Theory and Applications of Categories, Vol. 18, No. 3, 2007, pp. 102–117.

FUNCTIONAL ANALYSIS ON NORMED SPACES: THE BANACH SPACE COMPARISON

M. SIOEN, S. VERWULGEN

Abstract. It is the aim of this paper to compute the category of Eilenberg-Moore algebras for the monad arising from the dual unit-ball functor on the category of (semi)normed spaces. We show that this gives rise to a stronger algebraic structure than the totally convex one obtained from the closed unit ball functor on the category of Banach spaces.

1. Introduction and basic notations

It was shown in [5] that contravariant Hom-functors, which we will call dualization functors for the rest of the paper, often give rise to meaningful categorical dualities such as the celebrated Gelfand-Naimark duality. A beautiful way to infer the Gelfand-Naimark duality in a purely categorical way is given in [4]. Throughout functional analysis, dual spaces and dualization functors into the scalar field occur everywhere. Another famous example is provided by the Riesz representation theorem which describes the dual of the Banach space of all (necessarily) bounded continuous functions on a given compact Hausdorff space in measure-theoretic terms. In all what follows, all modules or vectorspaces will be considered over ℝ and we will write sNorm_{1} (resp. Ban_{1} ) for the category of seminormed (resp. Banach) spaces and non-expansive linear maps.

Moreover, it is well-known that the topological dual of a seminorned space comes equipped with a canonical dual norm, making it into a Banach space. This allows for the formation of (countably infinite) totally convex combinations on the closed dual unit ball, giving it the algebraic structure of a totally convex module in the sense of [7, 8]. Let us recall that in [7, 8] the authors showed that the category TC of totally convex modules and totally affine maps is the category of Eilenberg-Moore algebras for the monad arising from an adjunction having the closed unit ball-functor from Ban_{1} to Set as a right adjoint. Loosely speaking, this expresses the fact that the algebraic structure (i.e. describable in terms of generators and relations) which is intrinsically present on the closed unit ball of a Banach space is exactly the one of a totally convex module. Answering the analogous question for the closed unit ball-functor on the category sNorm_{1} instead of Ban_{1}, the authors obtained in [7, 8] the (larger) category AC of absolutely convex modules and absolutely affine maps as the category of Eilenberg-Moore algebras.

Received by the editors 2006-04-10 and, in revised form, 2007-01-26.

Transmitted by Walter Tholen. Published on 2007-02-18.

2000 Mathematics Subject Classification: 46B04, 46M99, 52A01.

Key words and phrases: Banach space, Monad, Totally convex module, Duality.

\begin{matrix}c \\ ◯ \end{matrix} M. Sioen, S. Verwulgen, 2007. Permission to copy for private use granted.

FUNCTIONAL ANALYSIS ON NORMED SPACES: THE BANACH SPACE COMPARISON 103

As shown in [12], the category AC arises also in so called “Quantified Functional Analysis”. We use this term for the theory of (locally convex) approach spaces as developed e.g. in [11, 12], where the key idea is to work with canonical numerical structures overlying (locally convex) vector topologies, instead of only considering the topological level. For more information we refer to [11, 12]. In [12] we proved that AC is also the category of Eilenberg-Moore algebras for the monad arising from a dual adjunction having the dualization functor Hom_{lcApVec}(−, ℝ) on lcApVec^{op} as a right adjoint. Here lcApVec denotes the category of locally convex approach spaces and contractive linear maps, and ℝ is equipped with the absolute value as norm.

If we define C : sNorm_{1}^{op} → Set to be the restriction of Hom_{lcApVec}(−, ℝ) to sNorm_{1}^{op}, then C is in fact the dualization functor Hom_{sNorm_{1}}(−, ℝ) which is nothing but the closed dual unit ball functor on sNorm_{1}^{op}. Again C is the right adjoint of a (dual) adjunction, giving rise to a monad. The question we address in this paper is finding a description of the category of Eilenberg-Moore algebras for this monad. Speaking more loosely as we did above this answers the question of which canonical algebraic structure is present on the closed dual unit ball of a seninormed space.

From the previous discussion one might be tempted to guess that this category of Eilenberg-Moore algebras is concretely isomorphic with TC. Quite surprisingly this is not the case as we shall prove. The resulting category SC of Eilenberg-Moore algebras has as objects sets which allow for an abstract integration with respect to certain finitely additive measures, also called charges, of total variation at most one in the sense of [9]. The crucial ingredient to obtain this result is a Riesz-type representation theorem for charges, to be found in [9] which we state in precise terms later on.

