Characterisation of the Berkovich Spectrum of

the Banach Algebra of Bounded Continuous Functions

Tomoki Mihara

Received: May 08, 2014 Communicated by Peter Schneider

Abstract. For a complete valuation field k and a topological space X, we prove the universality of the underlying topological space of the Berkovich spectrum of the Banach k-algebra C_{bd}(X, k) of bounded continuous k-valued functions on X. This result yields three applications: a partial solution to an analogue of Kaplansky conjecture for the automatic continuity problem over a local field, comparison of two ground field extensions of C_{bd}(X, k), and non-Archimedean Gel’fand theory. 2010 Mathematics Subject Classification: 11S80, 18B30, 46S10 Keywords and Phrases: Berkovich spectrum, Stone space, Banaschewski compactification, non-Archimedean Gel’fand–Naimark theorem, non-Archimedean Gel’fand theory, non-Archimedean Kaplansky conjecture

Contents

0 Introduction  770
1 Preliminaries  772
1.1 Berkovich Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
1.2 Stone Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
1.3 Universality of the Stone Space . . . . . . . . . . . . . . . . . . . . . 773
2 Main Result  777
2.1 Statement of the Main Theorem . . . . . . . . . . . . . . . . . . . . 777
2.2 Maximality of a Closed Prime Ideal . . . . . . . . . . . . . . . . . . 779
2.3 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . 779
3 Related Results  785
3.1 Another Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 785
3.2 Relation to the Stone–Čech compactification . . . . . . . . . . . . . . 788
4 Applications  789
4.1 Automatic Continuity Theorem . . . . . . . . . . . . . . . . . . . . . 789
4.2 Ground Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . 791
4.3 Non-Archimedean Gel’fand Theory . . . . . . . . . . . . . . . . . . 795

Acknowledgements  797
References  797

0 Introduction

A non-Archimedean analytic space plays an important role in various studies in modern number theory. There are several ways to formulate a non-Archimedean analytic space, and one of them is given by Berkovich in [Ber1] and [Ber2]. Berkovich introduced the spectrum ℳ_{k}(𝒜) of a Banach algebra 𝒜 over a complete valuation field k. The space ℳ_{k}(𝒜) is to a Banach algebra 𝒜 what Spec(A) is to a ring A. We note that ℳ_{k}(𝒜) is called the Berkovich spectrum in modern number theory, but the same notion is originally defined by Bernard Guennebaud in [Gue]. The class of Banach algebras topologically of finite type over a complete valuation field is significant in analytic geometry, just as the class of algebras of finite type over a field is significant in algebraic geometry. A Banach algebra topologically of finite type is called an affinoid algebra, and the Berkovich spectrum of an affinoid algebra is called an affinoid space. The space ℳ_{k}(𝒜) is a compact Hausdorff G-topological space. For the notion of G-topology, see [BGR]. Berkovich formulated an analytic space by gluing affinoid spaces with respect to a certain G-topology, just as Grothendieck did a scheme by gluing affine schemes with respect to the Zariski topology. We remark that an affinoid space is studied well, while few properties are known for the Berkovich spectrum of a general Banach algebra. Throughout this paper, X and k denote a topological space and a complete valuation field respectively. Here a valuation field means a field endowed with a valuation of height at most 1, and we allow the case where the valuation is trivial. We study the underlying topological space of the Berkovich spectrum BSC_{k}(X) of the Banach algebra C_{bd}(X, k) of bounded continuous k-valued functions on X. In Theorem 2.1, we prove that BSC_{k}(X) is naturally homeomorphic to the Stone space UF(X) associated to X, where UF(X) is a topological space under X (Definition 1.1) constructed using the set of ultrafilters of a Boolean algebra associated to X. This homeomorphism is significant because UF(X) is an initial object in the category of totally disconnected compact Hausdorff spaces under X (Definition 1.2). As a consequence, BSC_{k}(X) satisfies the same universality, and hence is independent of k. We note that Banaschewski proved the existence of such an initial object only for zero-dimensional spaces in [Ban] Satz 2, while we deal with a general topological space in this paper. We also remark that many of our results are verified by Alain Escassut and Nicolas Maïnetti in [EM1] and [EM2] under the assumption that X is metrisable by an ultrametric. Therefore our results are generalisations of some of their results. We have three applications of Theorem 2.1, which connects non-Archimedean analysis and general topology. First, C_{bd}(X, k) satisfies the weak version of the automatic continuity theorem if k is a local field (Theorem 4.6). Namely, for a Banach k-algebra 𝒜, every injective k-algebra homomorphism φ: C_{bd}(X, k) ↪ 𝒜 with closed image is continuous. In particular, it gives a criterion for the continuity of a faithful linear representation of C_{bd}(X, k) on a Banach space. Second, for an extension K/k of complete valuation fields, the ground field extension BSC_{K}(X) → BSC_{k}(X) induced by the inclusion C_{bd}(X, k) ↪ C_{bd}(X, K) is a homeomorphism (Proposition 4.9). There is another ground field extension K ˆ⊗_{k}C_{bd}(X, k) → C_{bd}(X, K) given by the universality of the complete tensor product ˆ⊗_{k} in the category of Banach k-algebras. We will see the difference of those two in Theorem 4.12. Finally, we show that the natural continuous map X → UF(X) is a homeomorphism onto the image if and only if X is zero-dimensional and Hausdorff (Lemma 4.13). We establish Gel’fand theory for totally disconnected compact Hausdorff spaces in this case (Theorem 4.19) using a non-Archimedean generalisation of Stone– Weierstrass theorem ([Ber1] 9.2.5. Theorem). Here, Gel’fand theory means a natural contravariant-functorial one-to-one correspondence between the collection 𝒞 (X) of equivalence classes of totally disconnected compact Hausdorff spaces which contain X as a dense subspace and the set 𝒞′(X) of closed k-subalgebras of C_{bd}(X, k) separating points of X. We remark that the Berkovich spectrum of a Banach algebra is analogous to the Gel’fand transform of a commutative C^{∗}-algebra. We study Berkovich spectra in this paper expecting that many facts for Gel’fand transforms also hold for Berkovich spectra. For example, it is well-known that an initial object in the category of compact Hausdorff spaces under X exists and is constructed as the Gel’fand transform ℳ_{}(C_{bd}(X, )) of the commutative C^{∗}-algebra C_{bd}(X, ) of bounded continuous - valued functions on X. Therefore our result for the universality of BSC_{k}(X) is a direct analogue of this fact. We recall another construction of an initial object in the category of compact Hausdorff spaces under X. The Stone–Čech compactification βX of X is constructed as a closed subspace of a direct product of copies of the closed unit disc ^{∘} ⊂ , and it admits a canonical continuous map X → βX such that every bounded continuous -valued function on X uniquely extends to a continuous function on βX. This extension property guarantees that βX is also an initial object in the category of compact Hausdorff spaces under X. One sometimes assumes that X is a completely regular Hausdorff space in the definition of βX so that X → βX is a homeomorphism onto the image, but we do not because we allow compactifications of X whose structure morphism is not injective. Imitating the construction of βX, we construct a compactification SC_{k}(X) of X as a closed subspace of a direct product of copies of the closed unit disc k^{∘} ⊂ k. We also compare BSC_{k}(X) and SC_{k}(X), and prove that they are naturally homeomorphic to each other under X when k is a local field or a finite field. In §1.1, we recall the definition of Berkovich spectra. In §1.2, we recall the Stone space UF(X) associated to X. In §1.3, we show the universality of UF(X). In §2.1, we state the main theorem (Theorem 2.1). In order to verify it, we construct two set-theoretical maps supp: BSC_{k}(X) → Spec(C_{bd}(X, k)) and Ch_{•}: Spec(C_{bd}(X, k)) → UF(X). We show that the composite Ch_{supp} Ch_{•} ∘ supp: BSC_{k}(X) → UF(X) is a homeomorphism. Its proof is not straightforward, and is completed in the following two subsections. In §2.2, we show that every closed prime ideal of C_{bd}(X, k) is maximal. In §2.3, we verify that the image of supp coincides with the subset of closed prime ideals, and we prove that the restriction of Ch_{•} on the image of supp is bijective. After that, we verify that Ch_{supp} is a homeomorphism, and this completes the proof of Theorem 2.1. In §3.1, we compare BSC_{k}(X) and SC_{k}(X) in the case where k is a local field or a finite field. In §3.2, we observe a connection between BSC_{_{p}} (X) and βX. We show that BSC_{_{p}} (X) is homeomorphic to βX for special X’s. In §4, we deal with the three applications of Theorem 2.1 mentioned above.

1 Preliminaries

In this section, we recall the definition of the Berkovich spectrum ℳ_{k}(𝒜) of a Banach algebra 𝒜, and the Stone space UF(X) associated to X. For more details, see [Ber1] and [Ber2] for Berkovich spectra, and see [Ban], [Joh], [Sto2], and [Sto3] for Stone spaces.

1.1 Berkovich Spectra

A Banach k-algebra means a pair (𝒜 , ∥ · ∥) of a unital associative commutative kalgebra 𝒜 and a complete submultiplicative non-Archimedean norm ∥ · ∥: 𝒜 → [0, ∞). We often write 𝒜 instead of (𝒜 , ∥ · ∥) for short. Let (𝒜 , ∥ · ∥) be a Banach k-algebra. Since 𝒜 is unital, it admits a canonical ring homomorphism k → 𝒜, and we also denote by a ∈ 𝒜 the image of a ∈ k. A map x: 𝒜 → [0, ∞) is said to be a bounded multiplicative seminorm of (𝒜 , ∥ · ∥) if the following conditions hold:

(i) x(f − g) ≤ max{x(f), x(g)} for any f, g ∈ 𝒜.

(ii) x(f g) = x(f)x(g) for any f, g ∈ 𝒜.

(iii) x(f) ≤ ∥f ∥ for any f ∈ 𝒜.

(iv) x(a) = |a| for any a ∈ k.

We denote by ℳ_{k}(𝒜) = ℳ_{k}(𝒜 , ∥ · ∥) the set of bounded multiplicative seminorms of (𝒜 , ∥ · ∥) endowed with the weakest topology for which for any f ∈ 𝒜, the map

f^{∗}: ℳ_{k}(𝒜)  → [0, ∞)
x ↦ x(f)

is continuous. We call ℳ_{k}(𝒜) the Berkovich spectrum of (𝒜 , ∥ · ∥). By [Ber1] 1.2.1. Theorem, ℳ_{k}(𝒜) is a compact Hausdorff space, and is non-empty if and only if 𝒜 0.

1.2 Stone Spaces

A U ⊂ X is said to be clopen if it is closed and open. We denote by CO(X) ⊂ 2^{X} the set of clopen subsets of X. A topological space X is said to be zero-dimensional if CO(X) forms an open basis of X. The space CO(X) possesses much information about the topology of X when X is zero-dimensional. The most elementary example of a zero-dimensional space is the underlying topological space of k. For each c ∈ k and ϵ > 0, the subsets of k of the forms {c′ ∈ k | |c′− c| < ϵ}, {c′ ∈ k | |c′− c| ≤ ϵ}, {c′ ∈ k | |c′− c| > ϵ}, and {c′ ∈ k | |c′− c| ≥ ϵ} are clopen. The set CO(X) is a Boolean algebra with respect to ∨, ∧, ¬, and ⊥ given by setting U∨V U∪V, U∧V U∩V, ¬U X\U, and ⊥ ∅ respectively for U, V ∈ CO(X). We recall the notion of an ultrafilter of a Boolean algebra. For readers who are not familiar with Boolean algebras and filters, [Joh] and [Sto3] might be helpful. For a Boolean algebra (A, ∨, ∧, ¬), an ℱ ⊂ A is said to be a filter of (A, ∨, ∧, ¬) if it satisfies the following:

(i) ¬ ⊥∈ ℱ.

(ii) a ∧ b ∈ ℱ for any a, b ∈ ℱ.

(iii) a ∨ b ∈ ℱ for any a ∈ A and b ∈ ℱ.

