ON THE FACTORIZATION FIELDS OF BANACH SPACES

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OF BANACH SPACES

ALEXANDRU MIHAIL and SPERANT¸ A VL ˘ADOIU

The aim of the paper is to give a necessary and sufficient condition for the factorization of a field of Banach spaces by a subfield.

AMS 2000 Subject Classification: 46B25.

Key words: field of Banach spaces, factorization.

1. PRELIMINARIES AND EXAMPLES

Throughout the paper, (X, τ) is a normal (and Hausdorff) topological space. The Banach spaces will be supposed to be complex, although this is not necessary. The results are also valid for real Banach spaces. For a Banach space E, S(E) is the set of Banach subspaces of E. For a Banach space E and a set A ⊂E, hAi is the Banach subspace generated by A. For a Banach spaceE and a topological spaceX, C(X, E) is the Banach space of continuous bounded functions fromX toE with the normkfk= sup

x∈X

kf(x)k_E. LetXbe a set and Y a subset of it. By iwe denote the inclusion between Y and X.

For a topological space X and a net (xν)ν we say that xν → ∞ if for every compact set K ⊂X there exists ν₀ such that x_ν ∈/ K for every ν ≥ν₀. We first give the classical definition of a field of Banach spaces:

Definition 1.1. Let X be a topological space and (E(x))x∈X a family of complex Banach spaces. An element ξ∈ Q

x∈X

E(x) is called a field of vectors.

Definition 1.2. Let X be a topological space. A pair ((E(x))x∈X,Γ) consisting of a family of complex Banach spaces and a subset of fields of vectors Γ⊂ Q

x∈X

E(x) is called a continuous field of Banach spaces if 1. Γ is a complex vectorial subspace in Q

x∈X

E(x);

2. for everyx∈X the setξ(x) for ξ∈Γ is dense in E(x);

3. for everyξ∈Γ the functionx→ kξ(x)kis continuous;

MATH. REPORTS10(60),3 (2008), 253–263

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4. ifη∈ Q

x∈X

E(x) is such that for everyx∈X and for everyε >0 there existsξ⁰ ∈Γ such thatkη(y)−ξ⁰(y)k< εon a neighborhood ofx,thenη∈Γ.

By a field of Banach spaces we will understood a continuous field of Banach spaces.

Let E = ((E(x))x∈X,Γ). The topological space X is named the basic space of the field of Banach spaces E. The Banach spaces E(x) are the fibres of E. The function π_x : Γ→E(x) given by π_x(ξ) =ξ(x) is the projection on E(x). For a closed set F ⊂X,E|_F = ((E(x))x∈F,Γ/∼_F) is the reduction of the field ((E(x))x∈X,Γ) on F, where ξ ∼_F ξ⁰ if and only if ξ(x) = ξ⁰(x) for every x∈F. There is a projection πF : Γ→ Γ/∼_F given by πF(ξ) =ξ. Web will also see π_F as a morphism from E into E|_F.

Definition 1.3. Let E = ((E(x))x∈X,Γ) be a continuous field of Banach spaces. On Γ we consider the generalizat norm kξk = sup

x∈X

kξ(x)k,that is we could have kξk = +∞. Let B(E) = {ξ ∈ E | for every ε > 0 there exists a compact setK ⊂X such thatkξ(x)k< εfor everyx /∈K}. B(E) is a Banach space with norm k·k called the Banach space associated with the fieldE.

Definition 1.4. LetE= ((E(x))x∈X,Γ) andE⁰= ((E⁰(x))x∈X,Γ⁰) be two fields of Banach spaces. E⁰ is a subfield of E if Γ⁰ ⊂Γ. ThenE⁰(x)⊂E(x) for every x∈X.

Lemma 1.1. Let E = ((E(x))x∈X,Γ)be a field of Banach spaces, E⁰ = ((E⁰(x))x∈X,Γ⁰)a subfield of it. If η(x)∈E⁰(x) for every x∈X,then η∈Γ⁰. Proof. Let x∈X be fixed. There existsξx∈Γ⁰ such that η(x) =ξx(x).

It follows that for every ε >0 there exists a neighborhood W of x such that kη−ξ_xk< ε onW. This imply (Definition 1.2.4) thatη ∈Γ⁰.