2. Basic definitions

Let S be a set and let F be a field of subsets of S, i.e. a collection F ⊂ 2^{S} satisfying the following axioms:

1. ∅, S ∈ F,

2. A ∈ F ⇒ A^{c} ∈ F,

3. A, B ∈ F ⇒ A ∪ B ∈ F.

A bounded charge on (S, F) is an additive map α : F → ℝ, i.e. a map satisfying α(A ∪ B) = α(A) + α(B) for all A, B ∈ F with A ∩ B = ∅ (note that then α(∅) = 0 follows), such that the total variation

||α|| := sup{∑_{i=1}^{n} |α(A_{i})| | {A_{1}, …, A_{n}} ⊂ F finite partition of S}

is finite. We write ba(S, F) for the space of all bounded charges on (S, F), equipped with the total variation norm defined above. To ease the notation we put ba(S) instead of ba(S, 2^{S}),

A simple function on (S, F) is a map f : S → ℝ for which there exist a finite partition {A_{1}, …, A_{n}} ⊂ F of S and scalars a_{1}, …, a_{n} ∈ ℝ, such that

f = ∑_{i=1}^{n} a_{i}1_{A_{i}}.

We put B(S, F) for the closure, with respect to the supremum–norm ∥f∥_{∞} := sup_{s∈S} |f(s)|, of the set of simple functions on (S, F) in the space of all bounded functions from S to ℝ. The space (B(S, F)) is equipped with the norm || − ||_{∞}). If F is the collection of all subsets of S, we write BS instead of B(S, F), which in this case equals the space of all bounded real valued functions on S.

For α ∈ ba(S, F) and a ∈ B(S, F) the integral ∫_{s∈S} a(s)dα(s) can be introduced as the limit of the integrals of a sequence of simple functions converging uniformly to a [9], where the integral of a simple function is defined in the obvious way:

∫_{s∈S} ∑_{i=1}^{n} a_{i}1_{A_{i}}(s)dα(s) := ∑_{i=1}^{n} a_{i}α(A_{i}).

We will use the notational convention to write OX for the closed unit ball of a seminormed space X and CX for the closed dual unit ball, i.e.

CX := {φ : X ⟶ ℝ |φ is linear and for all x ∈ X: |φ(x)| ≤ ∥x∥}.

In the sequel L_{C}X denotes the topological dual of X, equipped with the dual norm

∥φ∥_{C} := inf\{k > 0 | \frac{1}{k}φ ∈ CX\}.

The following representation theorem identifies the dual space of (B(S, F), || − ||_{∞}).

2.1. Theorem. [9, 1] The assignment

\begin{matrix}γ_{(S,F)} : ba(S, F) & ⟶ & L_{C}(B(S, F), || − ||_{∞}) \\ α & ⟼ & ∫_{s∈S} ev(−, s)dα(s), \end{matrix}  (1)

with ( ∫_{s∈S} ev(−, s)dα(s))(a) := ∫_{s∈S} a(s)dα(s) is a linear isometry.

Another notational convention is to put T S for the closed unit ball of ba(S). From Theorem 2.1 we deduce that the map

\begin{aligned}γ_{S} : & T S = Oba(S, 2^{S}) & ⟶ & CBS \\ & α & ⟼ & ∫_{s∈S} ev(−, s)dα(s) \end{aligned}  (2)

is a one-one correspondence.

It is easy to verify that we have an endofunctor

\begin{aligned}T : Set ⟶ Set : (S_{1} →^{f} S_{2}) ⟼ & (T S_{1} →^{T f} T S_{2}) \\ & α ⟼ α_{f}, \end{aligned}  (3)

with α_{f}(A) := α(f^{−1}(A)) (A ∈ F_{2}).

We put

η_{S} : S ⟶ T S : s ⟼ δ_{s},  (4)

with δ_{s}(A) := 1 if s ∈ A and δ_{s}(A) := 0 if s ∉ A.

Recall that for α ∈ ba(S, F) and a ∈ B(S, F), we have

|∫_{s∈S} a(s)dα(s)| ≤ ∫_{s∈S} |a(s)|d|α|(s),

where |α| is the total variation of α, given by |α| := α^{+}+α^{−}, with α^{+}(A) := sup{α(B) | B ∈ F, B ⊂ A} and α^{−} := − inf{α(B) | B ∈ F, B ⊂ A} for each A ∈ F. Also recall that we have the identity |α|(S) = ||α||. Let β : 2^{Oba(S,F)} → ℝ be in T Oba(S, F). Then the map Oba(S, F) ⟶ ℝ : α ⟼ α(A) is bounded, because for each α ∈ Oba(S, F) we have |α(A)| ≤ |α(A)| + |α(A^{c})| ≤ ||α|| ≤ 1. This implies that this map is integrable with respect to β, i.e. the integral