A filter ℱ of (A, ∨, ∧, ¬) is said to be an ultrafilter if ℱ A and if for any filter ℱ′ of (A, ∨, ∧, ¬), ℱ ⊂ ℱ′ A implies ℱ = ℱ′. It is equivalent to the condition that ⊥ ℱ and either a ∈ ℱ or ⊥ a ∈ ℱ holds for any a ∈ A. For each S ⊂ A, the smallest filter ℱ of (A, ∨, ∧, ¬) containing S exists. Then ℱ is a proper subset of A if and only if a_{1} ∧ ⋯ ∧ a_{n} ⊥ for any n ∈ \{0} and (a_{1}, …, a_{n}) ∈ A^{n}. For any filter ℱ of (A, ∨, ∧, ¬) with ℱ A, there exists an ultrafilter ℱ′of (A, ∨, ∧, ¬) containing ℱ by Boolean prime ideal theorem. The set of ultrafilters of (A, ∨, ∧, ¬) is endowed with the topology described in the following way: Its subset 𝒰 is open if and only if for any ℱ ∈ 𝒰 , there is an a ∈ ℱ such that 𝒢 ∈ 𝒰 for any ultrafilter 𝒢 of (A, ∨, ∧, ¬) containing a. Applying this construction to CO(X), we denote by UF(X) the resulting topological space, and we call it the Stone space associated to X. For example, the subset

ℱ (x) {U ∈ CO(X) |x ∈ U}⊂ CO(X)

is an ultrafilter of (CO(X), ∨, ∧, ¬) for any x ∈ X, and we call such an ultrafilter a principal ultrafilter.

1.3 Universality of the Stone Space

We denote by C(X, Y) the set of continuous maps f : X → Y for topological spaces X and Y, and by Top the category of topological spaces and continuous maps. We also deal with the full subcategory TDCHTop ⊂ Top of totally disconnected compact Hausdorff spaces.

Definition 1.1. For a category 𝒞, a full subcategory 𝒞′ ⊂ 𝒞, and an A ∈ ob(𝒞), a 𝒞′-object under A is a pair (B, f) of a B ∈ ob(𝒞′) and an f ∈ Hom_{𝒞} (A, B). Here we regard B as an object of 𝒞 through the inclusion 𝒞′ ↪ 𝒞. We call f the structure morphism of (B, f) or simply of B. We denote by A/𝒞′ the category of 𝒞′-objects under A and morphisms compatible with the structure morphisms.

In the case 𝒞 = ob(Top), for an X ∈ ob(Top) and a (Y, f) ∈ ob(X/𝒞′), we call f the structure map of Y. We often abbreviate (Y, f) to Y.

Definition 1.2. For a category 𝒞, an object I of 𝒞 is said to be initial if Hom_{𝒞} (I, A) consists of one morphism for any A ∈ ob(𝒞).

An initial object is unique up to a unique isomorphism if it exists. For example, for a category 𝒞, a full subcategory 𝒞′ ⊂ 𝒞, and an A ∈ ob(𝒞), a (B, ι) ∈ ob(A/𝒞′) is initial if and only if the map

Hom_{𝒞} (A, C) → Hom_{𝒞′} (B, C)
f ↦ f ∘ ι

is bijective for any C ∈ ob(𝒞′). In other words, B is an initial 𝒞′-object under A with respect to ι if and only if for any C ∈ ob(𝒞′) and any g: A → C, there exists a unique g̃: B → C such that g = g̃∘ ι.

Theorem 1.3. The correspondence X UF(X) gives a functor UF: Top → TDCHTop which is the left adjoint functor of the inclusion TDCHTop ↪ Top.

We remark that [BJ] Proposition 5.7.12 and the universality of the Stone–Čech compactification imply Theorem 1.3. We will prove Theorem 1.3 in an explicit way at the end of this subsection. For the proof, we prepare several lemmas and a proposition. We note that for a category 𝒞 and a full subcategory 𝒞′ ⊂ 𝒞, a functor J: 𝒞 → 𝒞′ is a left adjoint functor of the inclusion I: 𝒞′ ↪ 𝒞 if and only if there is a natural transform ι: id_{𝒞} → I ∘ J such that the induced map

Hom_{𝒞′} (J(A), B) → Hom_{𝒞} (A, I(B))
f ↦ f ∘ ι_{A}

is bijective for any A ∈ ob(𝒞) and B ∈ ob(𝒞′). This is equivalent to the condition that J(A) is initial in A/𝒞′ with respect to the adjunction ι_{A}: A → I(J(A)) = J(A) for any A ∈ ob(𝒞). In order to give a proof of Theorem 1.3, we show several fundamental properties of the Stone Space. We remark that this gives an alternative proof of Theorem 3.13 in [Tar]. An x ∈ X is said to be a cluster point of an ℱ ∈ UF(X) if ℱ contains all clopen neighbourhood of x. Each x ∈ X is a cluster point of the principal ultrafilter ℱ (x) ∈ UF(X). Unlike a set-theoretical ultrafilter, the existence of a cluster point gives a strict restriction to an ultrafilter as is shown in the following lemma. An ultrafilter consists of open subsets, and hence carries more information on the topology of X than a settheoretical ultrafilter does.

Lemma 1.4. If an ℱ ∈ UF(X) has a cluster point, then ℱ is a principal ultrafilter.

Proof. Let x ∈ X be a cluster point of ℱ. Then ℱ contains the principal ultrafilter ℱ (x), and hence coincides with ℱ (x) by the maximality of an ultrafilter.

For a non-empty family ℱ of sets, we set ⋂ ℱ ⋂ _{U∈ℱ} U. We give an explicit description of the set of cluster points of a filter.

Lemma 1.5. The set of cluster points of an ℱ ∈ UF(X) coincides with ⋂ ℱ.

Proof. For a cluster point x ∈ X of ℱ, one has x ∈ ⋂ ℱ (x) = ⋂ ℱ by Lemma 1.4. For an x ∈ ⋂ ℱ, assume that there is a U ∈ CO(X) such that x ∈ U ℱ. Then one obtains X\U ∈ ℱ, and it contradicts the condition x ∈ ⋂ ℱ. Thus x is a cluster point of ℱ.

Lemma 1.6. If X is a discrete infinite set, then UF(X) contains a non-principal ultrafilter.

Proof. The cardinality of the set of principal ultrafilters is at most #X, while #UF(X) coincides with 2^{2^{#X}} in the case where X is a discrete infinite set by [Eng] 3.6.11. Theorem.

Proposition 1.7. Suppose that X is zero-dimensional.

(i) X is compact if and only if every ultrafilter has at least one cluster point.

(ii) X is Hausdorff if and only if every ultrafilter has at most one cluster point.

(iii) X is a totally disconnected compact Hausdorff space if and only if every ultrafilter has precisely one cluster point.

The assertion is an analogue of the classical result for set-theoretic ultrafilters, and the following proof imitates the proof of it. For the classical result, see [Eng] 1.6.11. Proposition and 3.1.24. Theorem.

Proof. When X is zero-dimensional, X is Hausdorff if and only if X is totally disconnected, and therefore the criteria (i) and (ii) immediately imply the criterion (iii). If X is compact, an ultrafilter has a cluster point because the intersection ⋂ ℱ is non-empty by the finite-intersection property of a compact space. On the other hand, suppose that every ultrafilter has at least one cluster point. Assume that X is not compact. Since X is zero-dimensional, there is a clopen covering 𝒰 of X which has no finite subcovering. The set 𝒱 {U ∈ CO(X) | X\U ∈ 𝒰 } of complements satisfies ⋂ 𝒱 = ∅ and any finite intersection of clopen subsets in 𝒱 is non-empty. Therefore there is an ℱ ∈ UF(X) containing 𝒱 . One has ⋂ ℱ ⊂ ⋂ 𝒱 = ∅, which contradicts the assumption that every ultrafilter has at least one cluster point by Lemma 1.5. Thus X is compact. If X is Hausdorff, then the continuous map ℱ (·): X → UF(X) is injective because X is zero-dimensional. Suppose that every ultrafilter has at most one cluster point. Assume that X is not Hausdorff. There are two distinct points x, y ∈ X such that any clopen neighbourhoods of x and y have non-empty intersection. In other words, one has U ∩V ∅ for any (U, V) ∈ ℱ (x)×ℱ (y). Take a clopen neighbourhood U ∈ ℱ (x) of x. By the argument above, one has X\U ℱ (y), and hence U ∈ ℱ (y). It implies ℱ (x) ⊂ ℱ (y), and therefore ℱ (x) = ℱ (y) by the maximality of an ultrafilter. Both x and y are two distinct cluster points of ℱ (x) = ℱ (y), and it contradicts the assumption that every ultrafilter has at most one cluster point. Thus X is Hausdorff.

As a consequence, for a zero-dimensional space X, one obtains the following criteria.

(i)′ The space X is compact if and only if ℱ (·) is surjective.

(ii)′ The space X is Hausdorff if and only if ℱ (·) is injective.

(iii)′ The space X is a totally disconnected compact Hausdorff space if and only if ℱ (·) is bijective.

We remark that the bijectivity of ℱ (·) in (iii)′ can be replaced by the condition that ℱ (·) is a homeomorphism by the following three lemmas.

Lemma 1.8. The ℱ (·): X → UF(X) is continuous and its image is dense.

Proof. For a U ∈ CO(X), the pre-image of the open subset {ℱ ∈ UF(X) | U ∈ ℱ } is U ⊂ X itself. Therefore ℱ (·) is continuous. Let ℱ ⊂ CO(X) be an ultrafilter, and 𝒰 ⊂ UF(X) an open neighbourhood of ℱ. By the definition of the topology of UF(X), there is a U ∈ CO(X) such that U ∈ ℱ and 𝒱 {ℱ′ ∈ UF(X) | U ∈ ℱ′} ⊂ 𝒰 . Then U ∅ because ∅ ℱ, and hence ℱ (U) ∅. Since ℱ (U) ⊂ ℱ (X) ∩ 𝒱 ap ⊂ ℱ (X) ∩ 𝒰 , one concludes ℱ (X) ∩ 𝒰 ∅.

Lemma 1.9. The space UF(X) is a totally disconnected compact Hausdorff space.

This assertion is contained in the general fact of the Stone space in [Sto2] Theorem IV_{2}, but we give a proof for reader’s convenience.

Proof. For a U ∈ CO(X), one has

UF(X) = {ℱ ∈ UF(X) |U ∈ ℱ } ⊔ {ℱ ∈ UF(X) |X\U ∈ ℱ } ,

and hence CO(UF(X)) forms an open basis of UF(X). Therefore by Proposition 1.7 and Lemma 1.8, it suffices to show that UF(X) is compact and Hausdorff, because a continuous map from a compact space to a Hausdorff space is a closed map. For ℱ , 𝒢 ∈ UF(X) with ℱ 𝒢, take a U ∈ CO(X) contained in precisely one of them. Then the complement X\U is contained in the other one. Therefore the partition

UF(X) = {ℱ ∈ UF(X) |U ∈ ℱ } ⊔ {ℱ ∈ UF(X) |X\U ∈ ℱ }

by clopen subsets of UF(X) separates ℱ and 𝒢. Thus UF(X) is Hausdorff. Assume that UF(X) is not compact. There is a clopen covering 𝒰 of UF(X) which has no finite subcovering. In particular, the subset

𝒱 {U ∈ CO(UF(X)) |UF(X)\U ∈ 𝒰 }

satisfies ⋂ 𝒱 = ∅ and any finite intersection of clopen subsets belonging to 𝒱 is non-empty. Since the map ℱ (·) is continuous, the inverse image

ℱ (·)^{∗}𝒱 {ℱ (·)^{−1}(V) | V ∈ 𝒱 }

is a non-empty subset of CO(X) satisfying that ⋂ ℱ (·)^{∗}𝒱 = ∅ and any finite intersection of clopen subsets belonging to ℱ (·)^{∗}𝒱 is non-empty. Therefore there is an ℱ ∈ UF(X) containing ℱ (·)^{∗}𝒱 by the facts recalled in §1.2. Since 𝒰 covers UF(X), there is a U ∈ 𝒰 containing ℱ. The pre-image V ∈ ℱ (·)^{∗}𝒱 of the complement UF(X)\U ∈ 𝒱 is contained in ℱ because ℱ (·)^{∗}𝒱 ⊂ ℱ. By the definition of the topology of UF(X), there is a W ∈ ℱ such that W ∈ 𝒢 implies 𝒢 ∈ U for any 𝒢 ∈ UF(X). In particular, for any x ∈ W, W ⊂ ℱ (x) and hence ℱ (x) ∈ U. Therefore one obtains W ⊂ ℱ (·)^{−1}(U). Since V, W ∈ ℱ, one has V ∩ W ∈ ℱ and hence V ∩W ∅. Take an x ∈ V ∩W ⊂ X. Since V = ℱ (·)^{−1}(UF(X)\U), one has ℱ (x) U, which contradicts the condition x ∈ W ⊂ ℱ (·)^{−1}(U). Thus UF(X) is compact.

