The Banach space D(0, 1) is primary

Seria I: PRACE MATEMATYCZNE XLV (1) (2005), 107-124

Artur Michalak^∗

The Banach space D(0, 1) is primary

Abstract. We show that the Banach space D(0, 1) of all scalar (real or complex) functions on [0, 1) that are right continuous at each point of [0, 1) with left-hand limit at each point of (0, 1] equipped with the uniform convergence topology is primary.

1991 Mathematics Subject Classification: 46B20, 46B25, 46E15.

Key words and phrases: C(K)-spaces.

A Banach space X is said to be primary if for every continuous projection P : X → X, either P (X) or (I − P )(X) is isomorphic to X. Many classical C(K) spaces are known to be primary, for instance: C([0, 1]), C(βN \ N), C(α) for some ordinal numbers α (see [8], [15], [5], [1] and [2]). Besides, the spaces c₀and l_∞= C(βN) are prime i.e. every infinite dimensional and complemented subspace of these spaces is isomorphic to the whole space (see [9]). We show that the Banach space D(0, 1) of all scalar (real or complex) functions on [0, 1) that are right continuous at each point of [0, 1) with left hands limit at each point of (0, 1] equipped with the uniform con- vergence topology is also a primary Banach space. The space D(0, 1) is isometrically isomorphic to the space of all continuous scalar functions on the two arrows space.

This space has appeared in the literature (see e.g. [3], [13], [16]) as an example of a Banach space with some interesting properties. Thus, for example, H. Corson [3]

showed that the quotient space D(0, 1)/C([0, 1]) is isomorphic to c₀([0, 1]) and also that the space D(0, 1) is not normal in its weak topology. There exists a countable set of evaluation functionals separating points of D(0, 1). Consequently, it contains no isomorphic copy of c₀(Γ) for any uncountable set Γ. Every isometric copy of c₀in D(0, 1) is complemented (see [13]). Moreover, D(0, 1) contains no isomorphic copy of l∞ (see [4]).

The present paper continues research on properties of the space D(0, 1) started in [10] and continued in [11]. The space plays a crucial rule in the description of

∗The research was supported by Komitet Bada´n Naukowych (State Committee for Scientific Research), Poland, grant no. P 03A 022 25.

Banach spaces X for which there exists an increasing function from [0, 1] into X with uncountable set of points of discontinuity (see [10]). Other properties of the space were studied in [3], [13] and [16]. The paper is devoted to the proof of the primariness of D(0, 1). A partial result in this direction was shown by W. Patterson in [13]. She showed that whenever D(0, 1) has Banach direct sum decomposition D(0, 1) = X ⊕ Y where Y is isometrically isomorphic to c₀, then X is isomorphic to D(0, 1). We divide the proof of primariness of D(0, 1) into a few steps. Two of them seem to be crucial. The first one is to show that for every continuous linear operator J : D(0, 1) → D(0, 1) with nonseparable range there exists a subspace Y of D(0, 1) which is isomorphic to D(0, 1), the restriction J |_Y is an isomorphism, and there exists a closed and uncountable subset S of the two arrows space and continuous map η : S → (D(0, 1)^∗, w∗) such that sup{|η(l)(J (y))| : l ∈ S} = kyk for every y ∈ Y . The second one is to show that if J : C(S) → D(0, 1) is an isometric embedding where S is a closed and uncountable subset of the two arrows space, then there exists a closed subspace Z of J (C(S)) which is isomorphic to D(0, 1) and complemented in D(0, 1). This part of our considerations is closely related to a result of A. Pe lczy´nski [14, Thm. 1a].

1. Preliminaries. Throughout, the paper X will be a real or complex Banach space, X^∗its topological dual, B_X its closed unit ball and w∗ the weak∗ topology on X^∗. If K is a compact Hausdorff space, then C(K) denotes the Banach space (under the sup norm) of all continuous scalar functions on K. The Dirac measure on K concentrated at a point k ∈ K will be denoted by δ_k. If A is a subset of a normed space, lin(A) and lin(A) = lin(A) are the linear span and the closed linear span of A, respectively. If I is an isomorphic embedding of a Banach space X into a Banach space Y , then by I⁻¹ we denote the linear operator defined on I(X) with values in X such that I⁻¹(I(x)) = x for every x ∈ X.

We denote by L the space of all nondecreasing functions x : [0, 1] → {0, 1}

equipped with the pointwise convergence topology. The space L consists of functions y^γ₁ = χ_[γ,1] and y^γ₀ = χ_(γ,1] for γ ∈ [0, 1]. It has two isolated points, the constant functions y₀¹ = 0 and y₁⁰ = 1. The classical two arrows space is homeomorphic to L0 = L \ {y0¹, y⁰₁}. Consequently, the space L is Hausdorff, compact, sequentially compact, hereditarily separable and hereditarily Lindel¨of (see [6, p. 270]). It is easy to check that the operator J : C(L0) → D(0, 1) given by J (f )(γ) = f (y^γ₀) is an isomorphism and an isometry. We apply in our considerations so many times properties of the two arrows space that we will work with the space C(L) instead of D(0, 1).