∫_{α∈Oba(S,F)} α(A)dβ(α)

exists and is finite. One moreover shows, in the same way as in any standard measure theory course, that the assignment

∫_{α∈Oba(S,F)} α(−)dβ(α) : F ⟶ ℝ
A ⟼ ∫_{α∈Oba(S,F)} α(A)dβ(α).

is actually a bounded charge on Oba(S, F). We thus obtain a map

∫_{Oba(S,F)} : T Oba(S, F) ⟶ Oba(S, F)

β ⟼ ∫_{α∈Oba(S,F)} α(−)dβ(α). (5)

which in case F = 2^{S} is denote by

μ_{S} : T^{2}S ⟶ T S.  (6)

2.2. Definition. A space of charges is a pair (M, I_{M}) consisting of a set M and a structure map

I_{M} : T M ⟶ M : α ⟼ I_{M}(α)

which satisfies (SC1) for all x ∈ M : I_{M}(δ_{x}) = x,

(SC2) for all β ∈ T^{2}M : I_{M}(∫_{α∈T M} α(−)dβ(α)) = (I_{M} ∘ T I_{M})(β). A morphism between the spaces of charges (M_{1}, I_{M_{1}}) and (M_{2}, I_{M_{2}}) is a map f : M_{1} ⟶ M_{2} such that I_{M_{2}} ∘ T f = f ∘ I_{M_{1}} and the category of spaces of charges is denoted SC.

These axioms may seem to be far fetched at first sight. However, notice that similar laws already where obtained in [10], where it was shown that the category of compact convex sets is monadic over the category of compacta.

3. On the algebraic structure of linear contractions

On the opposite category of sNorm_{1}, the category of seminormed spaces with linear non–expansive maps, we consider the following dualisation functor:

C : sNorm_{1}^{op} ⟶ Set : (Y ←^{f} X) ⟼ (CY →^{Cf} CX),

with Cf(φ) := φ ∘ f (φ ∈ CY). It immediately follows from the fact that the product ∏_{s∈S}(ℝ, | |) in the category sNorm_{1} is equal to BS, that the map

η′_{S} : S ⟶ CBS : s ⟼ ev(−, s)  (7)

is universal for C. We therefore obtain a functor

B : Set ⟶ sNorm_{1}^{op} : (S_{1} →^{f} S_{2}) ⟼ (BS_{2} →^{Bf} BS_{1} : a ⟼ a ∘ f)

that is left adjoint to C. Let T′ := (T′, η′, μ′) be the monad induced by the adjunction B ⊣ C. So T′ := CB, η′ is defined above and if μ′ := Cϵ′B is applied on a set S we have

μ′_{S} : T′^{2}S ⟶ T′S : Ψ ⟼ μ′_{S}(Ψ),  (8)

with μ′_{S}(Ψ) : BS ⟶ ℝ : a ⟼ Ψ(ev(−, a)).

For each point x in a seminormed space X, the map ev(−, x) : CX ⟶ ℝ is bounded and therefore integrable w.r.t. any α ∈ T CX, i.e. the integral ∫_{φ∈CX} φ(x)dα(φ) exists and is finite. Moreover, we have the inequality

|∫_{φ∈CX} φ(x)dα(φ)| ≤ ||x||||α||.

and, since the assignment ∫_{φ∈CX} φ(−)dα(φ) : X ⟶ ℝ is linear, we therefore have an action

I_{CX} : T CX ⟶ CX : α ⟼ ∫_{φ∈CX} φ(−)dα(φ).

We now come to our main theorem.

3.1. Theorem. The pair ĈX := (CX, I_{CX}) is an object in SC. Moreover, SC is a representation of the Eilenberg-Moore category of the adjunction B ⊣ C and the resp. comparison functor is given by

Ĉ : sNorm_{1}^{op} ⟶ SC : (Y ←^{f} X) ⟼ (ĈY →^{Cf} ĈX).

Proof. The proof is built upon the following series of facts.

Fact 1. Let σ_{S} denote the inverse of the map γ_{S} given in (2). Then the collection σ := (σ_{S})_{S∈Set} defines a natural transformation σ : T′ ⟶ T.