Lemma 1.10. If X is a totally disconnect compact Hausdorff space, then ℱ (·): X → UF(X) is a homeomorphism.

In particular, ℱ (·): UF(X) → UF(UF(X)) is a homeomorphism without the assumption on X by Lemma 1.9.

Proof. The assertion immediately follows from Proposition 1.7 (iii), Lemma 1.8, and Lemma 1.9, because every continuous map between compact Hausdorff spaces is closed.

Proof of Theorem 1.3. By Lemma 1.8 and Lemma 1.9, (UF(X), ℱ (·)) is an object of X/TDCHTop. Let Y, Z ∈ Top and f ∈ C(Y, Z). For an ℱ ∈ UF(Y), the subset

UF(f)_{∗}ℱ {U ∈ CO(Y) | φ^{−1}(U) ∈ ℱ } .

is an ultrafilter of CO(Z). The map UF(f)_{∗}: UF(Y) → UF(Z) is continuous by the definition of the topologies of UF(Y) and UF(Z). The correspondences Y UF(Y) and f UF(f)_{∗} gives a functor UF: Top → TDCHTop. Therefore it suffices to show that (UF(X), ℱ (·)) is an initial object of X/TDCHTop. Let (Y, φ) be an object of X/TDCHTop. Since the image of X is dense in UF(X) by Lemma 1.8 and Y is Hausdorff, a continuous extension UF(φ): UF(X) → Y is unique if it exists. The diagram

X ⟶^{φ} Y
ℱ(·) ↓  ↓^{ℱ(·)}
UF(X) ⟶^{UF(φ)_{∗}} UF(Y)

commutes by the definitions of ℱ (·) and UF(φ)_{∗}, and the right vertical map is a homeomorphism by Lemma 1.10. Therefore one obtains a continuous extension ℱ (·)^{−1} ∘ UF(φ)_{∗}: UF(X) → Y of φ.

2 Main Result

2.1 Statement of the Main Theorem

We denote by C_{bd}(X, k) the Banach k-algebra of bounded continuous k-valued functions on X endowed with the supremum norm. We put BSC_{k}(X) ℳ_{k}(C_{bd}(X, k)).

Let ι_{k} denote the evaluation map

ι_{k}: X → BSC_{k}(X)
x ↦ (ι_{k}(x): f ↦ |f (x)|),

which is continuous by the definition of the topology of BSC_{k}(X).

Theorem 2.1. There is a natural homeomorphism BSC_{k}(X) UF(X) compatible with ι_{k} and ℱ (·).

In other words, there is a natural transform Φ: BSC_{k} → UF such that Φ(Y) lies in Hom_{Y/TDCHTop}((BSC(Y), ι_{k}), (UF(Y), ℱ (·))) for any topological space Y. In particular, it gives an isomorphism (BSC(X), ι_{k}) (UF(X), ℱ (·)) in X/TDCHTop, and hence (BSC(X), ι_{k}) satisfies the same universality as (UF(X), ℱ (·)) does.

Corollary 2.2. The space BSC_{k}(X) is initial in X/TDCHTop with respect to ι_{k}.

Corollary 2.3. The functor

BSC_{k}: Top → TDCHTop
X  BSC_{k}(X)

is a left adjoint functor of the inclusion of the full subcategory.

Corollary 2.4. The image of ι_{k}: X → BSC_{k}(X) is dense.

In order to prove Theorem 2.1, we introduce two set-theoretical maps supp and Ch_{•}. For an x ∈ BSC_{k}(X), its support supp(x) {f ∈ C_{bd}(X, k) | x(f) = 0} is a closed prime ideal. We call the map

supp: BSC_{k}(X) → Spec(C_{bd}(X, k))
x ↦ supp(x)

the support map. For an m ∈ Spec(C_{bd}(X, k)), the family Ch_{m} {U ∈ CO(X) | 1_{U} m} is an ultrafilter, where 1_{U}: X → k denotes the characteristic function of U ∈ CO(X). Indeed, Ch_{m} is stable under ∪ because m is an ideal, and is stable under ∩ because m is a prime ideal. The maximality of Ch_{m} follows from the property that either 1_{U} ∈ m or 1_{X\U} = 1 − 1_{U} ∈ m holds for any U ∈ CO(X) because m is a prime ideal. We call the map

Ch_{•}: Spec(C_{bd}(X, k)) → UF(X)
m ↦ Ch_{m}

the characteristic map. We put Ch_{supp} Ch_{•} ∘ supp: BSC_{k}(X) → UF(X).

Example 2.5. For an x ∈ X, supp(ι_{k}(x)) ⊂ C_{bd}(X, k) is the maximal ideal consisting of functions vanishing at x, and one has Ch_{supp(ι_{k}(x))} = ℱ (x). Thus Ch_{supp} is an extension of the continuous map ℱ (·): X → UF(X) via ι_{k}.

We prove that Ch_{supp} is a homeomorphism under X in three steps in §2.2 and §2.3. First, we show that every closed prime ideal of C_{bd}(X, k) is a maximal ideal. Second, we verify that the image of supp coincides with the subset of closed prime ideals, and study the restriction of Ch_{•} on the image of supp. Finally, we prove that Ch_{supp} is a homeomorphism.

2.2 Maximality of a Closed Prime Ideal

We prove that every closed prime ideal of C_{bd}(X, k) is a maximal ideal. We remark that this is proved by Alain Escassut and Nicolas Maïnetti in [EM1] Theorem 12 in the case where X is an ultrametric space. Here we assume nothing on X, and hence X is not necessarily metrisable.

Proposition 2.6. For any m_{1}, m_{2} ∈ Spec(C_{bd}(X, k)) with m_{1} ⊂ m_{2}, Ch_{m_{1}} = Ch_{m_{2}}.

Proof. The condition m_{1} ⊂ m_{2} implies Ch_{m_{2}} ⊂ Ch_{m_{1}} by definition. Since Ch_{m_{2}} is an ultrafilter, the inclusion guarantees Ch_{m_{1}} = Ch_{m_{2}}.

Proposition 2.7. For closed prime ideals m_{1}, m_{2} ⊂ C_{bd}(X, k), the equality Ch_{m_{1}} = Ch_{m_{2}} implies m_{1} = m_{2}.

Proof. Suppose Ch_{m_{1}} = Ch_{m_{2}} for closed prime ideals m_{1}, m_{2} ⊂ C_{bd}(X, k). It suffices to show m_{1} ⊂ m_{2}. Take an element f ∈ m_{1}. For a positive real number ϵ, we set U_{ϵ} {x ∈ X | |f (x)| < ϵ}, and then U_{ϵ} ⊂ X is a clopen subset, because it is preimage of the clopen subset {c ∈ k | |f (x) − c| < ϵ} by the continuous function f. Set f_{ϵ} (1 − 1_{U_{ϵ}})f ∈ C_{bd}(X, k). Since f ∈ m_{1}, one has f_{ϵ} ∈ m_{1}. The absolute value of f_{ϵ} + 1_{U_{ϵ}} ∈ C_{bd}(X, k) at each point in X has a lower bound min{ϵ, 1}, and hence its inverse is bounded and continuous. It implies that f_{ϵ} + 1_{U_{ϵ}} is invertible in C_{bd}(X, k), and therefore 1_{U_{ϵ}} m_{1}. One has U_{ϵ} ∈ Ch_{m_{1}} = Ch_{m_{2}}, and hence 1 − 1_{U_{ϵ}} = 1_{X\U_{ϵ}} ∈ m_{2}. Thus f_{ϵ} = (1−1_{U_{ϵ}})f ∈ m_{2}, and the inequality ∥f −f_{ϵ}∥ = ∥1_{U_{ϵ}}f ∥ ≤ ϵ guarantees f ∈ m_{2} by the closedness of m_{2}.

Proposition 2.8. Every closed prime ideal of C_{bd}(X, k) is a maximal ideal.

We note that for a Banach k-algebra 𝒜, every maximal ideal of 𝒜 is a closed prime ideal by [BGR] 1.2.4. Corollary 5, but the converse does not hold in general. For example, the Tate algebra k{T} has a non-maximal closed ideal {0} ⊂ k{T}.

Proof. For a closed prime ideal m_{1} ⊂ C_{bd}(X, k), take a maximal ideal m_{2} ⊂ C_{bd}(X, k) containing m_{1}. Then m_{2} is also a closed prime ideal by [BGR] 1.2.4. Corollary 5. The assertion immediately follows from Proposition 2.6 and Proposition 2.7.

2.3 Proof of the Main Theorem

Proposition 2.9. The image of supp is the subset of closed prime ideals.

Proof. Every closed prime ideal m ⊂ C_{bd}(X, k) is a maximal ideal by Proposition 2.8, and hence there is an x ∈ BSC_{k}(X) such that supp(x) = m by the argument in the proof of [Ber1] 1.2.1. Theorem.

Proposition 2.10. The restriction of Ch_{•} on the image of supp is bijective.

Proof. If X = ∅, then Spec(C_{bd}(X, k)) = UF(X) = ∅, and hence we may assume X ∅. By Proposition 2.7 and Proposition 2.9, it suffices to verify the surjectivity. Take an ℱ ∈ UF(X). Set

m {f ∈ C_{bd}(X, k) | inf_{U∈ℱ} sup_{x∈U}|f (x)| = 0} ⊂ C_{bd}(X, k).

Then m ⊂ C_{bd}(X, k) is an ideal, and 1 m because |1(x)| = 1 for any x ∈ X ∅. We verify that the map

∥ · ∥_{ℱ} : C_{bd}(X, k) → [0, ∞)
f ↦ inf_{U∈ℱ} sup_{x∈U}|f (x)| < ∥f ∥

is continuous. The map ∥ · ∥_{ℱ} is continuous at any f ∈ C_{bd}(X, k) with ∥f ∥_{ℱ} = 0 because for any g ∈ C_{bd}(X, k)\{f }, there is a U_{0} ∈ ℱ with sup_{x∈U} |f (x)| < ∥f − g∥ and hence

∥g∥_{ℱ} ≤ inf_{U∈ℱ} sup_{x∈U}|f (x) − (f − g)(x)|
≤ inf_{U∈ℱ} sup_{x∈U}max{|f (x)|, |(f − g)(x)|}≤ ∥f − g∥.

The map ∥ · ∥_{ℱ} is locally constant at any f ∈ C_{bd}(X, k) with ∥f ∥_{ℱ} 0 because for any g ∈ C_{bd}(X, k) with ∥f − g∥ < ∥f ∥_{ℱ} , we have

∥g∥_{ℱ} ≤ inf_{U∈ℱ} sup_{x∈U}|f (x) − (f − g)(x)| ≤ inf_{U∈ℱ} sup_{x∈U}max{|f (x)|, |(f − g)(x)|}
≤ inf_{U∈ℱ} max{sup_{x∈U}|f (x)|, ∥f − g∥} = ∥f ∥_{ℱ} .