Let Γ be a closed subset of [0, 1]. We put LΓ= {y_s^γ : γ ∈ Γ, s ∈ {0, 1}}. Then L = L[0,1]. Moreover, we define L^rΓ to be the set obtained from LΓby removing all the points y^γ₀, where γ is a right-hand side isolated point of Γ (also the point y₀¹= 0 in case 1 ∈ Γ). Observe that if γ ∈ Γ is a right-hand side isolated point of Γ, then y^γ₀ is an isolated point of LΓ. Consequently, L^rΓis a closed subset of L. Moreover, L^r_[0,1] = L \ {0} and m(LΓ) = m(L^rΓ) = Γ where m : L → [0, 1] is the continuous map given by m(y_s^γ) = γ.

The space of all scalar functions x on [0, 1] such that Var(x) = |x(0)|+var_[0,1](x) <

∞ equipped with the pointwise convergence topology will be denoted by V and its subset {x ∈ V : Var(x) 6 c} (c > 0) by cG. It is easy to check that for every c > 0 the space cG is Hausdorff, compact and sequentially compact. The last fact follows

from the Helly theorem (see [12, p. 196]). If c = 1 we briefly write G. It is clear that LΓ⊂ G. For α ∈ [0, 1] we denote by πα: V → C the function given by πα(x) = x(α).

We will use the same notation π_γ for the restrictions of π_γ to subsets of V such as cG, L, LΓ, L^rΓ. (In each case the domain should be clear from the context.) We use the following convention: if we consider real (resp., complex) Banach spaces, then G consists of real (resp., complex) functions.

Let Γ be a closed subset of [0, 1]. Then the function π_{max Γ}takes the value 1 on each element of L^rΓ. For every γ₁< γ₂< · · · < γ_nin Γ and c₁, . . . , c_n∈ C we have the following estimates:

n

X

j=1

c_jπ_γj C(L^rΓ)

>

n

X

j=1

c_jπ_γj(y^γ₁ⁱ)      

=      

n

X

j=i

c_j       .

If we consider the functionPn

i=1c_iπ_γion L and G, then

n

X

i=1

c_jπ_γj C(G)

>

n

X

i=1

c_iπ_γi C(L)

>

Xⁿ

j=1

c_j π_γ1+

n

X

i=2

Xⁿ

j=i

c_j

(π_γi− π_γi−1) C(L^r_[0,1])

= max

16i6n

n

X

j=i

c_j       .

The equality follows from the fact that in the sequence π_γ1(y_s^γ), (π_γ2− π_γ1)(y^γ_s), . . . , (π_γn− π_γn−1)(y_s^γ) of 0’s and 1’s the number 1 appears at most once for every y^γ_s ∈ L^r_[0,1]. On the other hand, for every x ∈ G

n

X

j=1

c_jπ_γj(x)      

=      

Xⁿ

j=1

c_j

 x(γ₁) +

n

X

i=2

Xⁿ

j=i

c_j



(x(γ_i) − x(γ_i−1)       6 max

16i6n





      

n

X

j=i

c_j      









|x(γ₁)| +

n

X

j=2

|x(γ_j) − x(γ_j−1|

6 max

16i6n





      

n

X

j=i

c_j      







|x(0)| + var_[0,1](x)
.

Consequently,

n

X

i=1

c_iπ_γi C(G)

=

n

X

i=1

c_iπ_γi C(L)

=

n

X

i=1

c_iπ_γi C(L^r_[0,1])

=

n

X

i=1

c_iπ_γi C(L^r_Γ)

= max

16i6n

n

X

j=i

c_j       .

Moreover, for every γ₁< γ₂< · · · < γ_nin [0, 1] and c₁, . . . , c_n, c_n+1∈ C we have the following equality:

c_n+1+

n

X

i=1

c_iπ_γi C(L)

= max

16i6n+1

n+1

X

j=i

c_j       .

The above estimates show that the family {π_γ : γ ∈ Γ} consists of linearly inde- pendent functions on L^rΓ. It is clear that the function π_γ separates points y₀^γ and y^γ₁. If α, β ∈ Γ, α < β, and α is not a right-hand side isolated point in Γ, then there exists γ ∈ Γ such that α < γ < β and the function π_γ separates points y_s^α and y_r^β for every r, s ∈ {0, 1}. If α is a right-hand side isolated point in Γ, then the function π_α separates points y₁^α and y^β_s for every s ∈ {0, 1}. We have thus verified that the family of functions {π_γ: γ ∈ Γ} separates the points of L^rΓfor every closed subset Γ of [0, 1]. In the case of the space L it is obvious that the family of functions {π_γ: γ ∈ [0, 1]} separates its points. Moreover, on L,

π_γ1π_γ2= π_min{γ1,γ2} for all γ₁, γ₂∈ [0, 1].

Hence, by the Stone-Weierstrass theorem, for every closed subset Γ of [0, 1] the closed linear span of {π_γ : γ ∈ Γ} in C(L^rΓ) coincides with C(L^rΓ), and the closed linear span of {π_γ: γ ∈ [0, 1]} ∪ {1} in C(L) coincides with C(L).

