{F}_n})_n$ converges to $\varphi_{\mathcal{G}}$ with respect to $\mathcal{T}_{\theta}$ but not with respect to $\mathcal{T}_{\Theta}$.

This proves that $\mathcal{T}_{\Theta}$ and $\mathcal{T}_{\theta}$ are not identical. $\square$

**Corollary 4.7:** The topology $\mathcal{T}_{\Theta}$ is strictly finer than $\mathcal{T}_s$ on Mult$(H(D), \| \cdot \|_0)$.

**Remark:** We noticed in section 2 that in some cases the metrics $\mathcal{D}_0$ and $\mathcal{D}_1$ are lower than $\Theta$ but it is not obvious in the general case. The comparison between these metrics $\mathcal{D}_i$ and $\Theta$ or $\theta$ on Mult$(H(D), \| \cdot \|_0)$ does not seem easier.

## 5. THE CASE OF ADMISSIBLE AND C-COMPATIBLE ALGEBRAS

Recall that every Banach $\mathbb{K}$-algebra $(A, \| \cdot \|)$ is provided with its spectral semi-norm defined as $\|x\|_{sp} = \lim_{n \to +\infty} \sqrt[n]{\|x^n\|}$. Actually we have:

$$ \|x\|_{sp} = \sup\{\phi(x) \mid \phi \in \mathrm{Mult}(A, \| \cdot \|)\} \quad [9], [13]. $$

Henceforth, $(\mathbb{E}, \delta)$ will denote an ultrametric space. We denote by $B(a, r)$ the associated closed ball centered at $a$ with radius $r$ and by $\mathcal{T}_{\delta}$ the topology induced by the metric $\delta$.

As usual, given subsets $F, G$ of $\mathbb{E}$, we put $\delta(F, G) = \inf\{\delta(x, y) \mid x \in F, y \in G\}$. A subset which is both closed and open is referred to as a *clopen subset* of $\mathbb{E}$.

For every subset $F$ of $\mathbb{E}$, its characteristic function $u_F$ is the function defined from $\mathbb{E}$ to $\mathbb{K}$ as $u_F(x) = 1$ if $x \in F$ and $u_F(x) = 0$ if $x \in \mathbb{E} \setminus F$.
Let $\mathcal{B}(\mathbb{E})$ be the Boolean ring of clopen subsets of $\mathbb{E}$ equipped with the usual laws $\Delta$ for the addition and $\cap$ for the multiplication. The *Stone space associated to* $\mathcal{B}(\mathbb{E})$ is a compactification of $(\mathbb{E}, \mathcal{T}_{\delta})$, it is usually called the *Banaschewski compactification of* $(\mathbb{E}, \mathcal{T}_{\delta})$ [10], [18] and it can be obtained as the space of non-zero ring homomorphisms from $\mathcal{B}(\mathbb{E})$ onto the field $\mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$, equipped with the topology of pointwise convergence.

Let $(\mathbb{T}, \| \cdot \|)$ be a unital Banach $\mathbb{K}$-algebra of bounded continuous functions from $\mathbb{E}$ to $\mathbb{K}$. We denote by $\| \cdot \|_0$ the usual norm of uniform convergence on $\mathbb{T}$.

Given any $a \in \mathbb{E}$, we denote by $\varphi_a$ the multiplicative semi-norm on $\mathbb{T}$ defined as $\varphi_a(f) = |f(a)|$ for every $f \in \mathbb{T}$ and we denote by $\mathrm{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$ the set of all $\varphi_a$ where $a \in \mathbb{E}$.

Recall (see [7]) that a *semi-admissible algebra* is a Banach algebra $(\mathbb{T}, \| \cdot \|)$ of bounded continuous functions from $\mathbb{E}$ to $\mathbb{K}$:

* which contains the characteristic function of each clopen subset of $\mathbb{E}$,

* and in which any function is invertible if and only if its minimum is strictly positive.

A semi-admissible $\mathbb{T}$ is called *admissible* if its spectral semi-norm is equal to $\| \cdot \|_0$.

The usual algebra $(\mathcal{A}, \| \cdot \|_0)$ of all bounded continuous functions from $\mathbb{E}$ to $\mathbb{K}$ is admissible and $\mathrm{Mult}(\mathcal{A}, \| \cdot \|_0)$ is a compactification of $(\mathbb{E}, \mathcal{T}_{\delta})$ which is equivalent to the Banaschewski compactification [12]. Actually this is true for every *admissible* algebra $(\mathbb{T}, \| \cdot \|)$ [7].