In order to show this, take a map f : S_{1} ⟶ S_{2}. We have to verify commutation of the diagram

\xymatrix{T′S_{1} \ar[r]^{T′f} \ar[d]_{σ_{S_{1}}} & T′S_{2} \ar[d]^{σ_{S_{2}}} \\ T S_{1} \ar[r]_{T f} & T S_{2},}

i.e. σ_{S_{2}} ∘ CBf = T f ∘ σ_{S_{1}}. Since σ_{S_{1}} and σ_{S_{2}} are bijections, this follows since, for each α ∈ T S_{1} and φ ∈ BS_{2}, the following string of equalities hold:

(CBf ∘ γ_{S_{1}})(α)(φ) = CBf(σ_{S_{1}}^{−1}(α))(φ)
= σ_{S_{1}}^{−1}(α)(φ ∘ f)
= ∫_{S_{1}} φ ∘ fdα
= ∫_{S_{2}} φdα_{f}
= γ_{S_{2}}(α_{f})(φ)
= (γ_{S_{2}} ∘ T f)(α)(φ).

Fact 2. The diagram

\xymatrix{S \ar[r]^{η_{S}} \ar[dr]_{η′_{S}} & T S. \\  & T′S \ar[u]_{σ_{S}}}  (9)

commutes. Indeed, for s ∈ S and φ ∈ T′S we have

(γ_{S} ∘ η_{S})(s)(φ) = γ_{S}(δ_{s})(φ)
= ∫_{t∈S} φ(t)dδ_{s}(t)
= φ(s)
= η′_{S}(s)(φ),

from which the desired identity follows since γ_{S} is a bijection.

Fact 3. The diagram

\xymatrix{T′^{2}S \ar[d]_{σ_{S}^{2}} \ar[r]^{μ′_{S}} & T′S \ar[d]^{σ_{S}} \\ T^{2}S \ar[r]_{μ_{S}} & T S}  (10)

is commutative.

Bearing in mind the factorization \xymatrix{T′^{2}S \ar[rr]^{σ_{S}^{2}} \ar[r]_{T′σ_{S}} & T′T S \ar[r]_{σ_{T S}} & T^{2}S} , we have to verify

that σ_{S} ∘ μ′_{S} = μ_{S} ∘ σ_{T S} ∘ T′σ_{S}. Take Φ : BCBS ⟶ ℝ in T′^{2}S and A ⊂ S arbitrary. We have

μ_{S}((σ_{T S} ∘ T′σ_{S})(Φ)) = ∫_{α∈T S} α(−)d(σ_{T S} ∘ T′σ_{S})(Φ)(α),

so

(μ_{S} ∘ σ_{T S} ∘ T′σ_{S})(Φ)(A) = ∫_{α∈T S} α(A)d(σ_{T S} ∘ T′σ_{S})(Φ)(α)
= ∫_{α∈T S} ev(−, A)(α)dσ_{T S}(T′σ_{S}(Φ))(α)
= T′σ_{S}(Φ)(ev(−, A))
= Φ(ev(−, A) ∘ σ_{S}).

On the other hand we have that

(σ_{S} ∘ μ′_{S})(Φ)(A) = ∫_{s∈S} 1_{A}(s)dσ_{S}(μ′_{S}(Φ))(s)
= μ′_{S}(Φ)(1_{A})
= Φ(ev(−, 1_{A})).

We are done if we show that

ev(−, A) ∘ σ_{S} = ev(−, 1_{A}).

This is true since for each φ : BS ⟶ ℝ in T′S,

(ev(−, A) ∘ σ_{S})(φ) = σ_{S}(φ)(A)
= ∫_{s∈S} 1_{A}(s)dσ_{S}(φ)
= φ(1_{A}).

Fact 4. For the moment we only know that T′ = (T′, η′, μ′) is a monad. However, from a lengthy yet straightforward categorical computation [13] it follows that this information, together with Facts 1—3 suffice for the triple T = (T, η, μ) to be a monad too. Moreover, σ : T′ → T is an isomorphism of monads.

Fact 5. It is another straightforward exercise in category theory to show that the assignment

I_{γ} : Set^{T} ⟶ Set^{T′} : ((X, h) →^{f} (Y, k)) ⟼ ((X, h ∘ γ_{X}) →^{f} (Y, k ∘ γ_{Y})) (11)

is an isomorphism of categories.

Fact 6. The category SC equals Alg^{T} by definition [3, 6].

Now we proceed as follows. Let K : sNorm_{1}^{op} → Alg^{T′} be the usual comparison functor. We should compose K with the concrete isomorphism I_{γ} given above (11), in order to get the comparison functor of the adjunction B ⊣ C in the SC-representation. From the definition of the comparison functor [3] we see that I_{γ} ∘ K is given by

I_{γ} ∘ K : sNorm_{1}^{op} ⟶ SC : X →^{f} Y ⟼ (CY, Cϵ′_{Y} ∘ γ_{CY}) →^{Cf} (CX, Cϵ′_{X} ∘ γ_{CX}),

so we are done if we show that, for every seminormed space X,

Cϵ′_{X} ∘ γ_{CX} = I_{CX}.