In the sequel we will apply the following consequence of the estimates above.

Let Γ₁, Γ₂, Γ₃, Γ₄be subsets of [0, 1] such that Γ₂ and Γ₄are closed, Γ₁⊂ Γ₂and Γ₃⊂ Γ₄. The estimates above show that if ϕ : Γ₁ → Γ₃ is an increasing function, then the linear operator

M_ϕ: lin{π_α: α ∈ Γ₁} → lin{π_α: α ∈ Γ₃}

given by M_ϕ(π_γ) = π_ϕ(γ) has norm kM_ϕk 6 1. Hence, it has an extension to a continuous linear operator

M_ϕ: lin{π_α: α ∈ Γ₁} → lin{πα: α ∈ Γ₃},

where the closures are taken in C(L^rΓ2) and C(L^rΓ4), respectively. If additionally ϕ is injective, M_ϕis an isometry. If ϕ is increasing and surjective, then by selecting an element ψ(γ) in ϕ⁻¹({γ}) for each γ ∈ Γ₃we obtain a strictly increasing function ψ : Γ₃ → Γ₁ and, consequently, the subspace lin({π_ψ(γ) : γ ∈ Γ₃}) of C(L^r_Γ2) is isometric to the subspace lin({π_γ : γ ∈ Γ₃}) of C(L^r_Γ4). If ϕ : Γ₁→ Γ₃is a strictly decreasing function, then for every γ₁< γ₂< · · · < γ_nin Γ₁and c₁, . . . , c_n∈ C we have the following estimates:

n

X

i=1

c_iπ_ϕ(γi) C(L)

= max

16i6n

i

X

j=1

c_j      

= max

16i6n

n

X

j=1

c_j−

n

X

j=i+1

c_j       6 2

n

X

i=1

c_iπ_γi C(L)

. Similar we show that

n

X

i=1

c_iπ_γi C(L)

6 2

n

X

i=1

c_iπ_ϕ(γi) C(L)

.

Consequently, if ϕ is strictly decreasing, then the operator M_ϕ: lin{π_α: α ∈ Γ₁} → lin{π_α: α ∈ Γ₃} given by M_ϕ(π_α) = π_ϕ(α)is an isomorphism where the closures are taken in C(L^rΓ2) and C(L^rΓ4), respectively.

We denote by C₀(L^rΓ) the subspace of C(L^rΓ) consisting of all x ∈ C(L^rΓ) such that x(y₁^{min Γ}) = 0. A functionPn

i=1g_iπ_γi for γ₁, . . . , γ_n∈ Γ and scalars g₁, . . . , g_n is a member of C₀(L^rΓ) if and only ifPn

i=1g_i= 0. If x ∈ C₀(L^rΓ), y =Pn

i=1g_iπ_γifor γ₁, . . . , γ_n∈ Γ and scalars g₁, . . . , g_nand kx − yk < ε, then |x(y^{min Γ}₁ ) − y(y₁^{min Γ})| = 

 Pn

i=1g_i

< ε. Consequently, kx − y − (Pn

i=1g_i)π_{min Γ}
k < 2ε. It shows that the family of functionsPn

i=1g_iπ_γi wherePn

i=1g_i= 0 and γ₁, . . . , γ_n∈ Γ forms a dense subset of C₀(L^rΓ). Let P_Γ, P : C(L^rΓ) → C(L^rΓ) be the linear operators given by P_Γ(π_α) = π_α− π_{min Γ} and P (π_α) = π_α− 1, respectively, for every α ∈ Γ. Then for every γ₁< γ₂< · · · < γ_nin Γ and scalars g₁, . . . , g_nwe have the following estimates:

P_Γ(

n

X

i=1

g_iπ_γi) C(L^rΓ)

=



−

n

X

i=1

g_i



π_{min Γ}+

Xⁿ

i=1

g_iπ_γi

 C(L^rΓ)

=







max_26i6n   

Pn j=ig_j

 if γ₁= min Γ

max n

|0|, max_16i6n  

Pn j=ig_j

   o

if γ₁> min Γ 6

n

X

i=1

g_iπ_γi C(L^rΓ)

and

P (

n

X

i=1

g_iπ_γi) C(L^rΓ)

=

Xⁿ

i=1

g_iπ_γi

−Xⁿ

i=1

g_i π_{max Γ}

C(L^rΓ)

= max

16i6n

n

X

j=i

g_j−

n

X

i=1

g_i      

6 2

n

X

i=1

g_iπ_γi C(L^rΓ)

.

Consequently, P_Γ and P are projections onto C₀(L^rΓ) and kP_Γk 6 1 and kP k 6 2.