Therefore ∥ · ∥_{ℱ} is continuous. Since {0} ⊂ [0, ∞) is closed, m is a closed ideal. For f, g ∈ C_{bd}(X, k) with f g ∈ m, suppose f m. We prove g ∈ m. If g = 0, then g ∈ m. Therefore we may assume g 0. Since f m, there is some ϵ_{0} > 0 such that the clopen subset V {x ∈ X | |f (x)| < ϵ} does not belong to ℱ for any 0 < ϵ < ϵ_{0}. Let 0 < ϵ < ϵ_{0}. The condition f g ∈ m implies that there is some U ∈ ℱ such that sup_{x∈U} |(f g)(x)| < ϵ^{2}. Since ℱ is an ultrafilter, one has X\V ∈ ℱ and hence U\V = U ∩ (X\V) ∈ ℱ. For an x ∈ U\V, the inequality |g(x)| = |f (x)|^{−1}|f (x)g(x)| < ϵ implies sup_{x∈U\V}|g(x)| < ϵ. One obtains ∥g∥_{ℱ} = 0, and hence g ∈ m. Therefore m is a closed prime ideal. Let U ∈ ℱ. One gets ∥1_{U}∥_{ℱ} = 1 by definition, and hence U ∈ Ch_{m}. It implies ℱ ⊂ Ch_{m}. Since ℱ is an ultrafilter, one concludes ℱ = Ch_{m}. Thus ℱ is contained in the image of Ch_{supp} by Proposition 2.9.

Proof of Theorem 2.1. The map Ch_{supp} is compatible with ι_{k} and ℱ (·) as is shown in Example 2.5. We prove that Ch_{supp} is a homeomorphism. We first prove the bijectivity. Since the restriction of Ch_{•} on the image of supp is bijective by Proposition 2.10, we have only to show that supp is injective. For that purpose, for a maximal ideal m ⊂ C_{bd}(X, k), we consider the relation between the quotient seminorm ∥ · +m∥ at m and the map ∥ · ∥_{Ch_{m}} defined in the proof of Proposition 2.10. For an f ∈ C_{bd}(X, k), one has

∥f + m∥ = inf_{g∈m}∥f − g∥ ≥ inf_{g∈m}∥f − g∥_{Ch_{m}} = inf_{g∈m}∥f ∥_{Ch_{m}} = ∥f ∥_{Ch_{m}}.

Take an r ∈ with ∥f ∥_{Ch_{m}} < r. Set

U {x ∈ X ||f (x)| > r}.

Then U ⊂ X is clopen by an argument similar to the one in the proof of Proposition 2.7. If U ∈ Ch_{m}, then one has

∥f ∥_{Ch_{m}} = inf_{V∈Ch_{m}} sup_{x∈V}|f (x)| ≥ inf_{V∈Ch_{m}} sup_{x∈U∩V} |f (x)| ≥ inf_{V∈Ch_{m}}r = r

and hence it contradicts the condition ∥f ∥_{Ch_{m}} < r. It implies U Ch_{m}, and therefore 1_{U} ∈ m. One obtains

∥f + m∥ ≤ ∥f − 1_{U}f ∥ = ∥1_{X\U}f ∥ ≤ r.

One gets ∥f + m∥ = ∥f ∥_{Ch_{m}}. Next, we prove that the map ∥·∥_{Ch_{m}} is a bounded multiplicative seminorm on C_{bd}(X, k). It is a bounded power-multiplicative seminorm by definition, and it suffices to show the multiplicativity. Let f, g ∈ C_{bd}(X, k) such that ∥f g∥_{Ch_{m}} < ∥f ∥_{Ch_{m}}∥g∥_{Ch_{m}}. In particular, ∥f ∥_{Ch_{m}}∥g∥_{Ch_{m}} 0 and f, g m. Take an ϵ > 0 such that ϵ < ∥f ∥_{Ch_{m}}, ϵ < ∥g∥_{Ch_{m}}, and ∥f g∥_{Ch_{m}} < (∥f ∥_{Ch_{m}} − ϵ)(∥g∥_{Ch_{m}} − ϵ). Set

V_{1} {x ∈ X | |f (x)| > ∥f ∥_{Ch_{m}} − ϵ}
V_{2} {x ∈ X | |g(x)| > ∥g∥_{Ch_{m}} − ϵ}.

Then V_{1}, V_{2} ⊂ X are clopen. If V_{1} Ch_{m}, then X\V_{1} ∈ Ch_{m}, but the inequality

∥f ∥_{Ch_{m}} ≤ sup_{x∈X\V_{1}}|f (x)| ≤ ∥f ∥_{Ch_{m}} − ϵ

contradicts the condition ϵ > 0. Therefore V_{1} ∈ Ch_{m}. Similarly, one obtains V_{2} ∈ Ch_{m}, and hence V_{1} ∩ V_{2} ∈ Ch_{m}. Then the inequality

∥f g∥_{Ch_{m}} < (∥f ∥_{Ch_{m}} − ϵ)(∥g∥_{Ch_{m}} − ϵ) ≤ inf_{W∈Ch_{m}} sup_{x∈V_{1}∩V_{2}∩W}|f (x)| |g(x)|

≤ inf_{W∈Ch_{m}}sup_{x∈W}|f (x)g(x)| = ∥f g∥_{Ch_{m}}

holds, and it is a contradiction. Thus ∥f g∥_{Ch_{m}} = ∥f ∥_{Ch_{m}}∥g∥_{Ch_{m}}. We conclude that the map ∥ · ∥_{Ch_{m}} is a bounded multiplicative seminorm, and hence corresponds to a point in BSC_{k}(X). Now take an x ∈ BSC_{k}(X). Since y ∥ · ∥_{Ch_{supp}(x)} ∈ BSC_{k}(X) coincides with the quotient seminorm ∥ · +supp(x)∥, one has x(f) ≤ y(f) for any f ∈ C_{bd}(X, k). It implies that x gives a bounded multiplicative norm of the complete residue field k(y) at y, because supp(y) is a maximal ideal. It implies x = y because y(f) = y(f^{−1})^{−1} ≤ x(f^{−1})^{−1} = x(f) for any f ∈ k(y)^{×}. Thus x is reconstructed from its image y by Ch_{supp}, and hence Ch_{supp} is injective. Finally, we verify the continuity of Ch_{supp}. Take a U ∈ CO(X), and set 𝒰 {ℱ ∈ UF(X) | U ∈ ℱ }. The pre-image of 𝒰 by Ch_{supp} is the subset

{x ∈ BSC_{k}(X) | U ∈ Ch_{supp}(x)}= {x ∈ BSC_{k}(X) | 1_{U} supp(x)}
= {x ∈ BSC_{k}(X) |x(1_{U}) > 0}⊂ BSC_{k}(X),

and it is open by the definition of the topology of BSC_{k}(X). Therefore Ch_{supp} is a continuous bijective map between compact Hausdorff spaces, and is a homeomorphism. This completes the proof.

We give several corollaries. These are generalisations of some of results in [EM1] and [EM2]. In those papers, Alain Escassut and Nicolas Maïnetti deal with ultrametric spaces, while we deal with general topological spaces. We remark that they deal with not only the class of bounded continuous functions, but also that of bounded uniformly continuous functions with respect to the uniform structure associated to the ultrametric.

Corollary 2.11. The map supp gives a bijective map from BSC_{k}(X) to the set of maximal ideals of C_{bd}(X, k), and every maximal ideal of C_{bd}(X, k) is the support of a unique bounded multiplicative seminorm on C_{bd}(X, k).

This is a generalisation of [EM1] Theorem 16 for the class of bounded continuous functions.

Proof. We proved that the injectivity of supp in the proof of Theorem 2.1, and the image of supp coincides with the subset of maximal ideals by Proposition 2.8 and Proposition 2.9. Thus the assertion holds.

Corollary 2.12. Every bounded multiplicative seminorm on C_{bd}(X, k) is of the form

∥ · ∥_{ℱ} : C_{bd}(X, k) → [0, ∞)
f ↦ inf_{U∈ℱ} sup_{x∈U}|f (x)|

for a unique ℱ ∈ CO(X).

Proof. Let x ∈ BSC_{k}(X). We proved the equality x = ∥ · ∥_{Ch_{supp}(x)} in the proof of Theorem 2.1. The uniqueness of an ℱ ∈ CO(X) follows from the surjectivity of Ch_{supp}.

We denote by UF(|X|) the set of set-theoretical ultrafilters of X. We compare UF(|X|) with UF(X) through the bijection Ch_{supp} in Theorem 2.1.

Corollary 2.13. The inclusion CO(X) ↪ 2^{X} is a Boolean algebra homomorphism, and induces a surjective map

(· ∩ CO(X)): UF(|X|) → UF(X)
𝒰 ↦ 𝒰 ∩ CO(X).

For 𝒰 , 𝒰′ ∈ UF(|X|), the equality lim_{𝒰} |f (x)| = lim_{𝒰′} |f (x)| holds for any f ∈ C_{bd}(X, k) if and only if 𝒰 ∩ CO(X) = 𝒰′∩ CO(X).

Proof. Let ℱ ∈ UF(X). Since ℱ is a family of subsets of X which is closed under intersections and satisfies ∅ ℱ, there is an 𝒰 ∈ UF(|X|) containing ℱ. It implies the surjectivity of the given correspondence. Let 𝒰 ∈ UF(|X|) and f ∈ C_{bd}(X, k). The limit lim_{𝒰} |f (x)| exists because the boundedness of f guarantees that f (X) is relatively compact in . Moreover, since 𝒰 ∩ CO(X) ⊂ 𝒰 , we have ∥f ∥_{𝒰 ∩CO(X)} = lim_{𝒰} |f (x)|. Thus the second assertion follows from the injectivity of the inverse map of Ch_{supp}: BSC_{k}(X) → UF(X).

Corollary 2.14. Every bounded multiplicative seminorm on C_{bd}(X, k) is of the form

C_{bd}(X, k) → [0, ∞)
f ↦ lim_{𝒰}|f (x)|

for a 𝒰 ∈ UF(|X|), where lim_{𝒰} |f (x)| denotes the limit of the -valued continuous function |f |: X → : x ↦ |f (x)| along 𝒰 for each f ∈ C_{bd}(X, k).

This together with Corollary 2.13 is a generalisation of [EM1] Corollary 16.3.

Proof. Every x ∈ BSC_{k}(X) is presented as ∥ · ∥_{ℱ} by ℱ Ch_{supp}(x). By Corollary 2.13, there is a 𝒰 ∈ UF(|X|) containing ℱ, and satisfies x(f) = ∥f ∥_{ℱ} = lim_{𝒰} |f (x)| for any f ∈ C_{bd}(X, k).

A topological space X is said to be strongly zero-dimensional if for any disjoint closed subsets F, F′ ⊂ X there is a U ∈ CO(X) such that F ⊂ U ⊂ X\F′. We note that every strongly zero-dimensional Hausdorff space is zero-dimensional. For example, every topological space metrisable by an ultrametric is a first countable strongly zerodimensional Hausdorff space.

Corollary 2.15. Suppose that X is strongly zero-dimensional. For 𝒰 , 𝒰′ ∈ UF(|X|), the equality lim_{𝒰} |f (x)| = lim_{𝒰′} |f (x)| holds for any f ∈ C_{bd}(X, k) if and only if F ∩ F′ ∅ for any closed subsets F, F′ ⊂ X with F ∈ 𝒰 and F′ ∈ 𝒰′.

This is a generalisation of [EM1] Theorem 4 for the class of bounded continuous functions, and together with Corollary 2.13 implies [EM1] Theorem 1. We remark if we removed the assumption of the strong zero-dimensionality, then there are obvious counter-examples. For example, a connected space is never strongly zero-dimensional unless it has at most one point, and every k-valued continuous function on a connected space is a constant function. In particular, every set-theoretical ultrafilter gives the same limit.