Hereto, fix α ∈ T CX and x ∈ X. Then we obtain that

(Cϵ′_{X} ∘ γ_{CX})(α)(x) = (Cϵ′_{X}(γ_{CX}(α)))(x)
= (γ_{CX}(α) ∘ ϵ′_{X})(x)
= γ_{CX}(α)(ev(−, x))
= ∫_{φ∈CX} ev(−, x)(φ)dα(φ),

which completes the proof.

4. The Banach space connection

We write

O : Ban_{1} → Set : (X →^{f} Y) ⟼ (OX →^{f|_{OX}} OY)

for the closed unit ball-functor. For a set S, put

l_{1}S := {a : S → ℝ | {a ≠ 0} is (at most) countable and ||a||_{1} < ∞},

equipped with the sum-norm

||a||_{1} := ∑_{s∈S} |a(s)|.

Then the functor

l_{1} : Set → Ban_{1} : (S_{1} →^{g} S_{2}) ⟼ (l_{1}S_{1} →^{l_{1}g} l_{1}S_{2}),

with l_{1}g(a) := ∑_{s∈S_{1}} a(s)δ_{g(s)}^{f}, a ∈ l_{1}S_{1}, is left adjoint to O [7]. The unit of the adjunction l_{1} ⊣ O is given by

η′′_{S} : S ↪ Ol_{1}S : s ⟼ δ_{s}^{f} (S ∈ |Set|),

where the Dirac function δ_{s}^{f} is given by

δ_{s}^{f}(t) := \{ \begin{matrix}1 if s = t \\ 0 if s ≠ t \end{matrix} .

The counit of this adjunction at a Banach space X is given by the assignment

ϵ′′_{X} : l_{1}OX → X : a ⟼ ∑_{x∈OX} a(x)x.

Let T′′ be the monad induced by the adjunction l_{1} ⊣ O. We will proceed by providing an explicit description of E. By definition, an object in Alg^{T′′} is a pair (N, E_{N}) consisting of a set N and a structure map

E_{N} : Ol_{1}N ⟶ N

that satisfies the following equations:

(TCF1) E_{N}(δ_{x}^{f}) = x,

(TCF2) E_{N}(∑_{a∈Ol_{1}N} B(a)a(−)) = E_{N} (∑_{x∈N} (∑_{x=E_{N} (a)^{a∈Ol_{1}N}} B(a)) δ_{x}^{f}(−)).

4.1. Definition. [7] A totally convex structure is a set M together with, for each α = (α_{i})_{i∈ℕ_{0}} ∈ Ol_{1}ℕ_{0}, an operation α̂ : M^{ℕ_{0}} → M, such that, with the notation

∑_{i=1}^{∞} α_{i}x_{i} := α̂((x_{i})_{i∈ℕ_{0}}),

the following identities are satisfied:

(TC1) ∑_{i=1}^{∞} δ_{i}^{f}(k)x_{i} = x_{k},

(TC2) ∑_{j=1}^{∞} β_{j}(∑_{i=1}^{∞} α_{ij}x_{i}) = ∑_{i=1}^{∞}(∑_{j=1}^{∞} β_{j}α_{ij})x_{i}.

A totally affine map f : M → N between totally convex modules is a map between the underlying sets such that, for all (α_{i})_{i} ∈ Ol_{1}ℕ_{0} and for all (x_{i})_{i} ∈ M^{ℕ_{0}},

f(∑_{i=1}^{∞} α_{i}x_{i}) = ∑_{i=1}^{∞} α_{i}f(x_{i}).

The category of totally convex modules is denoted by TC and we put U : TC ⟶ Set for the forgetful functor.

The closed unit ball of a Banach space X is in a pointwise way a totally convex module, which is denoted ÔX. Moreover, if f : X → Y is a linear non–expansive map between Banach spaces, then f|_{OX} : ÔX → ÔY is a totally affine map between totally convex modules.

Let M be a totally convex module. We put

E_{M}(a) := ∑_{i=1}^{∞} a(x_{i})x_{i}, (a ∈ Ol_{1}M)  (12)

where (x_{i})_{i∈ℕ_{0}} is a sequence in M such that {x_{i} | i ∈ ℕ_{0}} contains the support of a. Conversely, let (N, E_{N}) be in Alg^{T′′}. Then we define

∑_{i=1}^{∞} α_{i}x_{i} := E_{N}(a),  (13)

where for every (α_{i})_{i∈ℕ_{0}} such that ∑_{i=1}^{∞} |α_{i}| ≤ 1 and for every (x_{i})_{i} ∈ N^{ℕ_{0}}, we put

a(x) := \begin{cases}0 & if x ∉ \{x_{n} | n ∈ ℕ_{0}\} \\ ∑_{i:x_{i}=x} α_{i} & otherwise. \end{cases}

We now have the following result.