We gather basic properties of the spaces C(L^rΓ) in the following Proposition 1.1 With the above notations we have

a) For every closed subset Γ of [0, 1] there exists an isometry E_Γ: C(L^rΓ) → C(L^r[0,1]) such that E_Γ(π_γ) = π_γ for every γ ∈ Γ. Moreover, the opera- tor E_Γ◦ R_Γ: C(L^r_[0,1]) → C(L^r_[0,1]) is a projection onto E_Γ(C(L^r_[0,1])), where RΓ: C(L^r_[0,1]) → C(L^rΓ) is the restriction operator given by R_Γ(x) = x|_L^r

Γ (all isometries considered in the paper are linear).

b) For every c > 0 there exists a continuous linear operator E : C(L^r[0,1]) → C(cG) such that E (π_α) = π_α for every α ∈ [0, 1].

c) The space C(L^rΓ1t L^rΓ2) is isometric to a subspace of C(L^r[0,1]) for every closed subsets Γ₁, Γ₂of [0, 1] (where t denotes the disjoint sum).

d) The space L C(L0)

c0 is isomorphic to C(L0).

e) The spaces C(L), C(L^r_[0,1]) and C(L0) are isomorphic.

f ) The spaces C(LΓ) and C₀(L^rΓ) are isomorphic to C(L) for every closed un- countable subset Γ of [0, 1].

g) For every closed uncountable subset Γ of [0, 1], the space C₀(L^rΓ) contains a subspace isometric to C(L^rT) for every closed subset T of Γ \ {min Γ}.

Proof Applying facts preceding the proposition one can easily shows parts a) and b).

c) Let φ₁: Γ₁ → [0,¹₄] and φ₂: Γ₂ → [¹₂, 1] be strictly increasing continuous functions. The map Φ : L^rΓ1t L^r_Γ2→ L^r_[0,1] given by

Φ(y_s^γ) =

(ys^φ1(γ) if y_s^γ∈ L^r_Γ1 ys^φ2(γ) if y_s^γ∈ L^rΓ2

is a continuous injection. Consequently, the space C(L^rΓ1 t L^r_Γ2) is isometric to C(L^r_{φ1(Γ1)∪φ2(Γ2)}). According to a) the last space is isometric to a subspace of C(L^r[0,1]).

The proofs of part d) the reader may find in [13, Lemma 3.1] and [11, Prop. 1].

But for the sake of completeness we present it.

Let L₀([a, b]) = L[a,b]\ {y^a₁, y₀^b}. Then L₀([a, b]) is closed and open in L and homeomorphic to L0 = L \ {y⁰1, y¹₀}. Since L₀ = L₀([0,¹₂]) t L₀([¹₂, 1]), the space C(L0) is isometrically isomorphic to C(L0)⊕C(L0), where the direct sum is equipped with the max norm. The subsets L₀([2⁻ⁿ, 2⁻ⁿ⁺¹]) (n ∈ N) of L are pairwise disjoint, each of them is closed-open and homeomorphic to L0, and their union is L0\ {y₀⁰}.

From this it follows easily that the space L C(L0)

c0 is isometrically isomorphic to the subspace {x ∈ C(L0) : x(y₀⁰) = 0} of C(L0). Applying now the Pe lczy´nski decomposition method (see [9, p. 54]) one gets that the spaces L C(L0)

c0 and C(L0) are isomorphic.

e) From the proof above it is seen that C(L0) ∼ C(L0) ⊕ Y , where Y is the space of scalars, and ∼ stands for “is isomorphic to”. Consequently,

C(L) ∼ C(L0) ⊕ Y ⊕ Y ∼ C(L0) ⊕ Y ∼ C(L0) ∼ C(L^r[0,1]).

f) For every closed uncountable subset Γ of [0, 1] there exists continuous non- decreasing surjection ϕ : Γ → [0, 1]. It is the well known fact but for the sake of completeness we give a sketch of the construction of the function. Let P = {₂^kn : n ∈ N, k = 0, . . . , 2ⁿ}. By induction we construct a strictly increasing function σ : P → Γ such that σ(0) = min Γ, σ(1) = max Γ and the interval (σ(s), σ(t)) con- tains uncountably many elements of the set Γ for every s, t ∈ P , s < t. Then we define the function ϕ : Γ → [0, 1] by ϕ(γ) = sup{t ∈ P : σ(t) 6 γ}.

Let ψ(γ) = min{ϕ⁻¹({γ})}. By estimates preceding the proposition the linear operator M_ϕ: lin{π_ψ(γ) : γ ∈ [0, 1]} → C(L^r[0,1]) given by M_ϕ(π_γ) = π_ϕ(γ) is an isomorphic embedding where the closure is taken in C(L^rΓ). Since ϕ(Γ) = [0, 1] and lin{π_γ : γ ∈ [0, 1]} = C(L^r_[0,1]), M_ϕ(C(L^rΓ)) = C(L^r_[0,1]). Similar we show that the linear operator P : C(L^rΓ) → lin{π_ψ(γ) : γ ∈ [0, 1]} given by P (π_γ) = π_ψ(φ(γ)) for every γ ∈ Γ is a continuous projection. Thus we show that C(L^r_[0,1]) is isomorphic

to a complemented subspace lin{π_ψ(γ): γ ∈ [0, 1]} of C(L^rΓ). Applying a), d), e) and the Pe lczy´nski decomposition method we get that the space C(L^rΓ) is isomorphic to C(L^r[0,1]) for every closed uncountable subset Γ of [0, 1]. In view of e) also C₀(L^rΓ) is isomorphic to C(L^r[0,1]) for every closed uncountable subset Γ of [0, 1] (any two subspaces of C(L^rΓ) of codimension 1 are isomorphic).

g) Let T be a closed subset of Γ \ {min Γ}. For every α ∈ T π_α(y₁^{min Γ}) = 1.