Lemma1.2.Let Xbe a locally compact and paracompact topological space and Y is a closed set in X. Then

1. if ϕ : Y → (0,∞) is a continuous function such that ϕ(x) → 0 as x → ∞, then there exists a continuous function ϕ˜ : X → (0,∞) such that ϕ|˜_Y =ϕ and ϕ(x)˜ →0 as x→ ∞;

2. if ϕ : X → (0,∞) is a continuous function such that ϕ(x) → 0 as x→ ∞ on Y, then there exists a continuous function ϕ˜ : X → (0,∞) such that ϕ|˜_Y = 1 and ϕ(x)˜ ·ϕ(x)→0 as x→ ∞ on X.

Proposition 1.1. Let E = ((E(x))x∈X,Γ)be a continuous field of Ba- nach spaces.

1. Ifη∈Γand ifg:X →Cis a continuous function on,X thengη∈Γ.

2. For everyx0 ∈X and every v ∈ E(x0), there exists ξ ∈Γ such that ξ(x0) =v.

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3. Let ((E⁰(x))x∈X,Γ⁰) be a subfield of the field ((E(x))x∈X,Γ), Y a closed set in X, ϕ : X → (0,∞) a continuous function, and let ξ ∈ E|_Y be such that for every x∈Y there exists ξ⁰ ∈Γ⁰ for which kξ(x)−ξ⁰(x)k< ϕ(x).

Then there exists η ∈Γ⁰ such that kξ(x)−η(x)k< ϕ(x) for every x∈Y and kη(x)k<kξ(x)k+ϕ(x) for everyx∈X.

4. If ξ∈ E|_Y then there exists η∈Γsuch that η|_Y =ξ.

Proof. See [2] for 1, 2 and 4.

3.kξ(x)−ξx(x)k< ϕ(x). By the assumptions made there exists a neighborhood V_x of x such that kξ(y)−ξ_x(y)k< ϕ(y) for every y ∈V_x. It follows thatkξ_x(y)k<kξ(y)k+ϕ(y) for everyy∈V_x. The set{V_x |x∈Y} ∪ {X\Y} is an open local finite cover of X. Since Y is paracompact, there exists an open local finite cover (V_x_i)i∈I of Y. Then {V_x_i |i ∈I} ∪ {X\Y} is an open local finite cover of X. Let φi for Vxi, i ∈ I, and φ for X\Y a partition of unity subordinated to it. Let η = P

i∈I

φ_iξ^xⁱ ∈ Γ. Then kξ(x)−η(x)k < ϕ(x) for every x∈Y andkη(x)k<kξ(x)k+ϕ(x) for everyx∈X.

Corollary 1.1. Let E = ((E(x))x∈X,Γ) be a continuous field of Ba- nach spaces, ϕ:X→(0,∞) a continuous function and E⁰ = ((E⁰(x))x∈X,Γ⁰) a subfield of it. Let ξ ∈ E be such that for every x ∈ X there exists ξx ∈ Γ⁰ for which kξ(x)−ξ_x(x)k < ϕ(x). Then there exists η ∈ Γ⁰ such that kξ(x)−η(x)k< ϕ(x) for every x∈X.

Corollary 1.2. Let E⁰ = ((E⁰(x))x∈X,Γ⁰) be a subfield of the field E = ((E(x))x∈X,Γ)and Y a closed set in X. If ξ ∈B(E|_Y) then there exists η ∈B(Γ) such that η|_Y =ξ.

Proof. It results from Proposition 1.1, point 4, and Lemma 1.2.2.

LetE = ((E(x))x∈X,Γ) be a field of Banach spaces andE⁰= ((E⁰(x))x∈X, Γ⁰) a subfield of it. Let us consider the Banach space E(x)/E⁰(x) and the complex vectorial space Γ/Γ⁰. Let P : Γ → Γ/Γ⁰, P^b :B(Γ) ^P

b

−→B(Γ)/B(Γ⁰) and P_x : E(x) → E(x)/E⁰(x) be the factorization functions. There exists a commutative and exact on lines diagram, namely,

0 −→ B(Γ⁰) −→ B(Γ) −→^P^b B(Γ)/B(Γ⁰) −→ 0



 yi⁰



 yi



 ybı

0 −→ Γ⁰ −→ Γ −→^P Γ/Γ⁰ −→ 0



 yπ_x⁰



 yπ_x



 ybπ_x

0 −→ E⁰(x) −→ E(x) −→^P^x E(x)/E⁰(x) −→ 0.