Semi-compatible and C-compatible algebras of bounded uniformly continuous functions were introduced and studied in [6], it was shown that for C-compatible algebras $(\mathbb{T}, \| \cdot \|)$, the space $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$ is again a compactification of $(\mathbb{E}, \mathcal{T}_{\delta})$ which is equivalent to a Banaschewski-like compactification [6], [11]. We will define these algebras further.

Then for any admissible or C-compatible algebra, $(\mathbb{E}, \mathcal{T}_{\delta})$ is a topological subspace of the multiplicative spectrum (equipped as usual with the topology of pointwise convergence).

Actually this result has a general proof applying to quite general algebras $\mathbb{T}$. It applies for example to semi-admissible and semi-compatible algebras.

As in section 2, let us denote by $\mathcal{T}_s$ the pointwise convergence topology on $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$. For a given $\phi \in \mathrm{Mult}(\mathbb{T}, \| \cdot \|)$, a neighborhood basis of $\phi$ is the set of all $\mathcal{W}(\phi, f_1, ..., f_q, \epsilon)$ where $f_1, ..., f_q \in \mathbb{T}$ and $\epsilon > 0$ where

$$ \mathcal{W}(\phi, f_1, ..., f_q, \epsilon) = \{ \psi \in \mathrm{Mult}(\mathbb{T}, \| \cdot \|) \mid \forall j = 1, ..., q, \quad \left| \phi(f_j) - \psi(f_j) \right|_{\infty} \le \epsilon \} $$

**Theorem 5.1:** Let $(\mathbb{T}, \| \cdot \|)$ be a unital Banach $\mathbb{K}$-algebra of bounded continuous functions from $\mathbb{E}$ to $\mathbb{K}$ and $\mathcal{B}$ the family of those clopen sets $F \subset \mathbb{E}$ such that the characteristic function of $F$ belongs to $\mathbb{T}$.
If $\mathcal{B}$ is a basis for the topology $\mathcal{T}_\delta$ then the topological space $(\mathbb{E}, \mathcal{T}_\delta)$ is homeomor-
phic to the subspace $\mathrm{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$ of $(\mathrm{Mult}(\mathbb{T}, \| \cdot \|), \mathcal{T}_s)$.

As a consequence its closure $\bar{\mathbb{E}}$ is a compactification of $(\mathbb{E}, \mathcal{T}_\delta)$.

**Proof:** Since $\mathcal{B}$ is a basis for $\mathcal{T}_\delta$, the set $\mathbb{E}$ may be identified with $\mathrm{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$ because given $a \neq b$ in $\mathbb{E}$ we have $\varphi_a \neq \varphi_b$, just consider the characteristic function of any member $B$ of $\mathcal{B}$ containing $a$ which does not contain $b$. Let us check that the topology $\mathcal{T}_\delta$ and the restricted topology $\mathcal{T}_{s|\mathbb{E}}$ induced by $\mathcal{T}_s$ on $\mathbb{E}$ coincide comparing their neighborhood filters at some point $a \in \mathbb{E}$ identified with $\varphi_a$.

Consider any $B$ in $\mathcal{B}$ containing $a$, its characteristic function $u$ belongs to $\mathbb{T}$ and:

$$
\mathcal{W}(a, u, \frac{1}{2}) \cap \mathbb{E} = \left\{ b \in \mathbb{E} \mid \left| |u(a)| - |u(b)| \right|_{\infty} \le \frac{1}{2} \right\} = B
$$

since $u(a) = 1$. So $B$ is a neighborhood of $a$ with respect to the topology $\mathcal{T}_{s|\mathbb{E}}$.

Conversely, consider any $\epsilon > 0$ and $f_1, \dots, f_q \in \mathbb{T}$. Each $f_j$ is continuous at $a$ so there exists some $B \in \mathcal{B}$ containing $a$ such that for every $j = 1, \dots, q$ and every $b \in B$ we have: $\left| |f_j(a)| - |f_j(b)| \right|_{\infty} \le \epsilon$. That is to say that $B \subset \mathcal{W}(a, f_1, \dots, f_q, \epsilon) \cap \mathbb{E}$ and then any neighborhood of $a$ for $\mathcal{T}_{s|\mathbb{E}}$ is also a neighborhood for $\mathcal{T}_{\delta}$. $\square$

With the hypothesis of Theorem 5.1, we may identify any $a \in \mathbb{E}$ with the element $\varphi_a$ of $\mathrm{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$. Then if $X$ is a subset of $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$ we could write $\tilde{X} = X \cap \mathbb{E}$ for $\tilde{X} = X \cap \mathrm{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$.