Consequently, π_α− π_{min Γ} is a member of C₀(L^rΓ) for every α ∈ T . Let I : C(L^rT) → C₀(L^rΓ) be the linear operator given by I(π_α) = π_α− π_{min Γ} for every α ∈ T . For every γ₁< γ₂< · · · < γ_n in T and g₁, . . . , g_n∈ C we have the following equalities:

I(

n

X

i=1

g_iπ_γi) C(L^rΓ)

=



−

n

X

i=1

g_i



π_{min Γ}+

Xⁿ

i=1

g_iπ_γi

 C(L^rΓ)

= max





     

n

X

i=1

g_i−

n

X

i=1

g_i     

, max

16i6n





      

n

X

j=i

g_j      













=

n

X

i=1

g_iπ_γi C(L^rT)

.

Thus we show that I is an isometry. _

Let Γ be a closed subset of [0, 1]. For a function f : Γ → C we define

var_Γ(f ) = sup ( _n

X

k=0

|f (α_k+1− f (α_k)| : α₀< · · · < α_n, α₀, . . . , α_n∈ Γ, n ∈ N )

and Var_Γ(f ) = |f (min Γ)| + var_Γ(f ). We say that a function f : Γ → X has bounded variation on Γ with respect to a subset E of X^∗ if Var_Γ,E(f ) = sup{Var_Γ(x^∗◦ f ) : x^∗∈ E} < ∞. We will usually assume that E is a norming subset of X^∗(i.e. there exist constants C > c > 0 such that ckxk 6 sup{|x^∗(x)| : x^∗∈ E} 6 Ckxk for every x ∈ X). The relation between continuous linear operators on C(L^rΓ) and functions of bounded variation on Γ describes the following theorem. For the case Γ = [0, 1]

analogue result was shown in [11, Prop. 3].

Proposition 1.2 Let X be a Banach space and Γ a closed subset of [0, 1].

a) For every continuous linear operator S : C(L^rΓ) → X the function f : Γ → X given by

f (α) = S(π_α) for every α ∈ Γ

has bounded variation with respect to B_X∗ and Var_Γ,BX∗(f ) 6 kSk.

b) For every function f : Γ → X of bounded variation with respect to a norming subset E of X^∗there exists a continuous linear operator S : C(L^rΓ) → X such that

f (α) = S(π_α) for every α ∈ Γ

and kSk 6¹_cVar_Γ,E(f ) where c = inf{sup{|x^∗(x)| : x^∗∈ E} : kxk = 1}.

Proof a) For every γ1< γ₂< · · · < γ_n, γ₁, . . . , γ_n∈ Γ and x^∗∈ X^∗we have the following estimates

|x^∗(f (γ₁))| +

n−1

X

i=1

|x^∗(f (γ_i+1)) − x^∗(f (γ_i))|

6 max

|εi|=1

(    

S^∗(x^∗) ε₀π_γ1+

n−1

X

i=1

ε_i(π_γi+1− π_γi)
     )

6 kS^∗k kx^∗k max

|εi|=1{|ε_i|} 6 kS^∗k kx^∗k .

b) Since the family {π_α: α ∈ Γ} consists of linearly independent functions on L^rΓ, the linear operator S : C(L^rΓ) → X given by S(π_α) = f (α) is well defined. Moreover, for every γ₁< γ₂< · · · < γ_n, γ₁, . . . , γ_n∈ Γ, g₁, . . . , g_n∈ C we have the following estimates

S

n

X

i=1

g_iπ_γi

!

6 1 c sup

x^∗∈E

x^∗ S

n

X

i=1

g_iπ_γi

!!

    6 1

c sup

x^∗∈E





     

Xⁿ

i=1

g_i



x^∗(f (γ₁))     

+      

n

X

i=2

Xⁿ

j=i

g_j



(x^∗(f (γ_i) − f (γ_i−1)))      





 61

c

n

X

i=1

g_iπ_γi C(L^rΓ)

Var_Γ,E(f ).

Thus we show that S is a continuous linear operator on the closed linear hull of {π_α: α ∈ Γ} in C(L^rΓ). The last subspace coincides with C(L^rΓ). _

2. Main results.

Theorem 2.1 Let Γ be a closed and uncountable subset of [0, 1] and A a closed subset of LΓ. If Φ : A → L is a continuous map, then for every closed and uncount- able subset B of Φ(A) there exists a closed and uncountable subset T of Γ and a continuous strictly monotonic map ϕ : T → [0, 1] such that

a) LT ⊂ A,

b) Φ(LT) = Lϕ(T )⊂ B, c)

Φ(y_s^τ) =

(y_s^{ϕ(τ )} if ϕ is increasing y_1−s^{ϕ(τ )} if ϕ is decreasing for every τ ∈ T and s ∈ {0, 1}.