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One can define T : Γ/Γ⁰ → (E(x)/E⁰(x))x∈X = Q

x∈X

E(x)/E⁰(x) by T(bη) = (bπx(η))b x∈X. Let also T^b = T ◦bı : B(Γ)/B(Γ⁰) → Q

x∈X

E(x)/E⁰(x). T is an injective morphism of complex vectorial spaces as we will see in the proof of Theorem 4.1 and T^b is an injective morphism of Banach spaces as we will also see in the proof of Theorem 3.1. Then Γ/Γ⁰ can be seen as a subset of

Q

x∈X

E(x)/E⁰(x). A natural question that arises is whether there exists a field of Banach spaces Eb = ((E(x)/E⁰(x))x∈X,Γ/Γ⁰). The answer is not always positive as one can see from the example below.

Example 1.2. LetE = ((C)_x∈[0,1], C([0,1]) be the natural field associated with the C^∗-algebra of continuous functions on the unit interval with complex values and

E₀ = (

C ifx∈(0,1]

{0} ifx= 0

x∈[0,1]

, A={f ∈C([0,1])|f(0) = 0}

!

the natural field associated with the C^∗-algebra of continuous functions on the unit interval with complex values which vanishe at 0; E₀ is a subfield of E. For x 6= 0, E₀(x) = E(x) = C and E(x)/E₀(x) = {0} while for x = 0, E₀(0) ={0}, E(0) = Cand E(0)/E₀(0) =C. There is no continuous field on [0,1] with fibres

(

{0} if x∈(0,1]

C ifx= 0

x∈[0,1]

.

The purpose of the paper is to give a necessary and sufficient condition for the pair Eb= ((E(x)/E⁰(x))x∈X,Γ/Γ⁰) is to be a field of Banach spaces.

Section 2 gives an example where Eb= ((E(x)/E⁰(x))x∈X,Γ/Γ⁰) is a field of Banach spaces, the case of direct sum of two fields of Banach spaces. The necessary and sufficient condition is that E⁰ = ((E⁰(x))x∈X,Γ⁰) has a special property of closeness. We study two cases when a subfield can be called a closed subfield. These are contained in Sections 3 and 4. The main result will be given in the last section.

2. THE CASE OF DIRECT SUM

An example whenEbis a field of Banach spaces is the case where the field is the direct sum of two fields of Banach spaces and the subfield is one of the components.

Lemma 2.1. Let E₁ = ((E1(x))x∈X,Γ1)and E₂ = ((E2(x))x∈X,Γ2) be two fields of Banach spaces. Then E₁⊕ E₂ = ((E₁(x)⊕E₂(x))x∈X,Γ₁⊕Γ₂) is a continuous field of Banach spaces.

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Proof. Let us check Definition 1.2.

1) Γ₁⊕Γ₂ ⊂ Q

x∈X

(E₁(x)⊕E₂(x)) = Q

x∈X

(E₁(x))⊕ Q

x∈X

(E₂(x)).

2) Obvious.

3) The function x→ kξ₁(x)⊕ξ2(x)k= max(kξ₁(x)k,kξ₂(x)k) is continuous.

4) Let η ∈ Q

x∈X

(E1(x)⊕E2(x)) be such that for every x ∈X and every ε > 0 there exists an open neighborhood Vx of x and ξ₁^x ∈ Γ1, ξ^x₂ ∈ Γ2 for which kη(y)−ξ^x₁(y)⊕ξ^x₂(y)k < ε for every y ∈ V_x. Let (V_x_i)i∈I be an open local finite cover of X and (φi)i∈I a partition of unity subordinated to it. Let ξ₁ = P

i∈I

φ_iξ₁^xⁱ ∈ Γ₁ and ξ₂ = P

i∈I

φ_iξ₂^xⁱ ∈Γ₂. Thenkη(y)−ξ₁(y)⊕ξ₂(y)k< ε for every y∈Y. It follows that η∈Γ₁⊕Γ₂.

Corollary2.1. Let E = ((E(x))x∈X,Γ)be a field of Banach spaces and E₁ = ((E₁(x))x∈X,Γ₁) and E₂ = ((E₂(x))x∈X,Γ₂) two fields of Banach spaces such that E1(x)∩E2(x) = ∅ and E1(x)⊕E2(x) = E(x) for every x ∈ X.