The second part of the following result is similar to some result obtained in [13]
in another context. As a matter of fact it applies for any disjoint clopen subsets
of $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$ when $\mathbb{E}$ is dense in it.

**Proposition 5.2:** *Assume that $\mathbb{T}$ contains the characteristic function of every clopen set of $\mathbb{E}$.*

• The family of all clopen sets of E is the family of all X ∩ E where X is a clopen subset of the closure E of E in Mult(T, || . ||).

* If X, Y are clopen subsets of $\bar{\mathbb{E}}$ making a partition, then there exist idempotents $u$ and $v \in \mathbb{T}$ such that:

$$
\forall \phi \in X, \forall \psi \in Y \quad \phi(u) = 1, \psi(u) = 0, \phi(v) = 0, \psi(v) = 1
$$

**Proof:**

* Let $F$ be a clopen set of $\mathbb{E}$. Denote by $G$ the complement of $F$ in $\mathbb{E}$. We check that their closures $\bar{F}$ and $\bar{G}$ in $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$ make a partition of $\bar{\mathbb{E}}$. Of course $\bar{F} \cup \bar{G} = \bar{\mathbb{E}}$. Furthermore if $\phi$ is in the intersection, then each neighborhood of $\phi$ intersects both $F$ and $G$. Take the characteristic function $u$ of $F$. Then $u \in \mathbb{T}$ and the neighborhood $\mathcal{W}(\phi, u, \frac{1}{4})$ meets $F$ and $G$ so there exist $a \in F$ and $b \in G$ such that $\phi_a, \phi_b \in \mathcal{W}(\phi, u, \frac{1}{4})$, that is:

$$
|\phi(u) - u(a)|_{\infty} \le \frac{1}{4} \quad \text{and} \quad |\phi(u) - u(b)|_{\infty} \le \frac{1}{4}
$$
Then $|u(a) - u(b)|_{\infty} \le \frac{1}{2}$. Since $a \in F$ we conclude that $u(a) = u(b) = 1$ which is false since $b \notin F$.

Hence we conclude that $\bar{F}$ is a clopen of $\bar{\mathbb{E}}$ and we have $F = \bar{F} \cap \mathbb{E}$.

• Let $X$ and $Y$ be clopen subsets of $\bar{\mathbb{E}}$ making a partition. Put $\tilde{X} = X \cap \bar{\mathbb{E}}$ and $\tilde{Y} = Y \cap \bar{\mathbb{E}}$. These are disjoint clopen sets of $\bar{\mathbb{E}}$ and the characteristic functions $u$ and $v$ of $\tilde{X}$ and $\tilde{Y}$ are in $\mathbb{T}$. Since $X$ and $Y$ are open the closure of $\tilde{X}$ in $\bar{\mathbb{E}}$ is $X$ and the closure of $\tilde{Y}$ is $Y$.

Next for every $\varphi = \varphi_a \in \tilde{X}$ we have $\varphi(u) = 1$ and $\varphi(v) = 0$. Therefore for every $\phi \in X$, we have also $\phi(u) = 1$ and $\phi(v) = 0$. Similarly for every $\psi \in Y$, we have $\psi(u) = 0$ and $\psi(v) = 1$. $\square$

**From now on** we assume that $(\mathbb{T}, \| \cdot \|)$ satisfies the hypothesis of theorem 5.1. so the topological space $(\mathbb{E}, \mathcal{T}_{\delta})$ is homeomorphic to the subspace $\text{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$ of $(\text{Mult}(\mathbb{T}, \| \cdot \|), \mathcal{T}_s)$. Considering the uniform point of view, when is $(\mathbb{E}, \delta)$ a uniform subspace of the multiplicative spectrum?