Moreover, there exists a continuous strictly monotonic map ψ : ϕ(T ) → T such that ψ(ϕ(τ )) = τ for every τ ∈ T . Consequently, there exists a continuous map Ψ : Lϕ(T )→ LT such that Ψ(Φ(y_s^τ)) = y_s^τ for every τ ∈ T and s ∈ {0, 1}.

In the proof of the theorem above we will need the following properties of L:

1) (y^γ_sⁿ

n) converges to y₁^γ0, if (γ_n) is a strictly increasing sequence converging to γ₀ and s_n∈ {0, 1},

2) (y_s^γn

n) converges to y₀^γ0, if (γ_n) is a strictly decreasing sequence converging to γ₀ and s_n∈ {0, 1}.

Proof Let K = Φ⁻¹(B). Let m : L → [0, 1] be the continuous map given by m(y^γ_s) = γ. Then m(K) is a compact and uncountable subset of [0, 1]. Let T₁ be the set obtained from m(K) by removing all the points γ, where γ is a right- hand side or a left-hand side isolated point of m(K). It is clear that m(K) \ T₁ is a countable set. Consequently, T₁ is a G_δ subset of [0, 1]. Every element of T₁ is a cluster point of a strictly increasing sequence of elements of m(K) and also it is a cluster point of a strictly decreasing sequence of elements of m(K). Applying properties 1) and 2) of L we get that {y^γs : γ ∈ T₁, s ∈ {0, 1}} ⊂ K. For every ε > 0 we put S_ε = {γ ∈ T₁ : |m(Φ(y₁^γ)) − m(Φ(y^γ₀))| > ε}. The set Sε is finite.

Otherwise, we find a monotonic sequence (γ_n) ⊂ S_ε, but then lim y₀^γn= lim y₁^γn and

|m(Φ(y₁^γn)) − m(Φ(y₀^γn))| > ε}, it contradicts the continuity of m ◦ Φ. Consequently, we may define ϕ(γ) = m(Φ(y_s^γ)) for every γ ∈ T₂ = T₁\S∞

n=1S¹

n and s ∈ {0, 1}.

If a strictly monotonic sequence (γ_n) ⊂ T₂ converges to γ ∈ T₂ and (s_n) ⊂ {0, 1}, then sequences (m(Φ(y_s^γ_nⁿ))) converges to m(Φ(y^γ₁)) = m(Φ(y₀^γ)). Consequently, ϕ is a continuous map on T₂.

Applying 1) and 2) and continuity of Φ for every γ ∈ T₂ we find ε > 0 such that either ϕ(α) − ϕ(γ) 6 0 or ϕ(α) − ϕ(γ) > 0 for every α ∈ (γ − ε, γ] ∩ T2. Otherwise, we are able to find an increasing sequence (α_n) ⊂ T₂converging to γ such that ϕ(α_2n+1) is a strictly increasing sequence and ϕ(α_2n) is a strictly decreasing sequence and lim_nΦ(y^α₀²ⁿ⁺¹) = y₁^ϕ(γ)6= y^ϕ(γ)₀ = lim_nΦ(y₀^α2n). For every n ∈ N let C_n = {γ ∈ T₂: ϕ(α) − ϕ(γ) 6 0, for all α ∈ (γ −_n¹, γ] ∩ T₂} and Dn = {γ ∈ T₂: ϕ(α) − ϕ(γ) > 0, for all α ∈ (γ −_n¹, γ] ∩ T₂}. If a sequence (γ_n) ⊂ C_k converges to γ ∈ T₂, then ϕ(α) − ϕ(γ_n) 6 0 for every α ∈ (γ −_k¹, γ) ∩ T₂and n big enough.

Applying continuity of ϕ at γ we get that ϕ(α)−ϕ(γ) 6 0. Thus we show that Ckis a closed subset of T₂for every k ∈ N. Similar we show that also Dkis a closed subset of T₂for every k ∈ N. Consequently, Ckand D_kare G_δsubsets of [0, 1] for every k ∈ N.

Moreover, T₂=S∞

n=1C_n∪ D_n. It is clear that the set Φ({y^γ_s : γ ∈ T₂, s ∈ {0, 1}}) is an uncountable subset of B; only a countable number of elements of B has been removed. Hence, the set ϕ(T₂) is uncountable. Consequently, at least one of the sets ϕ(C_n), ϕ(D_n) is uncountable. Without lost of generality we may assume that there exists an uncountable set C_nsuch that also the set ϕ(C_n) is uncountable. We find c, d ∈ [0, 1] with 0 < d − c < _n¹ such that ϕ|_Cn∩[c,d] is an increasing function and the set ϕ(C_n∩ [c, d]) is uncountable. It is clear that the set C_n∩ [c, d] is a G_δ subset of [0, 1]. Consequently, it is homeomorphic to a complete separable metric space. Hence, ϕ|_Cn∩[c,d] is a continuous map between complete separable metric spaces with uncountable range. By theorem of Kuratowski (see [7, p. 351]) there exists a subset T of C_n∩ [c, d] homeomorphic to the Cantor set such that ϕ|T is a homeomorphism. Since ϕ is increasing on T , the map ϕ|_T is strictly increasing. Let ψ : ϕ(T ) → T be given by ψ(ϕ(t)) = t. It is clear that ψ is a continuous strictly increasing function. Applying once again facts 1) and 2) we see that Φ(y_s^τ) = ys^{ϕ(τ )}