Then B(E₁⊕ E₂) =B(E).

Corollary 2.2. Let E₁ = ((E₁(x))x∈X,Γ₁)and E₂ = ((E₂(x))x∈X,Γ₂) be two fields of Banach spaces and E= ((E(x))x∈X,Γ) =E₁⊕ E₂.Then E/E₁ is a field of Banach spaces and is isomorphic to E₂.

3. WEAK CLOSED SUBFIELDS

Definition 3.1. LetE = ((E(x))x∈X,Γ) be a field of Banach spaces and E⁰= ((E⁰(x))x∈X,Γ⁰) a subfield of E. E⁰ is said to be a weak closed subfield of E if and only if for every closed set F such that F =

◦

F and every η∈Γ such that η|_K ∈Γ⁰|_K for every compact setK ⊂

◦

F one hasη|_F ∈Γ|_F.

Lemma 3.1. Let E₁ = ((E1(x))x∈X,Γ1) and E₂ = ((E2(x))x∈X,Γ2) be two fields of Banach spaces and E = ((E(x))x∈X,Γ) = E₁⊕ E₂. Then E₁ is a weak closed subfield of E.

Proof. Let η ∈ Γ and F a closed set such that F =

◦

F and η|_K ∈ Γ₁|_K for every compact set K ⊂F^◦. Thenη=η₁⊕η₂, where η₁ ∈Γ₁ and η₂ ∈Γ₂. Since η|_K ∈Γ1|_K, for every compact setK ⊂

◦

F we have that η2|^◦

F = 0. This imply that η2|_F = 0.It is clear that η=η1 on F.

The proofs of Lemmas 3.2 and 3.3 below are similar to the proof of Lemma 3.1.

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Lemma 3.2. Let E = ((E(x))x∈X,Γ) be a field of Banach spaces, E⁰ = ((E⁰(x))x∈X,Γ⁰)a weak closed subfield of Eand E⁰⁰= ((E⁰⁰(x))x∈X,Γ⁰⁰)a weak closed subfield of E⁰. Then E⁰⁰ is a weak closed subfield of E⁰.

Lemma 3.3. Let E = ((E(x))x∈X,Γ) be a field of Banach spaces, E⁰ = ((E⁰(x))x∈X,Γ⁰) a weak closed subfield of E and F a closed set in X. Then E⁰|_F is a weak closed subfield of E|_F.

The property of being a weak closed subfield is not preserved under all the operations, as we can see in the case of the direct sum of two disjoint subfields.

Example 3.1. Let E = (

C⊕C ifx∈(0,1]

C(1⊕1) ifx= 0

x∈[0,1]

, A = {f ∈

C([0,1],C⊕C)|f(0)∈C(1⊕1)}

!

. LetE₁= (

C⊕0 ifx∈(0,1]

0 ifx= 0

x∈[0,1]

, A₁={f∈C([0,1],C⊕0)|f(0) = 0}

!

andE₂= (

0⊕C ifx∈(0,1]

0 ifx= 0

x∈[0,1]

,

A2={f ∈C([0,1],0⊕C) | f(0) = 0}

!

be two subfields of E. Then E₁ and E₂ are closed, but E₁⊕ E₂ is not closed. To see this let ξ ∈ E be given by ξ(x) = 1⊕1. Then ξ|_[ε,1] ∈ E₁⊕ E₂|_[ε,1] for every ε >0, but ξ /∈ E₁⊕ E₂. We also remark that A1 ⊂A,A2 ⊂A and A1∩A2={0}.

4. STRONG CLOSED SUBFIELDS

Definition 4.1. LetE = ((E(x))x∈X,Γ) be a field of Banach spaces and E⁰= ((E⁰(x))x∈X,Γ⁰) a subfield ofE. E⁰is said to be a strong closed subfield of E if for every η∈Γ,M >0, ε >0, and every convergent net (xν)ν,xν →x0, such that there exists ξ_ν ∈ Γ⁰ for which kη(x_ν)−ξ_ν(x_ν)k_E(x

ν) ≤ M, there exists ξ∈Γ⁰ with kη(x₀)−ξ(x₀)k_E(x

0)≤M+ε.