As in section 2, $\text{Mult}(\mathbb{T}, \| \cdot \|)$ being compact with respect to the topology $\mathcal{T}_s$, we denote again by $\mathcal{S}_s$ the unique uniformity inducing this topology. A basis for this uniformity is given by the family of all sets $U_{f_1, \dots, f_q, \epsilon}$ where

$$ U_{f_1, \dots, f_q, \epsilon} = \left\{ (\psi, \phi) \in \text{Mult}(\mathbb{T}, \| \cdot \|)^2 \mid \forall j = 1, \dots, q, \left| \phi(f_j) - \psi(f_j) \right|_{\infty} \le \epsilon \right\} $$ 
for every $\epsilon > 0$, $f_1, \dots, f_q \in \mathbb{T}$.

Therefore $\mathbb{E}$ comes equipped with two uniformities:

(a) the first one which we denote by $\mathcal{S}_\delta$ is the uniformity defined by $\delta$ and inducing the topology $\mathcal{T}_\delta$, it admits as a basis the family of all $(V_\epsilon)_{\epsilon>0}$, where:

$$ V_{\epsilon} = \{(x, y) \in \mathbb{E} \times \mathbb{E} \mid \delta(x, y) \le \epsilon\} $$

(b) the second one is the uniformity induced by the uniformity $\mathcal{S}_s$ of $\text{Mult}(\mathbb{T}, \| \cdot \|)$ on the subset $\mathbb{E}$ (identified with $\text{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$), this one will be denoted by $\mathcal{S}_{s|\mathbb{E}}$ and admits as a basis the family of all sets $U_{f_1, \dots, f_q, \epsilon} \cap (\mathbb{E} \times \mathbb{E})$ for $\epsilon > 0$, $f_1, \dots, f_q \in \mathbb{T}$. It induces the topology $\mathcal{T}_{s|\mathbb{E}}$ on $\mathbb{E}$ which coincides with $\mathcal{T}_\delta$.

**Proposition 5.3:** For any unital Banach $\mathbb{K}$-algebra $(\mathbb{T}, \| \cdot \|)$ we have: $\mathcal{S}_{s|\mathbb{E}} \subset \mathcal{S}_\delta$ if and only if every function $f \in \mathbb{T}$ is uniformly continuous.

**Proof:** Assume first that every function $f \in \mathbb{T}$ is uniformly continuous. Let $\epsilon > 0$ and $f_1, \dots, f_q$ in $\mathbb{T}$. We have to prove that $U_{f_1, \dots, f_q, \epsilon}$ contains some $V_\eta$ of $\mathcal{S}_\delta$ Since each function $f_j$ is uniformly continuous on $\mathbb{E}$, there exists $\eta > 0$ such that for all $(x, y) \in \mathbb{E} \times \mathbb{E}$ satisfying $\delta(x, y) \le \eta$ we have $\left| |f_j(x)| - |f_j(y)| \right|_{\infty} \le \epsilon$. Then the expected inclusion is true because $\left| |f_j(x)| - |f_j(y)| \right|_{\infty} = \left| \varphi_x(f_j) - \varphi_y(f_j) \right|_{\infty}$.

Conversely assume that $\mathcal{S}_{s|\mathbb{E}} \subset \mathcal{S}_\delta$. Let $f \in \mathbb{T}$ and $\epsilon > 0$. Then $U_{f, \epsilon}$ contains some $V_\eta$ of $\mathcal{S}_\delta$ which is to say that $f$ is uniformly continuous. $\square$
Recall now some vocabulary and also basic facts on the so-called *uniformly open sets* ([12], [6]).

**Definitions and notations:** A subset $F$ of $\mathbb{E}$ is said to be uniformly open if $\delta(F, \mathbb{E} \setminus F) > 0$. Notice that such a subset is clopen in $\mathbb{E}$ and that every open or closed ball of $\mathbb{E}$ is uniformly open, also the union of such balls with the same radius $r > 0$ is uniformly open.

We denote by $\mathbb{G}(\mathbb{E})$ the set of all uniformly open subsets of $\mathbb{E}$.

**Lemma 5.4** Let $F$ be a subset of $\mathbb{E}$ and let $u$ be its characteristic function. The following statements are equivalent:

1) $F$ is uniformly open,

2) $u$ is Lipschitz,

3) $u$ is uniformly continuous.

In particular the characteristic function of any ball is uniformly continuous.

**Lemma 5.5** Given two uniformly open subsets $F, G$ of $\mathbb{E}$, then $F \cup G, F \cap G, \mathbb{E} \setminus G, F \Delta G$ are uniformly open.