Proof. To begin with, suppose that the equality lim_{𝒰} |f (x)| = lim_{𝒰′} |f (x)| holds for any f ∈ C_{bd}(X, k). Then we have 𝒰 ∩ CO(X) = 𝒰′∩ CO(X) by Corollary 2.13. Let F, F′ ⊂ X be closed subsets with F ∈ 𝒰 and F′ ∈ 𝒰′. Assume F∩F′ = ∅. Then there is a U ∈ CO(X) such that F ⊂ U ⊂ X\F′because X is strongly zero-dimensional. We obtain U ∈ 𝒰 ∩ CO(X) and X\U ∈ 𝒰′∩ CO(X), and hence

lim_{𝒰}|1_{U}(x)| = ∥1_{U}∥_{𝒰 ∩CO(X)} = 1 0 = ∥1_{U}∥_{𝒰′∩CO(X)} = lim_{𝒰′}|1_{U}(x)|,

where 1_{U}: X → k denotes the characteristic function of U. It contradicts the assumption. Thus F ∩ F′ ∅. Next, suppose that F ∩ F′ ∅ for any closed subsets F, F′ ⊂ X with F ∈ 𝒰 and F′ ∈ 𝒰′. In order to verify 𝒰 ∩ CO(X) = 𝒰′ ∩ CO(X), it suffices to show 𝒰 ∩ CO(X) ⊂ 𝒰′ ∩ CO(X) by symmetry. Let U ∈ 𝒰 ∩ CO(X). Since U ∩ (X\U) = ∅, we have X\U 𝒰′ ∩ CO(X) by the assumption. Therefore U ∈ 𝒰′ ∩ CO(X) by the maximality of an ultrafilter. Thus 𝒰 ∩ CO(X) ⊂ 𝒰′ ∩ CO(X), and hence 𝒰 ∩ CO(X) = 𝒰′ ∩ CO(X). It implies that the equality lim_{𝒰} |f (x)| = lim_{𝒰′} |f (x)| holds for any f ∈ C_{bd}(X, k) by Corollary 2.13.

Corollary 2.16. The residue field of a maximal ideal of C_{bd}(X, k) is k if and only if it is a finite extension of k.

This is a generalisation of [EM2] Theorem 3.7 for the class of bounded continuous functions.

Proof. Let m ⊂ C_{bd}(X, k) be a maximal ideal whose residue field is a finite extension K of k. Take an arbitrary f ∈ K. Since K is a finite extension of k, f is algebraic over k. We prove f ∈ k. Assume f k. Let P(T) ∈ k[T] denote the minimal polynomial of f over k. Let L denote a decomposition field P, and fix an embedding K ↪ L. We endow L with a unique extension of the valuation of k. Since f k, P(T) is an irreducible polynomial over k with zeros f_{1}, …, f_{d} in L\k. Since L is a finite extension of k, k is closed in L. Therefore for any i ∈ ∩ [1, d], the map ξ_{i}: k ↦ [0, ∞): a ↦ |a − f_{i}| is a continuous map with ξ_{i}(a) ≥ r_{i} for any a ∈ k for some r_{i} ∈ (0, ∞). In particular, we have |P(a)| = ∏ _{i=1}^{d} ξ_{i}(a) ≥ ∏ _{i=1}^{d} r_{i} > 0. On the other hand, since K is the residue field of m, there is an F ∈ C_{bd}(X, k) whose image in K is f. Then F satisfies P(F) ∈ m. By the proof of Proposition 2.10, m coincides with the support of the bounded multiplicative seminorm ∥ · ∥_{Ch_{m}}, and hence there is a U ∈ Ch_{m} such that sup_{x∈U} |P(F)(x)| < ∏ _{i=1}^{d} r_{i} by the definition of ∥ · ∥_{Ch_{m}}. Since U ∅, there exists an x ∈ U. However, we have F(x) ∈ k, and hence |P(F)(x)| = |P(F(x))| ≥ ∏ _{i=1}^{d} r_{i}. It is a contradiction. Thus f ∈ k. We conclude that K = k.

Corollary 2.17. An ideal I ⊂ C_{bd}(X, k) coincides with C_{bd}(X, k) if and only if I satisfies

inf_{x∈X} sup_{f ∈S}|f (x)| > 0

for some non-empty finite subset S ⊂ I.

This is a generalisation of [EM1] Theorem 5 for the class of bounded continuous functions.

Proof. The sufficient implication is obvious because 1 ∈ C_{bd}(X, k). Suppose that I does not coincides with C_{bd}(X, k). Take a maximal ideal m ⊂ C_{bd}(X, k) containing I. Let S ⊂ m be a finite subset. Since ∥ · ∥_{Ch_{m}} satisfies ∥f ∥_{Ch_{m}} = 0 for any f ∈ m, we have that for any ϵ ∈ (0, ∞), there is a U ∈ Ch_{m} such that sup_{x∈U} |f (x)| < ϵ for any f ∈ S. In particular, we obtain inf_{x∈X}sup_{f ∈S} |f (x)| = 0 for any non-empty finite subset S ⊂ I.

We remark that Corollary 2.17is also verified in a direct way with no use of our results. Indeed, let I ⊂ C_{bd}(X, k) be an ideal such that there is a non-empty finite subset S ⊂ I with r inf_{x∈X}sup_{f ∈S} |f (x)| > 0. We put U_{f} {x ∈ X ||f (x)| ≥ r}∈ CO(X) for each f ∈ S. Then by the assumption, the family 𝒰 {U_{f} | f ∈ S} covers X. Taking a total order on S, we put S = {f_{0}, …, f_{d}}. Then setting U_{i} U_{f_{i}}\ ⋃ _{j=0}^{i−1} U_{f_{j}} for each i ∈ ∩[0, d], we obtain a refinement {U_{0}, …, U_{d}} of 𝒰 consisting of pairwise disjoint clopen subsets. For each i ∈ ∩ [0, d], we have |f (x)| ≥ r for any x ∈ U_{i}, and hence g_{i} (1 − 1_{U_{i}}) + 1_{U_{i}}f is an invertible element of C_{bd}(X, k) with ∥g_{i}^{−1}∥ ≤ max{r^{−1}, 1}, where 1_{U_{i}}: X → k denotes the characteristic function of U_{i}. We obtain

1 = ∑_{i=0}^{d} 1_{U_{i}} = ∑_{i=0}^{d} 1_{U_{i}}g_{i}^{−1}f_{i} ∈ I,

and thus I = C_{bd}(X, k).

3 Related Results

3.1 Another Construction

In the case where k is a local field or a finite field, we show that BSC_{k}(X) coincides with a space SC_{k}(X) defined in this section. Here a local field means a complete valuation field with non-trivial discrete valuation and finite residue field.

Definition 3.1. Denote by C_{bd}(X, k)(1) ⊂ C_{bd}(X, k) the subset C(X, k^{∘}) of bounded continuous k-valued functions on X which take values in the subring k^{∘} ⊂ k of integral elements, and consider the evaluation map

ι′_{k}: X → (k^{∘})^{C_{bd}(X,k)(1)}
x ↦ (f (x))_{f ∈C_{bd}(X,k)(1)}.

By the definition of the direct product topology, ι′_{k} is continuous. Denote by SC_{k}(X) ⊂ (k^{∘})^{C_{bd}(X,k)(1)}the closure of the image of ι′_{k}. We also denote by ι′_{k} the continuous map X → SC_{k}(X) induced by ι′_{k}.

If k is a local field or a finite field, then SC_{k}(X) is a totally disconnected compact Hausdorff space because so is k^{∘}.

Proposition 3.2. The space SC_{k}(X) satisfies the following extension property: For any f ∈ C_{bd}(X, k), there is a unique SC_{k}(f) ∈ C_{bd}(SC_{k}(X), k) such that f = SC_{k}(f)∘ι′_{k}. Moreover, the equality ∥f ∥ = ∥SC_{k}(f)∥ holds.

Proof. The uniqueness of SC_{k}(f) and the norm-preserving property is obvious because ι′_{k}(X) ⊂ SC_{k}(X) is dense and k is Hausdorff. We construct the extension SC_{k}(f). Note that |k| ⊂ [0, ∞) is bounded if and only if |k| = {0, 1}. Therefore |k| ⊂ [0, ∞) is unbounded or closed. It implies that that there is an a ∈ k^{×} such that ∥f ∥ ≤ |a|. For an x = (x_{g})_{g∈C_{bd}(X,k)(1)} ∈ SC_{k}(X), the value ax_{a^{−1}f} ∈ k is independent of the choice of an a ∈ k^{×}, and we set SC_{k}(f)(x) ax_{a^{−1}f}. Indeed, let a_{1}, a_{2} ∈ k^{×} and suppose ∥f ∥ ≤ min{|a_{1}|, |a_{2}|}. For any y ∈ X, one has ι′_{k}(y)_{a_{1}^{−1}f} = a_{1}^{−1}f (y) and ι′_{k}(y)_{a_{2}^{−1}f} = a_{2}^{−1}f (y). It implies a_{1}ι′_{k}(y)_{a_{1}^{−1}f} = a_{2}ι′_{k}(y)_{a_{2}^{−1}f} ∈ k. Since the image ι′_{k}(X) ⊂ SC_{k}(X) is dense, one obtains a_{1}x_{a_{1}^{−1}f} = a_{2}x_{a_{2}^{−1}f} ∈ k. By the discussion above, one gets SC_{k}(f) ∘ ι′_{k} = f. The map SC_{k}(f) is continuous by the definition of SC_{k}(X).

Corollary 3.3. For a (Y, φ) ∈ X/Top, there is a unique continuous map

SC_{k}(φ): SC_{k}(X) → SC_{k}(Y)

such that SC_{k}(φ) ∘ ι′_{k} = ι′_{k} ∘ φ.

Proof. The uniqueness of SC_{k}(φ) follows from the facts that X is dense in SC_{k}(φ) and that SC_{k}(Y) is Hausdorff. By Proposition 3.2, one has a unique continuous map

SC_{k}(φ): SC_{k}(X) → (k^{∘})^{C_{bd}(Y,k)(1)}

extending the composite

X −→^{φ} Y −→^{ι′_{k}} SC_{k}(Y) ↪ (k^{∘})^{C_{bd}(Y,k)(1)}.

Its image lies in the closed subspace SC_{k}(Y) because X is dense in SC_{k}(X). One obtains a continuous map SC_{k}(φ): SC_{k}(X) → SC_{k}(Y) such that SC_{k}(φ)∘ι′_{k} = ι′_{k}∘φ.

Thus one obtains a functor

SC_{k}:  Top  →  TDCHTop
Y  SC_{k}(Y)
(φ: Y → Z)  (SC_{k}(φ): SC_{k}(Y) → SC_{k}(Z))

with an obvious natural transform ι_{k}: id_{Top} → SC_{k}. We compare BSC_{k} with SC_{k} in the case where k is a local field or a finite field.

Lemma 3.4. Suppose that k is a local field or a finite field endowed with the trivial valuation, and that X is a totally disconnected compact Hausdorff space. Then ι′_{k}: X → SC_{k}(X) is a homeomorphism.

Proof. By the assumption of k, SC_{k}(X) is a totally disconnected compact Hausdorff space. Therefore it suffices to verify the injectivity of ι′_{k}, because a continuous map from a compact space to a Hausdorff space is a closed map. Let x, y ∈ X with x y. Since X is zero-dimensional and Hausdorff, there is a U ∈ CO(X) such that x ∈ U and y U. Then one has ι′_{k}(x)_{1_{U}} = 1 0 = ι′_{k}(y)_{1_{U}}, and hence ι′_{k}(x) ι′_{k}(y). Thus ι′_{k}: X → SC_{k}(X) is injective.

Proposition 3.5. Suppose that k is a local field or a finite field endowed with the trivial valuation. Then SC_{k}(X) is initial in X/TDCHTop with respect to ι′_{k}.

We remark that the assumption on the base field k is not necessary when X is compact. Analysis of continuous functions on a compact space is quite classical.

Proof. For a (Y, φ) ∈ ob(X/TDCHTop), we construct a continuous extension

ψ: SC_{k}(X) → Y

of φ in an explicit way. An extension ψ is unique if it exists, because the image of X is dense in SC_{k}(X) and Y is Hausdorff. Consider the commutative diagram

\begin{matrix}X & ⟶^{φ} & Y \\ ^{ι′_{k}}↓ & & ↓^{ι′_{k}} \\ SC_{k}(X) & ⟶_{SC_{k}(φ)} & SC_{k}(Y) \end{matrix}.