Let us note that the fact thatE⁰ is a strong closed subfield ofE is equivalent to the fact that the function x→d_E(x)(ξ(x), E⁰(x)) is lower semicontinuous, that is lim inf

x→x0

d_E(x)(ξ(x), E⁰(x))≥d_E(x₀₎(ξ(x0), E⁰(x0)) for everyx0∈X.

The following two lemmas are obvious.

Lemma 4.1. Let E = ((E(x))x∈X,Γ) be a field of Banach spaces, E⁰ = ((E⁰(x))x∈X,Γ⁰) a strong closed subfield of E and F a closed set in X. Then E⁰|_F is a strong closed subfield of E|_F.

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Lemma 4.2. Let E = ((E(x))x∈X,Γ) be a field of Banach spaces, E⁰ = ((E⁰(x))x∈X,Γ⁰) a strong closed subfield of E. Then E⁰ is a weak subfield of E.

The converse is not true as we can see from the example below.

Example 4.1. Let

E=

(

C⊕C ifx∈(0,1]

C(1⊕1) ifx= 0

x∈[0,1]

, A={f∈C([0,1],C⊕C)|f(0)∈C(1⊕1)}

! .

Let E⁰ = (

C[(sin(1/x)⊕(1−sin(1/x))] ifx∈(0,1]

0 ifx= 0

x∈[0,1]

, A = {f ∈

C([0,1],C⊕C)|f(0) = 0 andf(x)∈C[(sin(1/x)⊕(1−sin(1/x))] ifx∈(0,1]

! . Let e ∈ A be the function e(x) = (1,0). Then d(e(x),C[(sin(1/x)⊕(1− sin(1/x))]) = 1−sin(1/x) for x ∈ (0,1] and d(e(0),0) = 1. This shows that E⁰ is not a strong closed field. On the other hand, E⁰ is a weak closed field.

Indeed, let F be a closed set in [0,1] such that F =

◦

F. Let f ∈ A be such that f|_K ∈A⁰|_K for every compact set K ⊂F^◦. Let x₀ ∈∂F. There are two cases. First case x0 6= 0. Since f(x)∈C[(sin(1/x)⊕(1−sin(1/x))], we have f(x0)∈C[(sin(1/x0)⊕(1−sin(1/x0))]. Second case x0 6= 0. For x= _2nπ¹ we havef(x)∈0⊕C. Thenf(0)∈0⊕C. Similarly, forx= _2nπ+π/2¹ ,f(x)∈C⊕0 and f(0)∈C⊕0.Then f(0)∈0⊕0.This shows that f|_F ∈A⁰|_F.

Lemma 4.3. Let E₁ = ((E₁(x))x∈X,Γ₁)and E₂ = ((E₂(x))x∈X,Γ₂) be two fields of Banach spaces and E = ((E(x))x∈X,Γ) = E₁⊕ E₂. Then E₁ is a strong closed subfield of E.

Lemma 4.4. Let E = ((E(x))x∈X,Γ) be a field of Banach spaces, E⁰ = ((E⁰(x))x∈X,Γ⁰) a subfield of E. Then the function x→d_E(x)(ξ(x), E⁰(x)) is upper semicontinuous, that is,lim sup

x→x0

d_E(x)(ξ(x), E⁰(x))≤d_E(x₀₎(ξ(x₀), E⁰(x₀)) for every x₀∈X and every ξ∈Γ.

Proof. Letε >0 and ξ ∈Γ be fixed. There exists ηε∈B(Γ⁰) such that d_E(x₀₎(ξ(x₀), η_ε(x₀)) = kη_ε(x₀)−ξ(x₀)k_E(x

0) ≤ d_E(x₀₎(ξ(x₀), E⁰(x₀)) +ε/2.

Then there existsV_x∈ V_xsuch thatk(η_ε−η)(y)k_E(x₀₎< d_E(x₀₎(ξ(x₀), E⁰(x₀))+ε for everyy∈V_x. So,d_E(y)(ξ(y), E⁰(y))≤d_E(x₀₎(ξ(x₀), E⁰(x₀))+εfor everyy∈ Vx. It follows that lim sup

xν→x₀ d_E(x_v₎(ξ(xv), E⁰(xv))≤d_E(x₀₎(ξ(x0), E⁰(x0)) +εfor every ε >0, and so lim sup

xν→x0

d_E(x_v₎(ξ(x_v), E⁰(x_v))≤d_E(x₀₎(ξ(x₀), E⁰(x₀)).