**Corollary 5.6** $\mathbb{G}(\mathbb{E})$ is a Boolean ring with respect to the addition $\Delta$ and the multiplication $\cap$.

**Lemma 5.7** Given two subsets $F$ and $G$ of $\mathbb{E}$, there exists a uniformly open subset $C$ such that $F \subset C$ and $C \subset \mathbb{E} \setminus G$ if and only if $\delta(F, G) > 0$.

**Definitions [6], [11]:** A semi-compatible algebra is any unital Banach $\mathbb{K}$-algebra $(S, \| \cdot \|)$ of bounded uniformly continuous functions $f$ from $\mathbb{E}$ to $\mathbb{K}$ satisfying the two following properties:

(1) $S$ contains the characteristic function of every uniformly open subset of $\mathbb{E}$

(2) every function $f \in S$ such that $\inf_{x \in E} |f(x)| > 0$ is invertible in $S$.

Moreover, a semi-compatible $\mathbb{K}$-algebra is said to be *C-compatible* if its spectral norm is equal to the norm $\| \cdot \|_0$ of uniform convergence on $\mathbb{E}$.

Some examples of such algebras are studied in [6]. The algebras of all bounded uniformly continuous functions and the algebra of all bounded Lipschitzian functions are C-compatible algebras (when equipped with suitable norms). Notice that the letter “C” refers to the so-called *contiguity relation* between ultrafilters defined in [12].

Now let us prove the following theorem:

**Theorem 5.8:** Let $(\mathbb{T}, \| \cdot \|)$ be any unital Banach $\mathbb{K}$-algebra of bounded uniformly continuous functions from $\mathbb{E}$ to $\mathbb{K}$ containing the characteristic function of every ball $B(a, r)$ of $(\mathbb{E}, \delta)$ then $\mathfrak{S}_{s|\mathbb{E}} \subset \mathfrak{S}_{\delta}$ and we have:

$$ \mathfrak{S}_{s|\mathbb{E}} = \mathfrak{S}_{\delta} \text{ (i.e. } (\mathbb{E}, \delta) \text{ is a uniform subspace of } \text{Mult}(\mathbb{T}, \| \cdot \|)) $$

if and only if $(\mathbb{E}, \delta)$ is precompact (that is $\mathfrak{S}_{\delta}$ is totally bounded).

**Proof:** We get $\mathfrak{S}_{s|\mathbb{E}} \subset \mathfrak{S}_{\delta}$ from Proposition 5.3.

Next notice that $\mathfrak{S}_s$ is totally bounded since the topology $\mathcal{T}_s$ is compact on $\text{Mult}(\mathbb{T}, \| \cdot \|)$, therefore the induced uniformity on $\mathbb{E}$ is also totally bounded. So if $\mathfrak{S}_{\delta} \subset \mathfrak{S}_{s|\mathbb{E}}$ then $\mathfrak{S}_{\delta}$ is also totally bounded, that is $(\mathbb{E}, \delta)$ is precompact.
Conversely assume that $(\mathbb{E}, \delta)$ is precompact then $\mathfrak{S}_\delta$ is totally bounded. Take $V_\epsilon$ in $\mathfrak{S}_\delta$ for some $\epsilon > 0$. Since $(\mathbb{E}, \delta)$ is precompact, there exist $a_1, \dots, a_q \in \mathbb{E}$ such that $(B(a_j, \epsilon))_{j=1,\dots,q}$ is a partition of $\mathbb{E}$. It is easy to see that

$$V_{\epsilon} = \{(x, y) \in \mathbb{E} \times \mathbb{E} \mid \delta(x, y) \le \epsilon\} = \bigcup_{j=1}^{q} B(a_j, \epsilon) \times B(a_j, \epsilon)$$

Now consider the characteristic function $u_j$ of the ball $B(a_j, \epsilon)$, then $u_j$ is in $\mathbb{T}$ for each $j$ and $U_{u_1, \dots, u_q, \frac{1}{2}}$ is in $\mathfrak{S}_s$. We have:

$$U_{u_1, \dots, u_q, \frac{1}{2}} = \{(\psi, \phi) \in \mathrm{Mult}(\mathbb{T}, \| \cdot \|)^2 \mid \forall j = 1, \dots, q, \left|\phi(u_j) - \psi(u_j)\right|_{\infty} \le \frac{1}{2}\}$$

and then

$$U_{u_1, \dots, u_q, \frac{1}{2}} \cap (\mathbb{E} \times \mathbb{E}) = \{(x, y) \in \mathbb{E} \times \mathbb{E} \mid \forall j = 1, \dots, q, \left|u_j(x) - u_j(y)\right|_{\infty} \le \frac{1}{2}\} = V_{\epsilon}$$