By Lemma 3.4, the right vertical map is a homeomorphism, and one obtains a continuous map ψ ι′_{k}^{−1} ∘ SC_{k}(φ): SC_{k}(X) → Y.

Corollary 3.6. Suppose that k is a local field or a finite field endowed with the trivial valuation.

(i) The space SC_{k}(X) is homeomorphic to BSC_{k}(X) under X.

(ii) The space BSC_{k}(X) satisfies the extension property for a bounded continuous k-valued function on X in Proposition 3.2.

(iii) The natural homomorphism C(BSC_{k}(X), k) → C_{bd}(X, k) is an isometric isomorphism.

(iv) The space BSC_{k}(X) consists of k-rational points, and the residue field of any maximal ideal of C_{bd}(X, k) is k.

Proof. We deal only with (iv). Since every maximal ideal of C_{bd}(X, k) is the support of an x ∈ BSC_{k}(X) as is referred in the proof of Proposition 2.9, it suffices to verify the first assertion. We recall that for a Banach k-algebra 𝒜, an x ∈ ℳ_{k}(𝒜) is said to be a k-rational point if its support {f ∈ A | x(f) = 0} is a maximal ideal of 𝒜 whose residue field is k. The isomorphism C(BSC_{k}(X), k) → C_{bd}(X, k) in (iii) gives an identification BSC_{k}(X) = ℳ_{k}(C(BSC_{k}(X), k)). The assertion immediately follows from a non-Archimedean generalisation of Stone–Weierstrass theorem ([Ber1] 9.2.5. Theorem (i)) for C(BSC_{k}(X), k).

In particular, concerning Corollary 3.6 (iv), any point of BSC_{k}(X) is peaked in the sense that the complete residue field is a peaked Banach k-algebra ([Ber1] 5.2.1, Definition). A Banach k-algebra 𝒜 is said to be peaked if for any extension K/k of complete valuation fields, the norm on 𝒜 ˆ⊗_{k}K is multiplicative. The notion of a peaked point is useful when we consider the topology-theoretical multiplication of points of analytic group. Corollary 3.6 (iv) does not hold when k = _{p}. Indeed, consider the rigid analytic disc D^{1}( _{p}) ℳ_{_{p}} ( _{p}{z}). It admits a natural embedding _{p}^{∘} ↪ D^{1}( _{p}) into a dense subset. The bounded _{p}-algebra homomorphism φ: _{p}{z} → C_{bd}( _{p}^{∘}, _{p}) sending the variable z to the coordinate function z: _{p}^{∘} ↪ _{p} induces a continuous map φ^{∗}: BSC_{_{p}} ( _{p}^{∘}) → D^{1}( _{p}) under _{p}^{∘}. Since BSC_{_{p}} ( _{p}^{∘}) and D^{1}( _{p}) are compact Hausdorff spaces, the image of φ^{∗} contains the closure of the dense subset _{p}^{∘} ⊂ D^{1}( _{p}). Therefore φ^{∗} is surjective. Since _{p} is not spherically complete, there is a y ∈ D^{1}( _{p}) of type 4. Take an x ∈ BSC_{_{p}} ( _{p}^{∘}) in φ^{−1}(y). The induced bounded _{p}-algebra homomorphism φ_{x}: _{p}{z}/supp(y) → C_{bd}( _{p}^{∘}, _{p})/supp(x) gives an extension of fields transcendental over _{p} because y is of type 4. Thus the point x ∈ BSC_{_{p}} ( _{p}^{∘}) is not _{p}-rational.

3.2 Relation to the Stone–Čech compactification

A compact Hausdorff space is totally disconnected if and only if it is zero-dimensional, and hence under X, the notion of an initial totally disconnected compact Hausdorff space is equivalent to that of an initial zero-dimensional compact Hausdorff space. Banaschewski constructed a zero-dimensional compact Hausdorff space ζX under X in the case where X is zero-dimensional in [Ban], and proved that ζX is initial. In the case where X is a zero-dimensional Hausdorff space, the structure map X → ζX is a homeomorphism onto the image and ζX is sometimes called the Banaschewski compactification. Let p be a prime number. By the argument above, BSC_{_{p}} (X) is one of the generalisation of ζX. The Stone–Čech compactification βX has a universality as an initial object in the category of compact topological spaces under X, and hence there is a unique continuous map βX → BSC_{_{p}} (X) under X. Then a natural question arises: “When is the map βX → BSC_{_{p}} (X) is a homeomorphism?” In other words, “When is βX totally disconnected?” In such a case, BSC_{_{p}} (X) satisfies the extension property for an Archimedean bounded continuous function on X. This connection between the Archimedean analysis and the non-Archimedean analysis looks interesting. It is well-known that if X is an infinite discrete space, then βX is totally disconnected. In particular, one has β BSC_{_{p}} ( ). Furthermore, Banaschewski proved βX is homeomorphic to ζX if X is a second countable zero-dimensional Hausdorff space ([Ban] Satz 6. Korollar 2). In this case, βX is totally disconnected and hence is homeomorphic to BSC_{_{p}} (X). For example, the closed unit disc _{ℓ}^{∘} ⊂ _{ℓ} for a prime number ℓ ∈ is a second countable zero-dimensional Hausdorff space, and hence one has β _{ℓ}^{∘} BSC_{_{p}} ( _{ℓ}^{∘}). We do not know whether there is an example of a totally disconnected or zerodimensional space X such that the map βX → BSC_{_{p}} (X) is not a homeomorphism. Banaschewski gave the following necessary condition for the bijectivity of βX → BSC_{_{p}} (X) in [Ban] Satz 2. For a normal zero-dimensional space X, if the continuous map βX → BSC_{_{p}} (X) is a homeomorphism, then X is of Čech-dimension 0, i.e. any finite open covering of X admits a finite clopen refinement. Therefore if there is a normal zero-dimensional space of positive Čech-dimension, then it is an example.

4 Applications

4.1 Automatic Continuity Theorem

One of important classical problems for a commutative C^{∗}-algebra is Kaplansky Conjecture on automatic continuity problem, which claims that every injective - algebra homomorphism φ: C_{bd}(X, ) ↪ 𝒜 is continuous for any Banach -algebra 𝒜. Consider the following weak version: every injective -algebra homomorphism φ: C_{bd}(X, ) ↪ 𝒜 with closed image is continuous for any Banach -algebra 𝒜. It was proved by Kaplansky, and Solovay showed that the conjecture is independent of the axiom of ZFC. For more details about the automatic continuity problem, see [Dal]. [Sol], and [Woo]. Now it is natural to consider an analogous question in the non-Archimedean case, and we prove its weak version in this subsection.

Definition 4.1. Let 𝒜 be a Banach k-algebra, and m ⊂ 𝒜 a maximal ideal whose residue field K is a finite extension of k. For each f ∈ 𝒜, we denote by f (m) the image of f in K, and by |f (m)| the norm of f (m) with respect to a unique extension of the norm of k.

Lemma 4.2. Let 𝒜 be a Banach k-algebra. For a maximal ideal m ⊂ 𝒜 whose residue field is k, the canonical projection 𝒜 𝒜 /m _{k} k gives a decomposition 𝒜 = k ⊕ m as the orthogonal direct sum, i.e. the equality

∥a + g∥ = max{|a|, ∥g∥}

holds for any a ∈ k and g ∈ m.

Proof. Since the composite k ↪ 𝒜 𝒜 /m is a bijective k-linear homomorphism, one obtains a decomposition 𝒜 = k ⊕ m as the direct sum of k-vector spaces. Take an element f ∈ 𝒜, and denote by f (m) ∈ k the image of f in the quotient 𝒜 /m _{k} k. In order to prove the orthogonality of the direct sum, it suffices to show ∥f ∥ = max{|f (m)|, ∥f − f (m)∥}. The inequality ≤ is obvious. If |f (m)| ∥f − f (m)∥, the equality follows from the general property of a non-Archimedean norm, and hence we may assume |f (m)| = ∥f − f (m)∥. If f (m) = 0, then one has ∥f − f (m)∥ = 0 and therefore f = f (m) + (f − f (m)) = 0. Suppose f (m) 0. Assume ∥f ∥ < |f (m)|. Then one has ∥f (m)^{−1}f ∥ < 1, and hence

f − f (m) = −f (m)(1 − f (m)^{−1}f) ∈ k^{×}𝒜^{×} = 𝒜^{×}

by [BGR] 1.2.4. Proposition 4. It contradicts the fact f − f (m) ∈ m, and thus ∥f ∥ = |f (m)| = max{|f (m)|, ∥f − f (m)∥}.

We may apply Lemma 4.2 to any maximal ideal of C_{bd}(X, k) by Corollary 3.6 (iv).

Corollary 4.3. Suppose that k is a local field or a finite field endowed with the trivial norm. For any maximal ideal m ⊂ C_{bd}(X, k), the canonical projection C_{bd}(X, k) C_{bd}(X, k)/m gives a decomposition C_{bd}(X, k) = k ⊕ m as the orthogonal direct sum of k-Banach spaces.

This is a partial generalisation of [EM1] Theorem 7, which states that if X is an ultrametric space and k is locally compact, then the same holds.

Proposition 4.4. Suppose that k is a local field or a finite field endowed with the trivial norm. Let Max(C_{bd}(X, k)) ⊂ Spec(C_{bd}(X, k)) denote the subset of maximal ideals. Then the equality

∥f ∥ = sup_{m∈Max(C_{bd}(X,k))}|f (m)|

holds for any f ∈ C_{bd}(X, k). In particular, the norm of C_{bd}(X, k) is determined by the algebraic structure of it.

Proof. Since the norm of C_{bd}(X, k) is power-multiplicative, the equality

∥f ∥ = sup_{x∈BSC(X)}x(f)

holds for any f ∈ C_{bd}(X, k) by [Ber1] 1.3.1. Theorem. This gives the assertion because supp: BSC_{k}(X) → Spec(C_{bd}(X, k)) is bijective onto Max(C_{bd}(X, k)) by the proof of Theorem 2.1, and because the condition (iv) in the definition of a bounded multiplicative seminorm in §1.1 guarantees x(f) = |f (supp(x))| by Corollary 3.6 (iv).

Proposition 4.5. Suppose that k is a local field. Then every complete norm on the underlying k-algebra of C_{bd}(X, k) is equivalent to each other.

Proof. Since k is a local field, the norm of k is not trivial, and hence the boundedness of a k-linear homomorphism between normed k-vector spaces is equivalent to the continuity. Therefore it suffices to show that the identity id: C_{bd}(X, k) → C_{bd}(X, k) is a homeomorphism with respect to the metric topologies given by an arbitrary complete norm ∥ · ∥′ of the source and the supremum norm ∥ · ∥ of the target. By Lemma 4.2 applied to (C_{bd}(X, k), ∥ · ∥′) and Corollary 4.4, id is a k-linear contraction map, and hence is continuous. Moreover, since the norm of k is not trivial, the open mapping theorem holds by [BGR] 2.8.1. Theorem, and therefore id is an open map. Thus id is a homeomorphism.

Theorem 4.6. Suppose that k is a local field, and let 𝒜 be a Banach k-algebra. Then any injective k-algebra homomorphism φ: C_{bd}(X, k) ↪ 𝒜 whose image is closed is continuous.

Proof. Since the underlying metric spaces of C_{bd}(X, k) and 𝒜 are complete, it suffices to show that φ sends a Cauchy sequence in C_{bd}(X, k) to a Cauchy sequence in 𝒜. Let ∥ · ∥′: C_{bd}(X, k) → [0, ∞) denote the composite of φ and the norm of 𝒜. Then since φ is a bijective homomorphism of k-algebras, ∥ · ∥′ is a norm of the k-algebra C_{bd}(X, k). Since the image of φ is closed, ∥ · ∥′is complete, and hence is equivalent to the supremum norm by Proposition 4.5. We conclude that φ is continuous.