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Proposition 4.1. Let E = ((E(x))x∈X,Γ) be a field of Banach spaces, E⁰= ((E⁰(x))x∈X,Γ⁰) a subfield of E. Then E⁰ is a strong closed subfield of E if and only if the function x→d(ξ(x), E⁰(x))is continuous for every ξ ∈Γ.

Proof. It follows from the definition of the fact that E⁰ is a strong closed subfield of E, which is equivalent to the fact that the function x → d(ξ(x), E⁰(x)) is lower semicontinuous, and from Lemma 4.4.

5. MAIN RESULT

Theorem5.1. Let E = ((E(x))x∈X,Γ) be a field of Banach spaces and E⁰ = ((E⁰(x))x∈X,Γ⁰) a subfield of E. Then there exists a field of Banach spaces E/E⁰ = ((E(x)/E⁰(x))x∈X,Γ/Γ⁰) if and only if E⁰ is a strong closed subfield of E. In this case there exists a commutative and exact on lines diagram, namely,

0 −→ B(Γ⁰) −→ B(Γ) −→^P^b B(Γ)/B(Γ⁰) −→ 0



 yi⁰



 yi



 ybı

0 −→ Γ⁰ −→ Γ −→^P Γ/Γ⁰ −→ 0



 yπ_x⁰



 yπ_x



 ybπ_x

0 −→ E⁰(x) −→ E(x) −→^P^x E(x)/E⁰(x) −→ 0

and B(Γ)/B(Γ⁰) is isomorphic to B(E/E⁰) troughbı. We say that there exists the exact diagram of fields 0→ E⁰→ E → E/E⁰ →0.

Proof. Let E = ((E(x))x∈X,Γ) be a field of Banach spaces and E⁰ = ((E⁰(x))x∈X,Γ⁰) a subfield of E such that E/E⁰ = ((E(x)/E⁰(x))x∈X,Γ/Γ⁰) is a field of Banach spaces. We will first prove that E⁰ is a weak closed subfield of E and then thatE⁰ is a weak closed subfield ofE.

Let us suppose to reach a contradiction that there is a closed set F and η ∈Γ⁰ such that F =

◦

F and η|_K∈Γ⁰|_K andη|_F ∈/Γ⁰|_F for every compact set K ⊂F^◦.

Since a subset which contain a single point is compact, we haveπb_x(η) = P_x(η(x)) = 0 for every x ∈ F^◦. Because η|_F ∈/ Γ⁰|_F, there exists x₀ ∈ ∂F such that πbx0(η) = Px0(η(x0)) 6= 0 (by Lemma 1.1). It follows that the functionx→ kπb_x(η)kb is not continuous at x₀, which contradicts the fact that E/E⁰ = ((E(x)/E⁰(x))x∈X,Γ/Γ⁰) is a field.

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Let η ∈ Γ, ε > 0, M > 0, and a convergent net (xν)ν, xν → x0

be such that there exists ξ_ν ∈ Γ⁰ for which kη(x_ν)−ξ_ν(x_ν)k ≤ M. Then d_E(x_v₎(η(xν), E⁰(xv)) ≤ kη(x_ν)−ξν(xν)k ≤ M. It follows that d_E(x₀₎(η(x0), E⁰(x0))≤M. This implies that there existsξ ∈Γ⁰such thatkη(x₀)−ξ(x0)k ≤ M +ε.

LetE = ((E(x))x∈X,Γ) be a field of Banach spaces andE⁰= ((E⁰(x))x∈X, Γ⁰) a strong closed subfield of E, that is, for every η ∈Γ, ε >0,M >0, and every convergent net (xν)ν,xν →x0, such that there exists ξν ∈Γ⁰ for which kη(x_ν)−ξ_ν(x_ν)k ≤M, there existsξ∈Γ⁰ withkη(x₀)−ξ(x₀)k ≤M +ε.

For everyx ∈X there exists a commutative diagram which is exact on lines, namely,

0 −→ B(Γ⁰) −→ B(Γ) ^P

b

−→ B(Γ)/B(Γ⁰) −→ 0



 yi⁰



 yi



 ybı

0 −→ Γ⁰ −→ Γ −→^P Γ/Γ⁰ −→ 0



 yπ_x⁰



 yπx



 ybπx

0 −→ E⁰(x) −→ E(x) −→^P^x E(x)/E⁰(x) −→ 0 (see Section 1 for the notation in the diagram).