(if ϕ|_T would be decreasing, then Φ(y_s^τ) = y^{ϕ(τ )}_1−s) and Φ(LT) = Lϕ(T ). Since ψ is

continuous and strictly increasing, the map Ψ : Lϕ(T )→ LT given by Ψ(y_s^τ) = y^{ψ(τ )}s

is continuous and Ψ(Φ(y^τ_s)) = y^τ_s for every τ ∈ T and s ∈ {0, 1}. _

Remark 2.2 If we combine the first part of the proof above (for B = Φ(A)) with the theorem of Kuratowski (see [7, p. 351]) we get the following fact:if Γ is a closed and uncountable subset of [0, 1] and A is a closed subset of LΓ and Φ : A → L is a continuous map with an uncountable range, then there exists a closed subset T of Γ homeomorphic to the Cantor set such that LT ⊂ A and Φ|_LT is a homeomorphism.

It is shown in [11, Thm. 14] that every continuous linear operator J : C(L) → C(L) with nonseparable range fix a copy of C(L). Moreover, every positive continuous linear operator from C(L) into a Banach lattice X with nonseparable range also fix a copy of C(L) (it is a consequence of [10, Thm. 4c], [11, Thm. 5] and Proposition 1.2).

We need to have for any projection P on C(L) with nonseparable range a ”good”

situated copy of C(L^r_[0,1]) fixed by P . We show a little bit more that every such an operator J is an isomorphism on a subspace E_U(C₀(L^rU)) for some closed and uncountable subset U of [0, 1] where E_Uis the isometry defined in Proposition 1.1 a).

Theorem 2.3 If a bounded linear operator J : C(L^r_[0,1]) → C(L^r[0,1]) has nonsepara- ble range, then there exist closed and uncountable subsets T, U of [0, 1] such that the operator P_T◦ R_T◦ J ◦ E_U is an isomorphism between the subspace C₀(L^rU) of C(L^rU) and the subspace C₀(L^rT) of C(L^rT), where P_T: C(L^rT) → C₀(L^rT) is the projection given by P_T(π_α) = π_α− πmin T for every α ∈ T .

Proof Let f : [0, 1] → C(L^r_[0,1]) be given by f (α) = J (π_α). Since J has non- separable range, also f has nonseparable range. It follows from the fact that J (C(L^r[0,1])) ⊂ linf ([0, 1]). Let E = {δ_l : l ∈ L^r[0,1]}. Then (E, w∗) is homeo- morphic to L^r[0,1]. By Proposition 1.2 Var_[0,1],E(f ) 6 kJk. Let Φ : (E, w∗) → kJkG be the function given by Φ(δ_l)(α) = f (α)(l) for every α ∈ [0, 1] and l ∈ L^r[0,1]. It is clear that Φ is continuous. Hence, the set G_f = Φ(E) is a closed subset of kJ kG.

Let C_Φ: C(G_f) → C(L^r[0,1]) be the isometry given by C_Φ(h)(l) = h(Φ(δ_l)) for every l ∈ L^r[0,1] and h ∈ C(G_f). Then C_Φ(π_α)(l) = π_α(Φ(δ_l)) = Φ(δ_l)(α) = f (α)(l) for every l ∈ L^r[0,1] and α ∈ [0, 1]. Hence f (α) = C_Φ(π_α) for every α ∈ [0, 1]. Conse- quently, f ([0, 1]) ⊂ C_Φ(C(G_f)). Therefore, G_f is a nonmetrizable subset of kJ kG.

Otherwise, the space C(G_f) is separable and f has separable range. Let Q be a countable dense subset of [0, 1]. Let Γ₁ be the set of all γ ∈ [0, 1] \ Q such that there exist x₀, x₁∈ G_f with x₀(α) = x₁(α) for every α ∈ Q and x₀(γ) 6= x₁(γ). Let us note that at least one of functions x₀and x₁ is discontinuous at γ. The family of functions π_α for α ∈ Q ∪ Γ₁ separates points of G_f. Since G_f is nonmetrizable, the set Γ₁is uncountable. Since every scalar function of bounded variation has left and right hands limits at each point, we may find an uncountable subset Γ₂ of Γ₁, a number ε > 0 and {x^γ : γ ∈ Γ₂} ⊂ Gf such that (at least) one of the following cases holds:

1) lim_α→γ−x^γ(α) = lim_α→γ+x^γ(α) and |x^γ(γ) − lim_α→γx^γ(α)| > ε, 2) | lim_α→γ−x^γ(α) − lim_α→γ+x^γ(α)| > ε

for every γ ∈ Γ₂. For every γ ∈ Γ₂there exists δ_γ> 0 such that o_{[γ−δγ,γ+δγ]\{γ}}(x^γ) 6 ε

64, where

o_[α,β]\{γ}(f ) = lim

ε→0⁺

var_[α,γ−ε](f ) + var_[γ+ε,β](f )

for 0 6 α < γ < β 6 1. It is clear that there exists an uncountable subset Γ3of Γ₂ and δ > 0 such that δ_γ> δ for every γ ∈ Γ₃. Since [0, 1] can be covered by finitely many intervals of length ₂^δ, there exist 0 6 c < d 6 1, d − c < δ such that the set Γ₄= Γ₃∩ [c, d] is uncountable. From this point on we consider two cases.