The automatic continuity theorem immediately yields a criterion for the continuity of a faithful representation over a local field.

Corollary 4.7. Suppose that k is a local field. Let V be a Banach k-vector space and ρ: C_{bd}(X, k)×V → V a k-linear representation of a k-algebra C_{bd}(X, k) satisfying the following conditions:

(i) The k-linear operator ρ̃_{f} : V → V: v ↦ ρ(f, v) is bounded for any f ∈ C_{bd}(X, k).

(ii) The k-linear representation ρ is faithful, i.e. the equality ρ_{f} = 0 implies f = 0 for any f ∈ C_{bd}(X, k).

(iii) The image of the induced k-algebra homomorphism ρ̃_{·}: C_{bd}(X, k) → ℬ_{k}(V) is closed, where ℬ_{k}(V) is the Banach k-algebra of bounded operators on V.

Then ρ̃_{·}is bounded, and in particular, ρ: C_{bd}(X, k) × V → V is continuous.

4.2 Ground Field Extensions

We study the ground field extensions of C_{bd}(X, k). We note that there are two distinct notions of the ground field extensions. One is given by extending the scalar of functions, and the other is given by tensoring the scalar.

Proposition 4.8. Let K and L be complete valuation fields. Then there exists a unique homeomorphism BSC_{K}(X) BSC_{L}(X) compatible with the evaluation maps.

We remark that we do not assume that K and L contains the same base field k, and hence, for example, it is possible to choose _{p} and _{ℓ}((T)) for K and L respectively.

Proof. The assertion holds because BSC_{K}(X) and BSC_{L}(X) are initial objects with respect to the evaluation maps in X/TDCHTop by Corollary 2.2.

Proposition 4.9. Let K/k be an extension of complete valuation fields. Then the ground field extension BSC_{K}(X) → BSC_{k}(X) associated with the natural embedding C_{bd}(X, k) ↪ C_{bd}(X, K) is a homeomorphism.

Proof. The ground field extension above is compatible with the evaluation maps, and coincides with the unique homeomorphism in Proposition 4.8.

Now we consider the other ground field extension, namely, the canonical K-algebra homomorphism K ˆ⊗_{k}C_{bd}(X, k) → C_{bd}(X, K) induced by the universal property of the complete tensor product in the category of Banach k-algebras. In fact, the ground field extension is not an isomorphism in general, and it yields a criterion for a topological property of X and the valuation of k.

Lemma 4.10. The k-subalgebra of C_{bd}(X, k) consisting of locally constant bounded functions is dense.

Proof. Take an f ∈ C_{bd}(X, k). If f = 0, then f is locally constant. Suppose f 0. For an ϵ > 0, the pre-image of every open disc of radius ϵ in k by f is clopen. Therefore one obtains a pairwise disjoint clopen covering 𝒰 of X such that the image f (U) is contained in an open disc of radius ϵ in k for any U ∈ 𝒰 . Fix an a_{U} ∈ f (U) for each U ∈ 𝒰 . The infinite sum g ∑ _{U∈𝒰} a_{U}1_{U}: X → k convergences pointwise to a locally constant continuous function. The obvious inequality |g(x)| ≤ ∥f ∥ holds for any x ∈ X, and hence g is bounded. One has ∥f − g∥ ≤ ϵ by the definition of the disjoint clopen covering 𝒰 , and hence the k-subalgebra of locally constant functions is dense in C_{bd}(X, k).

Lemma 4.11. Suppose that k is spherically complete ([BGR] 2.4.4. Definition 1). Let K/k be an extension of complete valuation fields. Then the natural bounded K-algebra homomorphism ι_{K/k}: K ˆ⊗_{k}C_{bd}(X, k) → C_{bd}(X, K) is an isometry.

For example, a local field and every field endowed with the trivial norm are spherically complete. We will use this lemma for _{p} endowed with the trivial norm, endowed with the trivial norm, and _{p}.

Proof. Take an f ∈ K ˆ⊗_{k}C_{bd}(X, k). If f = 0, then ∥ι_{K/k}(f)∥ = 0 = ∥f ∥, and hence we assume f 0. In particular, X ∅ and both of K ˆ⊗_{k}C_{bd}(X, k) and C_{bd}(X, K) are non-zero Banach K-algebras. Therefore the norm of the bounded K-algebra homomorphism ι_{K/k} is 1 because ι_{K/k}(1) = 1 and the power-multiplicativity of the norm of C_{bd}(X, K) guaranties that ι_{K/k} is submetric. Set ϵ ∥f ∥/2, and take an element g = ∑ _{i=1}^{n} a_{i} ⊗ g_{i} ∈ K ⊗_{k} C_{bd}(X, k) with ∥f − g∥ < ϵ in K ˆ⊗_{k}C_{bd}(X, k). We may assume a_{i} 0 for any i = 1, …, n without loss of generality. By Lemma 4.10, there is a locally constant bounded k-valued function g′_{i} ∈ C_{bd}(X, k) such that ∥g_{i} − g′_{i}∥ < |a_{i}|^{−1}ϵ for each i = 1, …, n. In particular, setting g′ ∑ _{i=1}^{n} a_{i} ⊗ g′_{i} ∈ K ⊗_{k} C_{bd}(X, k), one has

∥f − g′∥ = ∥(f − g) + (g − g′)∥ ≤ max {∥f − g∥, ∥∑_{i=1}^{n} a_{i}(g_{i} − g′_{i})∥}

≤ max{∥f − g∥, max _{i=1}^{n}|a_{i}|∥g_{i} − g′_{i}∥} < ϵ < ∥f ∥,

and hence ∥g′∥ = ∥f ∥. Since k is spherically complete, the finite dimensional normed k-vector subspace ka_{1} + ⋯+ ka_{n} ⊂ K is k-Cartesian by [BGR] 2.4.4. Proposition 2, and hence there is an orthogonal basis b_{1}, …, b_{m} ∈ K of ka_{1} + ⋯+ ka_{n}. Expressing a_{1}, …, a_{n} as a k-linear combination of b_{1}, …, b_{m}, one obtains an expression g′ = ∑ _{i=1}^{m} b_{i}g′′_{i}by a unique system g′′_{1}, …, g′′_{m}∈ C_{bd}(X, k) of k-valued locally constant functions. For any x ∈ X, one has

|ι_{K/k}(g′)(x)| = |∑_{i=1}^{m} b_{i}g′′_{i}(x)|= max _{i=1}^{m}|g′′_{i}(x)||b_{i}|

by the orthogonality of b_{1}, …, b_{m}, and hence

∥ι_{K/k}(g′)∥ = sup_{x∈X}|ι_{K/k}(g′)(x)| = sup_{x∈X}max _{i=1}^{m}|b_{i}||g′′_{i}(x)| = max _{i=1}^{m}|b_{i}| sup_{x∈X}|g′′_{i}(x)|

= max _{i=1}^{m}|b_{i}|∥g′′_{i}∥ ≥ ∥g′∥.

Since ι_{K/k}is a K-linear contraction map, one gets ∥ι_{K/k}(g′)∥ = ∥g′∥. We conclude

∥ι_{K/k}(f − g′)∥ ≤ ∥f − g′∥ < ϵ < ∥f ∥ = ∥g′∥ = ∥ι_{K/k}(g′)∥

and thus

∥ι_{K/k}(f)∥ = ∥ι_{K/k}(f − g′) + ι_{K/k}(g′)∥ = ∥ι_{K/k}(g′)∥ = ∥g′∥ = ∥f ∥.

We denote by ⊂ k the topological closure of the field F of fractions of the image of the canonical ring homomorphism → k. We remark that F is _{p} if and only if k is of characteristic p > 0, and is otherwise. In the former case, is _{p} endowed with the trivial valuation. In the latter case, is endowed with the trivial norm if and only if k is of equal characteristic (0, 0), and is _{p} if and only if k is of mixed characteristic (0, p). In particular, is spherically complete. We determine when ι_{k/}: k ˆ⊗_{}C_{bd}(X, ) → C_{bd}(X, k) is an isomorphism. The following shows that C_{bd}(X, k) is “naive” enough if and only if X is compact or k is sufficiently small in some sense.

Theorem 4.12. Suppose that X is zero-dimensional and Hausdorff. Then the following are equivalent:

(i) The space X is compact, or k is a local field or a finite field endowed with the trivial norm.

(ii) The k-subalgebra of C_{bd}(X, k) generated by idempotents is dense.

In addition if , then (i) and (ii) are equivalent to the following:

(iii) The homomorphism ι_{k/}: k ˆ⊗_{}C_{bd}(X, ) → C_{bd}(X, k) is an isometric isomorphism.

(iv) The space BSC_{k}(X) consists of k-rational points.

(v) The map ι_{k}: X ↪ BSC_{k}(X) induces an isometric isomorphism C(BSC_{k}(X), k) → C_{bd}(X, k).

(vi) The space BSC_{k}(X) satisfies the extension property for a bounded continuous k-valued functions in Proposition 3.2.

We remark that a similar relation between (i) and (iv) is also verified by Alain Escassut and Nicolas Maïnetti in [EM1] in the case where X is an ultrametric space. For example, they proved in Theorem 7 that if X is an ultrametric space and k is locally compact, then (iv) holds.

Proof. Assume (i). We verify (ii). Take an f ∈ C_{bd}(X, k). If k is a local field or a finite field, the closed disc {a ∈ k | |a| ≤ ∥f ∥} ⊂ k is compact. Otherwise X is compact. Therefore for an ϵ > 0, there is a finite pairwise disjoint clopen covering 𝒰 of X such that the image f (U) ⊂ k is contained in an open disc of radius ϵ in k for any U ∈ 𝒰 . Fixing an a_{U} ∈ f (U) for each U ∈ 𝒰 , one has ∥f − ∑ _{U∈𝒰} a_{U}1_{U}∥ < ϵ. Thus k-subalgebra of C_{bd}(X, k) generated by idempotents is dense. Assume (ii). We verify (iii). Since ι_{k/} is an isometry by Lemma 4.11, it suffices to show that the image of ι_{k/} is dense. An idempotent of C_{bd}(X, k), which is a characteristic function on a clopen subset of X, is contained in the subset C_{bd}(X, ) ⊂ C_{bd}(X, k). Therefore the image of the natural homomorphism k⊗_{}C_{bd}(X, ) → C_{bd}(X, k) is dense by (ii), and hence the image of ι_{k/} is dense. Assume (iii). We verify (iv) in the case . For an x ∈ BSC_{k}(X), consider the composite x′: C_{bd}(X, ) → k(x) of x and the natural embedding C_{bd}(X, ) ↪ C_{bd}(X, k), where k(x) is the completed residue field at x. Since is contained in k, the character x′ defines an element x′ ∈ BSC_{}(X). Recall that = _{p} or _{p} now. Since BSC_{}(X) consists of -rational points by Corollary 3.6 (iv), the image of x′ is contained in ⊂ k. Therefore (iii) guarantees that the image of x is contained in the closure of the k-vector subspace of k(x) generated by ⊂ k, namely, the 1-dimensional vector subspace k ⊂ k(x). It implies k(x) = k. Assume (iv). We verify (v) in the case , and hence suppose that BSC_{k}(X) consists of k-rational points. Then the Gel’fand transform C_{bd}(X, k) → C(BSC_{k}(X), k) is an isometric isomorphism by [Ber1] 9.2.7. Corollary (ii), and coincides with the bounded k-algebra homomorphism induced by ι_{k}. Assume (v). We verify (vi) in the case , and hence suppose that ι_{k} induces an isometric isomorphism C(BSC_{k}(X), k) → C_{bd}(X, k). Take an f ∈ C_{bd}(X, k). The extension of f on BSC_{k}(X) is unique because the image of ι_{k} is dense by Corollary 2.4 and k is Hausdorff. Since ι_{k} induces an isomorphism C(BSC_{k}(X), k) → C_{bd}(X, k), there is an f′ ∈ C(BSC_{k}(X), k) whose image is f, or in other words, f′is the extension of f on BSC_{k}(X). Assume that (vi) and hold, or that (ii) and = hold. We verify (i). Suppose that X is non-compact and k is neither a local field nor a finite field. Since X is a zero-dimensional and non-compact, there is an ℱ ∈ UF(X) without a cluster point by Proposition 1.7. In particular, ℱ contains an infinite descending chain X = U_{0} U_{1} · · · . Indeed, for any U ∈ ℱ and x ∈ U ∅, there is a V ∈ CO(X) such that V ∈ ℱ, V ⊂ U, and x V because x is not a cluster point of ℱ. One obtains an infinite set 𝒰 = {U_{i}\U_{i+1} | i ∈ } of pairwise disjoint clopen subsets of X. If the residue field k̃ of k is an infinite field, set Y k̃ and take a set-theoretical lift φ: Y ↪ k^{∘} of the canonical projection k^{∘} Y. Otherwise, the image |k^{×}| ⊂ (0, ∞) is dense because k is neither a local field nor a finite field. Set Y |k^{×}| ∩ (1/2, 1) ⊂ (0, ∞), and take a set-theoretical lift φ: Y ↪ k^{∘} of the norm | · |: k → [0, ∞). Since Y is dense in (1/2, 1), it is an infinite set. In both cases, endow Y with the discrete topology. Since Y is an infinite set, there is an injective map ψ: ↪ Y. The composite φ∘ψ: ↪ k^{∘} is an injective continuous map, and the image is a closed discrete subspace because |φ(y) − φ(y′)| > 1/2 for any y, y′ ∈ Y. Since 𝒰 ⊂ CO(X) is an infinite covering of X, there is an injective map Ψ: ↪ 𝒰 . Then the pointwise convergent infinite sum