Therefore $V_\epsilon \in \mathfrak{S}_{s|\mathbb{E}}$ and $\mathfrak{S}_\delta \subset \mathfrak{S}_{s|\mathbb{E}}$. This ends the proof. $\square$

In particular the preceding Theorem applies to any algebra $(\mathbb{T}, \| \cdot \|)$ of bounded uniformly continuous functions containing the characteristic function of every uniformly open set of $(\mathbb{E}, \delta)$, to semi-compatible algebras for example. Next results hold also for any such algebra $\mathbb{T}$ and not only for semi-compatible algebras. **From now on**, if no more is specified, we consider a $\mathbb{K}$-algebra $(\mathbb{T}, \| \cdot \|)$ of bounded uniformly continuous functions containing the characteristic function of every uniformly open set of $(\mathbb{E}, \delta)$.

**Theorem 5.9:** Let $F$ and $G$ be subsets of $\mathbb{E}$. Identifying $F$ and $G$ with the corresponding subsets of $\mathrm{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$, then: $\delta(F, G) = 0$ if and only if their closures $\bar{F}$ and $\bar{G}$ in $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$ (or in $\bar{\mathbb{E}}$) have a non-empty intersection.

**Proof:** Assume first that $\bar{F} \cap \bar{G} \ne \emptyset$ and take $\phi$ in the intersection. Then each neighborhood of $\phi$ intersects both $F$ and $G$. Take any uniformly open set $O$ of $\mathbb{E}$ containing $F$ and let us prove that $O \cap G \ne \emptyset$. Consider the characteristic function $u$ of $O$, then $u$ is in $\mathbb{T}$ and the particular neighborhood $W(\phi, u, \frac{1}{4})$ meets $F$ and $G$. This implies that there exist $a \in F$ and $b \in G$ such that $|u(a) - u(b)|_{\infty} \le \frac{1}{2}$. Since $u(a) = 1$ because $a \in F \subset O$ we conclude that $u(b) = 1$ and then $b \in G \cap O$.

Then we have proved that each uniformly open set containing $F$ intersects $G$, so $\delta(F, G) = 0$ by lemma 5.7.

On another hand, assume that $\bar{F} \cap \bar{G} = \emptyset$. Since $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$ is compact, using Proposition 2.4, there exists $U \in \mathfrak{S}_s$ such that $U[\bar{F}] \cap \bar{G} = \emptyset$. Therefore we have also $U[F] \cap G = \emptyset$. According to Theorem 5.3, the element $U \cap (\mathbb{E} \times \mathbb{E})$ of $\mathfrak{S}_{s|\mathbb{E}}$ is in $\mathfrak{S}_\delta$, so it contains some $V_\epsilon$ of $\mathfrak{S}_\delta$ with $\epsilon > 0$ and we have again $V_\epsilon[F] \cap G = \emptyset$. This is to say $\delta(F, G) \ge \epsilon > 0$. $\square$

Theorem 5.10 is analogous to Proposition 5.2, it derives from Theorem 5.9. We use again the following notation: given a subset $X$ of $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$, the set $X \cap \mathrm{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$ is denoted by $\tilde{X}$ and we even write: $\tilde{X} = X \cap \mathbb{E}$.
**Theorem 5.10:**

• The family of all uniformly open sets of $\mathbb{E}$ is the family of all $X \cap \mathbb{E}$ where $X$ is a clopen subset of the closure $\bar{\mathbb{E}}$ of $\mathbb{E}$ in $\text{Mult}(\mathbb{T}, \| \cdot \|)$.