4.2. Theorem. [7] The correspondences M ⟼ (M, E_{M}) defined in (12) extends to a concrete isomorphism TC ≃ Alg^{T′′}, the inverse of which is described in (13). Moreover, Ô : Ban_{1} → TC is the respective comparison functor.

Another important issue in the theory of totally convex modules is the fact that Ô has a left adjoint S : TC → Ban_{1}, such that

S ∘ Ô ≃ id_{Ban_{1}}.  (14)

For a more detailed account, we refer to [7].

Note that there is a dualisation functor into the category of Banach spaces:

L_{C} : sNorm_{1}^{op} ⟶ Ban_{1} : (X →^{f} Y) ⟼ (L_{C}Y →^{L_{C}f} L_{C}X),

with L_{C}f(φ) := φ ∘ f (φ ∈ L_{C}Y). Thus, neglecting the dashed arrow, there is a commutative diagram

\xymatrix{Ban_{1} \ar[r]^{Ô} & TC \ar[dr]^{U} \\ _{sNorm{op \atop 1}} \ar[u]^{L_{C}} \ar[r]_{Ĉ} & SC \ar@{.>}[u]_{E} \ar[r]_{V} & Set,}  (15)

with V : SC → Set the canonical forgetful functor. Suppose that we could find a concrete functor E : SC → TC with the additional property that the square formed in diagram (15) commutes, then from (14) we see that

L_{C} ≃ S ∘ E ∘ Ĉ.

In other words, such E would yield a factorization of the dualisation L_{C} via the natural dual algebraic structure. The sequel of this section is devoted to the comparison of SC, the algebraic theory of dual unit balls, with TC, the algebraic component of Banach spaces.

Let (M, I_{M}) be a space of charges and fix a ∈ Ol_{1}M. Then we put

\overline{a} := ∑_{x∈M} a(x)δ_{x} ∈ T M.

As the closed unit ball of ba(M), T M carries a natural totally convex structure, so this assignment is well defined. Now we define

E^{I_{M}} (a) := I_{M}(\overline{a}),

so we have a map E^{I_{M}} : Ol_{1}M ⟶ M.

4.3. Theorem. The pair (M, E^{I_{M}}) is a totally convex module. Moreover, the assignment (M, I_{M}) ⟼ (M, E^{I_{M}}) defines a concrete functor

E : SC → TC

such that the square formed in the diagram (15) commutes.

Proof. First we verify that (M, E^{I_{M}}) satisfies the axioms for the formal representation of a totally convex module.

(TCF1) follows trivially from (SC1).

In order to obtain (TCF2), fix B ∈ (Ol_{1})^{2}M and define

β := ∑_{a∈Ol_{1}M} B(a)δ\overline{a} ∈ T^{2}M.

We then obtain that

\overline{∑_{a∈Ol_{1}M} B(a)a} = ∑_{x∈M} ∑_{a∈Ol_{1}M} B(a)a(x)δ_{x}
= ∑_{a∈Ol_{1}M} B(a)(∑_{x∈M} a(x)δ_{x})
= ∑_{a∈Ol_{1}M} B(a)\overline{a}

= ∫_{α∈T M} α(−)dβ(α).

We also have that for all A ∈ 2^{M}

T I_{M}(β)(A) = β(I_{M}^{−1}(A))
= ∑_{a∈Ol_{1}M} B(a)δc ; \overline{a}(I_{M}^{−1}(A))
= ∑_{x∈M} ∑_{I_{M} (\overline{a})=x^{a∈Ol_{1}M}} B(a)δ\overline{a}(I_{M}^{−1}(A))

= ∑_{x∈M} ∑_{I_{M} (\overline{a})=x^{a∈Ol_{1}M}} B(a)δ_{x}(A)

= ∑_{x∈M}( ∑_{E^{I_{M}} (a)=x^{a∈Ol_{1}M}} B(a))δ_{x}(A).

Hence

E^{I_{M}} ( ∑_{a∈Ol_{1}M} B(a)a) = I_{M}(\overline{∑_{a∈Ol_{1}M} B(a)a})

= I_{M} (∫_{α∈T M} α(−)dβ(α))
= I_{M}(T I_{M}(β))
= I_{M}(∑_{x∈M} ∑_{E^{I_{M}} (a)=x^{a∈Ol_{1}M}} B(a)δ_{x})

= I_{M}(\overline{∑_{x∈M}( ∑_{E^{I_{M}} (a)=x^{a∈Ol_{1}M}} B(a))δ_{x}^{f}})

= E^{I_{M}} (∑_{x∈M}( ∑_{E^{I_{M}} (a)=x^{a∈Ol_{1}M}} B(a))δ_{x}^{f} ) .