Let also T : Γ/Γ⁰ → (E(x)/E⁰(x))x∈X = Q

x∈X

E(x)/E⁰(x) be given by T(bη) = (bπx(bη))x∈X andT^b =T◦bı:B(Γ)/B(Γ⁰)→ Q

x∈X

E(x)/E⁰(x). We study the properties of T and prove thatE/E⁰ is a field (by checking Definition 1.2).

1. T is injective: T(ηb₁) =T(ηb₂) impliesbπ_x(ηb₁) =bπ_x(ηb₂) for everyx∈X.

That is,bπ_x(η\₁−η₂) = 0 for everyx∈X. This means that (η₁−η₂)(x)∈E⁰(x) for every x∈X, which shows thatη1−η2∈Γ⁰ (by Lemma 1.1).

2. We havekηk ≥ kb bπ_x(η)k for everyx∈X and everyη∈B(Γ), that is, T is bounded onB(Γ). Indeed,

kηkb = inf

ξ∈B(Γ⁰)

kη−ξk ≥ inf

ξ∈B(Γ⁰)

k(η−ξ)(x)k= inf

α∈E⁰(x)

kη(x)−αk=kbπ_x(η)k. 3. kηkb = sup

x∈X

kπb_x(η)kb for everyη ∈B(Γ), that is,T is an isometric morphism ofB(Γ) into the Banach space of bounded elements from Q

x∈X

E(x)/E⁰(x).

To reach a contradiction we suppose that there existsε >0 such thatkπbx(η)k+b ε≤ kηkb for everyx∈X. Let us remark thatkbηk= inf

ξ∈B(Γ)kη−ξk. For every x ∈ X there exists ηx ∈ B(Γ⁰) such that k(η_x−η)(x)k < kbπx(η)kb +ε/2 ≤

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kbηk − ε/2. By Proposition 1.1, point 3, there exists ξ ∈ B(Γ⁰) such that kη−ξk<kηk −b ε/3. Then kηk ≤ kb ηk −b ε/3, which is a contradiction.

4. The function x → kbπx(η)kb is upper semicontinuous, that is, lim sup

xν→x₀

kπbxν(η)k ≤ kb bπx0(η)kb (for the proof see also Lemma 4.4). Letε >0 be fixed. There existsη_ε ∈B(Γ⁰) such thatk(η_ε −η)(x)k<kbπ_x₀(η)k+ε/2 . Thenb there exists Vx ∈ V_x such thatk(η_ε−η)(y)k<kbπx0(bη)k+εfor every y∈Vx. So,kbπ_y(bη)k ≤ kbπ_x₀(η)k+εb for everyy∈V_xand lim sup

xν→x0

kπb_x_ν(bη)k ≤ kbπ_x₀(η)k+εb for every ε >0. It follows that lim sup

xν→x0

kπb_x_ν(bη)k ≤ kbπ_x₀(η)k.b

5. The functionx→ kbπ_x(η)kb is continuous (Definition 1.3). This results from point 4 and Proposition 1.4 because kbπx(η)kb =d_E(x)(ξ(x), E⁰(x)).

6. Letξ ∈ Q

x∈X

E(x)/E⁰(x) be such that for every ε >0 and everyx∈X there exist η_x,ε ∈ Γ and V_x,ε ∈ V_x∩τ such that kηb_x,ε(y)−ξ(y)k < ε. By replacing Vx,ε by a smaller open neighborhood of x, one can suppose that kη_x,ε(y)k <kξ(y)k+ 2ε for every y ∈ V_x,ε; (V_x,ε)x∈X is an open cover of X.