• If $X, Y$ are clopen subsets of $\bar{\mathbb{E}}$ making a partition, then there exist idempotents $u$ and $v \in \mathbb{T}$ such that:

$$ \forall \phi \in X, \forall \psi \in Y \quad \phi(u) = 1, \psi(u) = 0, \phi(v) = 0, \psi(v) = 1 $$

**Proof:**

• Let $X$ be a clopen subset of $\bar{\mathbb{E}}$. Denote by $Y$ its complement in $\bar{\mathbb{E}}$ and consider $\tilde{X} = X \cap \mathbb{E}$ and $\tilde{Y} = Y \cap \mathbb{E}$. Since $X$ and $Y$ are both closed and open, the closure of $\tilde{X}$ in $\bar{\mathbb{E}}$ is $X$ and the closure of $\tilde{Y}$ is $Y$. Then we have $\tilde{X} \cap \tilde{Y} = \emptyset$. So using Theorem 5.9, we get $\delta(\tilde{X}, \tilde{Y}) > 0$, which is to say that $\tilde{X}$ and $\tilde{Y}$ are uniformly open since they make also a partition of $\mathbb{E}$.

Conversely let $F$ be uniformly open in $\mathbb{E}$ and denote by $G$ its complement $\mathbb{E} \setminus F$. We have $\delta(F, G) > 0$ so by Theorem 5.9 we get $\bar{F} \cap \bar{G} = \emptyset$ in $\bar{\mathbb{E}}$. But $F \cup G = \mathbb{E}$ so $\bar{F} \cup \bar{G} = \bar{\mathbb{E}}$. Thus $\bar{F}$ and $\bar{G}$ are clopens of $\bar{\mathbb{E}}$ making a partition of $\bar{\mathbb{E}}$ and we have $F = \bar{F} \cap \bar{\mathbb{E}}$.

• We just have to proceed as for the second part of Proposition 5.2. From the above result, we get that $\tilde{X}$ and $\tilde{Y}$ are uniformly open sets of $\mathbb{E}$ so the characteristic functions $u$ and $v$ of $\tilde{X}$ and $\tilde{Y}$ are in $\mathbb{T}$. $\square$

**Remark:** Theorem 5.9 gives another direct proof to Theorem 5.8 for algebras $\mathbb{T}$ containing the characteristic function of every uniformly open set of $\mathbb{E}$. This proof is based on links between compactifications, uniformities and proximities.

Recall the following known facts ([4], [19]). There is a one-to-one correspondence between compactifications and the so-called proximities (in the sense of Efremovič) inducing a given completely regular topological space $(\mathbb{E}, \mathcal{T})$ and also with the totally bounded uniformities inducing $\mathcal{T}$.

More precisely a proximity $\Pi$ is any relation on subsets of $\mathbb{E}$ satisfying some properties. It induces a topology $\mathcal{T}$ on $\mathbb{E}$ in the following sense: $x$ is close to $A$ with respect to $\mathcal{T}$ if and only if $\{x\} \Pi A$.

Given any proximity $\Pi$ on $\mathbb{E}$ inducing a topology $\mathcal{T}$, the associated compactification $\mathbb{F}$ of $(\mathbb{E}, \mathcal{T})$ is the only one which satisfies:

$A \Pi B$ if and only if the closures of $A$ and $B$ in $\mathbb{F}$ intersect.

Then the associated totally bounded uniformity $\mathcal{S}_{\Pi}$ is the uniformity on $\mathbb{E}$ induced by the unique compatible uniformity on $\mathbb{F}$. This uniformity is characterized to be the smallest uniformity $\mathcal{S}$ on $\mathbb{E}$ inducing $\Pi$ in the following sense:

$A \Pi B$ if and only if for every $V \in \mathcal{S}$ we have $V[A] \cap B \neq \emptyset$

Also $\mathcal{S}_{\Pi}$ is the greatest totally bounded uniformity on $\mathbb{E}$ coarser than any uniformity $\mathcal{S}$ inducing $\Pi$ [19].

In our context, let us consider the particular proximity $\Pi$ defined by:

$$ A \Pi B \iff \delta(A, B) = 0 $$
According to the preceding Theorem 5.9 the closure of $\mathbb{E}$ in $(\mathrm{Mult}(\mathbb{T}, \| \cdot \|), \mathcal{T}_s)$ is the corresponding compactification of $(\mathbb{E}, \mathcal{T}_\delta)$. Next the corresponding totally bounded uniformity is $\mathcal{S}_{s|\mathbb{E}}$. It is the only totally bounded uniformity inducing $\Pi$ and it is contained in any other uniformity inducing $\Pi$. Now we just have to notice that the metric uniformity $\mathcal{S}_\delta$ induces $\Pi$ since:

$$ \delta(A, B) = 0 \quad \text{if and only if for every } V \in \mathcal{S}_\delta \text{ we have } V[A] \cap B \neq \emptyset $$

Therefore we get that $\mathcal{S}_{s|\mathbb{E}} \subset \mathcal{S}_\delta$ and equality if and only if $\mathcal{S}_\delta$ is totally bounded, which is to say that $(\mathbb{E}, \delta)$ is precompact. $\square$

To conclude we give without proofs the following results obtained in [6], [7], [12] concerning the special cases of admissible and C-compatible algebras to which the above results apply. More specifically their multiplicative spectrum are homeomorphic to some Boolean space. Recall first the following known facts about Stone spaces (see [18] for a bibliography).