To finish the proof, we have to check that for an SC–morphism f : (M, I_{M}) ⟶ (N, I_{N}), automatically f : (M, E^{I_{M}}) ⟶ (N, E^{I_{N}}) is a TC–morphism. That is, from the commutation of the diagram

\xymatrix{T M \ar[r]^{I_{M}} \ar[d]_{T f} & M \ar[d]^{f} \\ T N \ar[r]_{I_{N}} & N}

we have to show the commutation of

\xymatrix{Ol_{1}M \ar[r]^{E^{I_{M}}} \ar[d]_{Ol_{1}f} & M \ar[d]^{f} \\ Ol_{1}N \ar[r]_{E^{I_{N}}} & N}.

Take a ∈ Ol_{1}M. Then

(f ∘ E^{I_{M}})(a) = (f ∘ I_{M})(\overline{a})
= I_{N}(T f(\overline{a}))
= I_{N}(\overline{Ol_{1}f(a)})
= (E^{I_{N}} ∘ Ol_{1}f)(a),

where the last but one equality is true because, as one easily verifies, we have T f(\overline{a}) = \overline{Ol_{1}f(a)}.

4.4. Corollary. If (M, I_{M}) and (N, I_{N}) are isomorphic spaces of charges then (M, E_{I_{M}}) and (N, E_{I_{N}}) are isomorphic totally convex structures.

From the above remark, in combination with the following theorem, it is noted that the SC theory is of strictly stronger nature than the theory of totally convex modules.

4.5. Theorem. The categories SC and TC (with their canonical forgetful functors) are not concretely isomorphic.

Proof. It is well–known from the general theory of monads (see e.g. [3, 6]) that the assignment F S := (T S, μ_{S}) (S ∈ |Set|) defines a functor F : Set ⟶ SC that is left adjoint to V : SC ⟶ Set. Now suppose SC and TC were concretely isomorphic. Since adjunctions are determined up to natural isomorphism, this would imply that the underlying sets of the free TC–object on ℝ (i.e. Ôl_{1}ℝ) and the free SC–object on ℝ (i.e. F ℝ) would have the same cardinality. Now on the one hand we see that

#Ol_{1}ℝ = #l_{1}ℝ ≤ #(ℝ^{ℕ} × ℝ^{ℕ}) = #ℝ.

On the other hand, we can define for every ultrafilter U on ℝ a charge α_{U} that is an element of T ℝ by

α_{U}(A) := \begin{cases}1 & if A ∈ U, \\ 0 & otherwise, \end{cases}

and it is easy to see that α_{U} ≠ α_{V} if U ≠ V. If ℝ is equipped with the discrete topology, it therefore follows from [2], Theorem 9.2, that

#T ℝ = #Oba(ℝ, 2^{ℝ}) ≥ #β(ℝ) = #2^{2^{ℝ}} > #2^{ℝ}.

Hence #Ol_{1}ℝ < #T ℝ, yielding a contradiction.

It is always nice to have the dual of a normed space represented by a concrete Banach space. This representation puts, often in a canonical way, an SC–structure on the closed unit ball of that Banach space. If we apply the forgetful functor E, we see from the commutation of diagram (15) that we then recover the pointwise TC–structure on the closed unit ball of the Banach space. As an example we reconsider the representation theorem 2.1 in this context.

4.6. Theorem. The pair (Oba(S, F), ∫_{Oba(S,F)}) is an SC–object and

γ_{(S,F)} : (Oba(S, F), ∫_{Oba(S,F)}) ⟶ ĈB(S, F)

is an SC–isomorphism.

Proof. We only have to show that γ_{(S,F)} is an SC–morphism, that is, we have to show the commutation of the diagram

\xymatrix{T Oba(S, F) \ar[r]^{^{∫^{Oba(S,F)}}} \ar[d]_{T γ_{(S,F)}} & Oba(S, F) \ar[d]^{γ_{(S,F)}} \\ T CB(S, F) \ar[r]_{I_{CB(S,F)}} & CB(S, F).}

Take β ∈ T Oba(S, F) and a ∈ B(S, F). Then on the one hand we have that

(I_{CB(S,F)}(T γ_{(S,F)}(β))) (a)  (16)
= (I_{CB(S,F)}(β_{γ_{(S,F)}})) (a)  (17)
= ∫_{φ∈CB(S,F)} ev(−, a)(φ)dβ_{γ_{(S,F)}}(φ)  (18)
= ∫_{α∈T Oba(S,F)} ev(−, a) ∘ γ_{(S,F)}(α)dβ(α)  (19)
= ∫_{α∈T Oba(S,F)} (∫_{s∈S} a(s)dα(s)) dβ(α).  (20)