There exists an open local finite cover (Vxi,ε)i∈I of X. Let φi be a partition of unity subordinated to it. Let η = P

i∈I

φiηxi,ε ∈ Γ. Then kbη−ξk < ε and kη(y)k<kξ(y)k+ 2εfor everyy∈X.There areηb_nsuch thatkηb_n−ξk<1/2ⁿ. Then kηbn−ηbn+1k ≤ kηbn−ξk+kηbn+1−ξk<1/2ⁿ+ 1/2ⁿ⁺¹ = 3/2ⁿ⁺¹.Next, there areξn∈B(Γ⁰) such thatkη_n−ηn+1−ξn+1k<1/2ⁿ⁺¹+kηbn−ηbn+1k<

1/2ⁿ⁻¹.Hence P

n≥2

kη_n−ηn−1−ξ_nk<∞. Letµ_n=η₁+ P

k=2,n

η_k−η_k−1+ξ_kand µ=η1+ P

n≥2

ηn−ηn−1+ξn. Thenµ= lim

n→∞µn inB(Γ) andkµbn−ξk= bη1+ P

k=1,n

ηk+1\−ηk+ξk−ξ =

bη1+ P

k=1,n

ηbk+1−ηbk−ξ

=kbηn+1−ξk<1/2ⁿ⁺¹. So, kµb−ξk= 0.It remains to prove thatB(Γ)/B(Γ⁰) is isometric isomorphic to B(Γ/Γ⁰). Note that T^b = T ◦bı : B(Γ)/B(Γ⁰) → Q

x∈X

E(x)/E⁰(x) is an isometric morphism. Let ξ ∈B(Γ). The fact that kbπx(ξ)k ≤ kξ(x)k implies that T(ξ)∈B(Γ/Γ⁰). We have thus proved thatT^b(B(Γ)/B(Γ⁰))⊂B(Γ/Γ⁰).

Let ξb∈ B(Γ/Γ⁰). Then d(ξ(x), E⁰(x)) → 0 as x → ∞. By Proposition 1.1, point 3, there exists η∈Γ⁰ such that d(ξ(x), η(x))→0 asx → ∞. It follows thatξ−η∈B(Γ) . This completes the proof becauseξb=ξ[−η=P(ξ−η).

Proposition5.1. Let X be a locally compact and paracompact topological space, E = ((E(x))x∈X,Γ)a field of Banach spaces, E⁰ = ((E⁰(x))x∈X,Γ⁰) a strong closed subfield of E and E/E⁰ = ((E(x)/E⁰(x))x∈X,Γ/Γ⁰) the factor space. Let F be a closed set in X. Then (E|_F)/(E⁰|_F) is isomorphic to

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(E/E⁰)|_F and there exists the exact on lines commutative diagram 0 −→ B(Γ⁰) −→ B(Γ) −→^P^b B(Γ)/B(Γ⁰) −→ 0



 yi⁰



 yi



 ybı

0 −→ Γ⁰ −→ Γ −→^P Γ/Γ⁰ −→ 0



 yπ⁰_F



 yπ_F



 ybπ_F

0 −→ Γ⁰|_F −→ Γ|_F −→^P^F (Γ|_F)/(Γ⁰|_F) −→ 0.

Proof. All the follow result from the Theorem 5.1 and the existence and the properties of bπ_F. The existence of bπ_F follows from the fact that if ξ ∈Γ and P(ξ) = 0, then ξ ∈ Γ⁰, which implies that ξ|_F ∈ Γ⁰|_F and PF(ξ|_F) = 0.

x∈F

kπb_F,x(ξ(x))k

= sup

x∈F

kbπ_x(ξ(x))k=kP(ξ)|_Fkfor every ξ∈Γ.

It remains to prove thatπbF(B(Γ/Γ⁰)) =B(Γ|_F/Γ⁰|_F). Letξ∈Γ be such thatP_F(ξ|_F)∈B(Γ|_F/Γ⁰|_F). That is,d(ξ(x), E⁰(x))→0 asx→ ∞on F. By Proposition 1.1 there existsη ∈Γ⁰such thatd(ξ(x), η(x)) =kξ(x)−η(x)k →0 asx→ ∞onF. By Corollary 1.2 there exists ˜ξ ∈B(Γ) such that ˜ξ|_F =ξ−η.

Then P_F( ˜ξ|_F) =P_F(ξ−η|_F) =P_F(ξ|_F).

REFERENCES

[1] J.L. Kelly,General Topology. D. Van Nostrand Co. Inc., Princeton, NJ, 1955.

[2] J. Dixmier,C^∗-algebras.North-Holland Publishing Co, 1982.

[3] A. Mihail,On inductive limits of fields ofC^∗-algebras. Rev. Roumaine Math. Pures Appl.

47(2002),3, 349–371.

Received 21 May 2007 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest 1, Romania