Let $\mathcal{B}(\mathbb{E})$ be the ring of clopen sets of $\mathbb{E}$. To any subring $\mathcal{C}$ of $\mathcal{B}(\mathbb{E})$ one associates its so-called *Stone space* which we denote by $\mathrm{St}(\mathcal{C})$. This space $\mathrm{St}(\mathcal{C})$ can be obtained as the set of all non-zero ring homomorphisms from $\mathcal{C}$ onto $\mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$, equipped with the topology of pointwise convergence. When the subring $\mathcal{C}$ is a basis for $(\mathbb{E}, \mathcal{T}_\delta)$ then $\mathrm{St}(\mathcal{C})$ is a 0-dimensional compactification of $(\mathbb{E}, \mathcal{T}_\delta)$ and $\mathcal{C}$ turns out to be the family of all $X \cap \mathbb{E}$ where $X$ is a clopen set of $\mathrm{St}(\mathcal{C})$. When $\mathcal{C} = \mathcal{B}(\mathbb{E})$, the space $\mathrm{St}(\mathcal{B}(\mathbb{E}))$ is the usually called *Banaschewski compactification of $(\mathbb{E}, \mathcal{T}_\delta)$*.

**Theorem 5.11:** ([6], [7]) *When $(\mathbb{T}, \| \cdot \|)$ is an admissible or a C-compatible algebra then $\mathrm{Mult}_{\mathbb{E}}(\mathbb{T}, \| \cdot \|)$ is dense in $(\mathrm{Mult}(\mathbb{T}, \| \cdot \|), \mathcal{T}_s)$ so that $(\mathrm{Mult}(\mathbb{T}, \| \cdot \|), \mathcal{T}_s)$ is a compactification of the topological space $(\mathbb{E}, \mathcal{T}_\delta)$.*

**Theorem 5.12:** ([7]) *When $(\mathbb{T}, \| \cdot \|)$ is an admissible algebra then $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$ is a compactification of $(\mathbb{E}, \mathcal{T}_\delta)$ which is equivalent to the Banaschewski compactification of $\mathbb{E}$.*

In particular Theorems 5.11 and 5.12 are true for the algebra $(\mathcal{A}, \| \cdot \|_0)$ of all bounded continuous functions from $\mathbb{E}$ to $\mathbb{K}$, this recovers results of [12].

**Theorem 5.13:** ([6]) *When $(\mathbb{T}, \| \cdot \|)$ is a C-compatible algebra then $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$ is a compactification of $(\mathbb{E}, \mathcal{T}_\delta)$ which is equivalent to the compactification given by the Stone space associated to the subring $\mathcal{G}(\mathbb{E})$ of uniformly open sets.*

Then if $(\mathbb{T}, \| \cdot \|)$ is admissible or C-compatible, $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$ is homeomorphic to the Stone space associated to the ring $\mathcal{B}(\mathbb{E})$ or to the subring $\mathcal{G}(\mathbb{E})$. So we recover results of Proposition 5.2 and Theorem 5.10 for these particular algebras:

**Corollary 5.14:** *When $(\mathbb{T}, \| \cdot \|)$ is an admissible algebra then $\mathcal{B}(\mathbb{E})$ is the family of all $X \cap \mathbb{E}$ where $X$ is a clopen set of $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$.*

*When $(\mathbb{T}, \| \cdot \|)$ is a C-compatible algebra then $\mathcal{G}(\mathbb{E})$ is the family of all $X \cap \mathbb{E}$ where $X$ is a clopen set of $\mathrm{Mult}(\mathbb{T}, \| \cdot \|)$.*
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UNIVERSITÉ CLERMONT AUVERGNE, UMR CNRS 6620, LMBP, F-63000 CLERMONT-FERRAND, FRANCE.

Email address: monique.chicourrat@uca.fr, alain.escassut@uca.fr