f ∑_{n∈} φ(ψ(n))1_{Ψ(n)} : X → k

determines a locally constant bounded function on X. There is a non-principal ultrafilter ℱ ∈ CO( ) by Lemma 1.6. Now assume . By the conditions (vi), there is a continuous extension BSC_{k}(f): BSC_{k}(X) → k of f. Moreover, taking a representative x_{n} ∈ Ψ(n) for each n ∈ , one obtains a continuous map x: → X ↪ BSC_{k}(X). Since BSC_{k}(X) is an object of X/TDCHTop, a unique continuous extension BSC_{k}(x): UF( ) BSC_{k}( ) → BSC_{k}(X) of x exists. The composite BSC_{k}(f) ∘ BSC_{k}(x): UF( ) → BSC_{k}(X) → k is a continuous extension of the composite f ∘ x = φ ∘ ψ: → k. In particular, BSC_{k}(f) ∘ BSC_{k}(x) is continuous at ℱ ∈ UF( ), but it contradicts the fact that φ ∘ ψ is an injective map whose image is a closed discrete subspace. An injective net whose image is discrete and closed never has a limit. It is a contradiction. Therefore one obtains = , and (ii) holds by the assumption. Take a k-linear combination g = ∑ _{i=1}^{n} a_{i}1_{U_{i}} ∈ C_{bd}(X, k) of idempotents with ∥f − g∥ < 1. Now the image of g contains at most n points, and hence there is an integer m ∈ such that g(x) ψ(m) for any x ∈ X identifying the cosets k̃ as a family of disjoint clopen subsets of k^{∘} in the tautological sense. Then one has |f (x_{m}) − g(x_{m})| = 1, and it contradicts the condition ∥f − g∥ < 1. Thus X is compact, or k is a local field or a finite field.

Since we did not use the assumption that in the proof that (ii) with implies (iii), the condition that ι_{k/} is an isometric isomorphism is weaker than (ii).

4.3 Non-Archimedean Gel’fand Theory

We establish non-Archimedean Gel’fand theory for a zero-dimensional Hausdorff space. We recall that a completely regular Hausdorff space is a topological space which can be embedded in a direct product of copies of the closed unit disc ^{∘} ⊂ as a subspace. On the other hand, a non-Archimedean counterpart of a completely regular Hausdorff space over k is a topological space which can be embedded in a direct product of copies of the closed unit disc k^{∘} ⊂ k as a subspace. We call such a topological space a non-Archimedean completely regular Hausdorff space over k. A direct product of copies of k^{∘} is a zero-dimensional Hausdorff space. A subset of a zero-dimensional Hausdorff space is again a zero-dimensional Hausdorff space, and so is a non-Archimedean completely regular Hausdorff space over k. Now we verify that the converse also holds in the case where k is a local field or a finite field.

Lemma 4.13. Suppose that k is a local field or a finite field endowed with the trivial norm. The following are equivalent:

(i) The space X is zero-dimensional and Hausdorff.

(ii) The space X is Hausdorff, and bounded continuous k-valued functions separates a point and a disjoint closed subset of X, i.e. for any x ∈ X and any closed subset F ⊂ X with x F, there is an f ∈ C_{bd}(X, k) such that f (x) = 0 and f (y) = 1 for any y ∈ F.

(iii) The continuous map ι′_{k}: X → SC_{k}(X) is a homeomorphism onto the image.

(iv) The space X is embedded in a direct product of copies of k^{∘} as a subspace.

We note that the description of (ii)-(iv) seem to depend on the base field k while (i) does not. Therefore the notion of “a non-Archimedean completely regular Hausdorff space” is independent of the base field.

Proof. Recall that SC_{k}(X) is a closed subspace of a direct product of copies of k^{∘}, and hence (iii) implies (iv). Moreover, (iii) implies (i) as we mentioned in the beginning of this section. Assume (i). We verify (ii). Let x ∈ X and F ⊂ X be a closed subset with x F. Since X is zero-dimensional, there is a clopen neighbourhood U ⊂ X of x contained in the open subset X\F ⊂ X, and the characteristic function 1_{X\U} separates x and F. Assume (ii). We verify (iii). Since X is Hausdorff, a point of X is closed. For x, y ∈ X with x y, take an f ∈ C_{bd}(X, k) which separates x and y. Then f 0. Since the valuation of k is discrete or trivial, the image |k| ⊂ [0, ∞) is closed. By the definition of the supremum norm, ∥C_{bd}(X, k)∥ ⊂ [0, ∞) is contained in the closure of |k| ⊂ [0, ∞), and hence ∥C_{bd}(X, k)∥ ⊂ |k|. Therefore there is an a ∈ k^{×} such that 0 ∥f ∥ = |a|. Then one has ∥a^{−1}f ∥ = 1 and a^{−1}f ∈ C_{bd}(X, k)(1) separates x and y. Therefore one has ι′_{k}(x) ι′_{k}(y) comparing their (a^{−1}f)-th entry, and ι′_{k} is injective. In order to prove that ι′_{k} is an open map onto the image, take an open subset U ⊂ X. For an x ∈ U, take an f ∈ C_{bd}(X, k) such that f (x) = 0 and f (y) = 1 for any y ∈ X\U. By the same argument as above, there is an a ∈ k^{×} such that ∥f ∥ = |a|. Then the pre-image by ι′_{k} of the open subset V ⊂ SC_{k}(X) given by the condition that the (a^{−1}f)-th entry is contained in the open neighbourhood k\{a^{−1}} ⊂ k of 0 ∈ k is an open neighbourhood of x contained in U. Therefore the image ι′_{k}(U) contains the open neighbourhood V ∩ ι′_{k}(X) of ι′_{k}(x) in ι′_{k}(X), and thus ι′_{k}(U) ⊂ ι′_{k}(X) is open. We conclude that ι′_{k} is a homeomorphism onto the image.

Proposition 4.14. The map ι_{k}: X → BSC_{k}(X) is a homeomorphism onto the image if and only if X is zero-dimensional and Hausdorff.

Proof. By Proposition 4.8, it is reduced to the case where k = _{p} for a prime number p ∈ . The assertion immediately follows from Corollary 3.6(i) and Lemma 4.13.

Definition 4.15. Let 𝒜 ⊂ C_{bd}(X, k) be a closed k-subalgebra. For x, x′ ∈ X, we write x ∼_{𝒜} x′ if f (x) = f (x′) for any f ∈ 𝒜. The binary relation ∼_{𝒜} is an equivalence relation, and we denote by X/𝒜 the quotient space X/ ∼_{𝒜} . We say 𝒜 separates points of X if the condition x ∼_{𝒜} x′implies x = x′for any x, x′ ∈ X.

Lemma 4.16. The map ι′_{k}: X → SC_{k}(X) uniquely factors through the canonical projection X X/C_{bd}(X, k), and the induced map X/C_{bd}(X, k) → SC_{k}(X) is an injective continuous map.

Proof. It immediately follows from the definitions of ∼_{C_{bd}(X,k)}and SC_{k}(X).

Lemma 4.17. Suppose that X is zero-dimensional and Hausdorff. Then C_{bd}(X, k) separates points of X.

Proof. By Lemma 4.13and Lemma 4.16, the projection X X/C_{bd}(X, k) is injective.

Definition 4.18. A topological space (Y, f) under X is said to be faithful if f : X → Y is a homeomorphism onto the image, to be full if f (X) is dense in Y, and to be fully faithful if it is full and faithful. We denote by X/TDCHTop_{ff} ⊂ X/TDCHTop the full subcategory of fully faithful totally disconnected compact Hausdorff spaces under X.

By Lemma 4.13, X/TDCHTop_{ff} is a non-empty category if and only if X is zerodimensional and Hausdorff. Suppose that X is zero-dimensional and Hausdorff. The isomorphism relation in X/TDCHTop_{ff} is an equivalence relation in a class. We denote by 𝒞 (X) the class (X/TDCHTop_{ff})/ of equivalence classes. The class 𝒞 (X) is not a proper class. Indeed, for any (Y, f) ∈ ob(X/TDCHTop_{ff}), f extends to f̃: UF(X) → Y by Theorem 1.3. Since UF(X) is compact and Y is Hausdorff, f̃(UF(X)) ⊂ Y is a closed subspace containing the dense subspace f (X) ⊂ Y, and hence f̃is a surjective closed map. Therefore (Y, f) is obtained as a quotient of UF(X), and 𝒞 (X) admits a set-theoretical representative.

Theorem 4.19. Suppose that k is a local field or a finite field endowed with the trivial norm, and that X is zero-dimensional and Hausdorff. Then there is a contravariant-functorial one-to-one correspondence between 𝒞 (X) and the set of closed k-subalgebras of C_{bd}(X, k) separating points of X.

Proof. Denote by 𝒞′(X) the set of closed k-subalgebras of C_{bd}(X, k) separating points of X. The correspondences are given in the following way:

𝒞 (X)  ⟷  𝒞′(X)
[f : X ↪ Y]  Im(f^{a}: C(Y, k) ↪ C_{bd}(X, k))
[X ↪ BSC_{k}(X) ℳ_{k}(𝒜)]  (𝒜 ⊂ C_{bd}(X, k)).

They are the inverses of each other by the generalised Stone–Weierstrass theorem ([Ber1] 9.2.5. Theorem). We remark that for any fully faithful totally disconnected compact Hausdorff space (Y, f) under X, the associated bounded homomorphism f^{a}: C(Y, k) ↪ C_{bd}(X, k) is an isometry because X is a dense subspace of Y, and hence its image is closed. On the other hand for any closed k-subalgebra 𝒜 of C_{bd}(X, k) separating points of X, the associated continuous map X ↪ BSC_{k}(X) ℳ_{k}(𝒜) is a homeomorphism onto the image which is a dense subspace, because X is a dense subspace of BSC_{k}(X), the condition that 𝒜 separates points ofX guarantees the injectivity, and every continuous map between compact Hausdorff spaces BSC_{k}(X) and ℳ_{k}(𝒜) is a closed map.

Acknowledgements I would like to thank Takeshi Tsuji. I appreciate having gracious discussions in seminars. I am profoundly grateful to Atsushi Matsuo for their great helps and instructive advices on writing this paper. I express my gratitude to Atsushi Yamashita for joining in seminars on this paper. I am thankful to my friends for devoting time to frequent discourse. I acknowledge my family’s deep affection. I am a research fellow of Japan Society for the Promotion of Science. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan.

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Tomoki Mihara 800

Documenta Mathematica 19 (2014)