Calculating the other way around the diagram, we obtain that

((γ_{(S,F)} ∘ ∫_{Oba(S,F)})(β)) (a)  (21)
= (γ_{(S,F)}(∫_{α∈Oba(S,F)} α(−)dβ(α))) (a)  (22)
= ∫_{s∈S} a(s)d (∫_{α∈Oba(S,F)} α(−)dβ(α)) (s).  (23)

We are therefore done if we show that (20) and (23) are equal. Let (a_{n})_{n} be a sequence of F–simple functions, converging uniformly to a. We write a_{n} = ∑_{i=1}^{m_{n}} a_{i}^{n}1_{A_{i}^{n}}, with all a_{i}^{n} ∈ ℝ and all A_{i}^{n} ∈ F such that all {A_{1}^{n}, …, A_{m_{n}}^{n}} are a partition of S. Then

∫_{s∈S} a(s)d (∫_{α∈Oba(S,F)} α(−)dβ(α)) (s)
= lim_{n→∞} ∫_{s∈S} a_{n}(s)d (∫_{α∈Oba(S,F)} α(−)dβ(α)) (s)

= lim_{n→∞} ∑_{i=1}^{m_{n}} a_{i}^{n} ∫_{α∈Oba(S,F)} α(A_{i}^{n})dβ(α)

= lim_{n→∞} ∫_{α∈Oba(S,F)} (∑_{i=1}^{m_{n}} a_{i}^{n}α(A_{i}^{n})) dβ(α)

= lim_{n→∞} ∫_{α∈Oba(S,F)} (∫_{s∈S} a_{n}(s)dα(s)) dβ(α)
= ∫_{α∈Oba(S,F)} (∫_{s∈S} a_{n}(s)dα(s)) dβ(α),

where the last step is valid because the sequence (α ⟼ ∫_{s∈S} a_{n}(s)dα(s))_{n} is uniformly convergent to α ⟼ ∫_{s∈S} a(s)dα(s).

That Theorem 4.6 is indeed a strengthening of the Riesz–type representation theorem 2.1 is easily seen now since the latter can be obtained as a simple corollary using the commutative diagram 15.

4.7. Corollary. (ba(S, F), ∥ ∥) and L_{C}B(S, F) are isomorphic Banach spaces.

This is of course not surprising since Corollary 4.7 has served as a starting point for our theory.

References

[1] N. Dunford and J. T. Schwartz. Linear operators Part I, volume VII of Pure and applied mathematics. Interscience publishers, 1971.

[2] L. Gillman and M. Jerison. Rings of continuous functions. The university series in higher mathematics. Van Nostrand company, 1966.

[3] S. Mac Lane. Categories for the Working Mathematician. Springer, 1998.

[4] J. W. Negrepontis. Duality in analysis from the point of view of triples. Journal of Algebra, 19:228–253, 1971.

[5] H. E. Porst and W. Tholen. Concrete dualities. Category Theory at Work, pages 111–136, 1991.

[6] D. Pumplün. Eilenberg-moore algebras revisited. Seminarberichte, 29:57–144, 1988.

[7] D. Pumplün and H. Röhrl. Banach spaces and totally convex spaces i. Communications in Algebra, 12(8):953–1019, 1984.

[8] D. Pumplün and H. Röhrl. Banach spaces and totally convex spaces ii. Communications in Algebra, 13(5):1047–1113, 1985.

[9] K. B. Rao and M. B. Rao. Theory of charges. Pure and Applied Mathematics. Academic Press, 1983.

[10] Z. Semadeni. Monads and their Eilenberg–Moore algebras in functional analysis. Queen’s papers in pure and applied mathematics, 33:1–98, 1973.

[11] M. Sioen and S. Verwulgen. Locally convex approach spaces. Applied General Topology, 4(2):263–279, 2003.

[12] M. Sioen and S. Verwulgen. Quantified functional analysis and seminormed spaces: a dual adjunction. Journal of pure and applied algebra, 207:675–686, 2006.

[13] S. Verwulgen. A Categorical approach to quantified functional analysis. PhD thesis, University of Antwerp, 2003.

Mark Sioen Department of Mathematics, VUB Pleinlaan 2 B-1050 Brussels

Stijn Verwulgen Department of Mathematics and Computer science, University of Antwerp Middelheimlaan 1 B-2020 Antwerp

Email: msioen@vub.ac.be
      stijn.verwulgen@ua.ac.be

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