First we show that the case 1) cannot occur. It is clear that there exists an uncountable subset Γ₅ of Γ₄ and a scalar e with |e| 6 supγ∈Γ4 Var_[0,1](x^γ) 6 kJk such that for every γ ∈ Γ₅

α→γlimx^γ(α) − e    6 ε

8.

Let ψ : C → [0, 1] be a continuous function such that ψ({z ∈ C : |z − e| 6 ^ε₄}) = 0 and ψ(C \ {z ∈ C : |z − e| > ^ε₂}) = 1. The function Ψ : G_f → [0, 1]^[c,d]given by Ψ(x) = ψ ◦ x|_[c,d]is continuous. Moreover,

Ψ(x^γ)(α) =

(1 for α = γ

0 for α ∈ [c, d] \ {γ}.

Hence, {Ψ(x^γ) : γ ∈ Γ₅} is an uncountable subset of [0, 1]^[c,d]discrete in its subspace topology. Consequently, the subset {x^γ : γ ∈ Γ₅} of Gf is discrete in its subspace topology. Since the function Φ is continuous, there exists an uncountable subset of L discrete in its subspace topology. But L is hereditarily separable (see [6, p. 270]).

We have arrived at a contradiction.

Let us consider now the case 2). Then there exist an uncountable subset Γ₅of Γ₄ and a, b ∈ C such that |a| 6 supγ∈Γ4Var_[0,1](x^γ) 6 kJk, |b| 6 supγ∈Γ4Var_[0,1](x^γ) 6 kJ k, |b − a| > ^ε₂ and | lim_α→γ−x^γ(α) − a| 664^ε, | lim_α→γ+x^γ(α) − b| 6 64^ε for every γ ∈ Γ₅. Let Γ₆ be the set obtained from the closure of Γ₅ in [0, 1] by removing all right-hand side and all left-hand side isolated point of Γ₅. It is clear that Γ₆ is an uncountable G_δ subset of [0, 1]. For every γ ∈ Γ₆ there exist a strictly increasing sequence and a strictly decreasing sequence of elements of Γ₅converging to γ. For each γ ∈ Γ₆, let x^γ₁ (resp., x^γ₀) be any cluster points of a sequence (x^γn) where (γ_n) ⊂ Γ₅is a strictly increasing (resp., decreasing) sequence converging to γ. Then points x^γ₁ and x^γ₀are members of G_f and

|x^γ₀(α) − a| 6 ₃₂^ε if c 6 α 6 γ, |x^γ₀(α) − b| 6 ₃₂^ε if γ < α 6 d,

|x^γ₁(α) − a| 6 ₃₂^ε if c 6 α < γ, |x^γ₁(α) − b| 6 ₃₂^ε if γ 6 α 6 d, o_[c,d]r{γ}(x^γ₀) 6 ₆₄^ε, o_[c,d]r{γ}(x^γ₁) 6 ₆₄^ε

for every γ ∈ Γ₆. The last fact is a straightforward consequence of the following inequality o_[c,d]\{γ}(x^γ_s) 6 supυ>η>0{lim sup_n(var_[c,γ−η]+ var_[γ+η,d])(x^γn)}, where υ = min{γ − c, d − γ}. There exists a subset Γ of Γ₆ homeomorphic to the Cantor set (see [7, p. 351]). It is clear that Γ ⊂ (c, d). Let Z be the closure of the set {x^γ_s : γ ∈ Γ, s ∈ {0, 1}} in G_f. Let θ : C → [0, 1] be a continuous function such that

{z ∈ C : |z − a| 6 ₃₂^ε} = θ⁻¹({0}) and {z ∈ C : |z − b| 6 ₃₂^ε} = θ⁻¹({1}). Let Θ : kJ kG → [0, 1]^[c,d]be given by Θ(x) = θ ◦ x|_[c,d]. It is clear that Θ is a continuous map. Moreover, Θ(x^γ_s) = y^γ_s|_[c,d] for every γ ∈ Γ and s ∈ {0, 1}. Consequently, L˜Γ = {y_s^γ|_[c,d]: y_s^γ ∈ LΓ} ⊂ Θ(Φ(E)). The subset ˜LΓof [0, 1]^[c,d] is homeomorphic to LΓ (the map y_s^γ → y_s^γ|_[c,d] from LΓ to ˜LΓ is a continuous bijection). Hence Θ(Z) = ˜LΓ Now we apply Theorem 2.1 to the map Θ ◦ Φ|_Φ−1(Z). According to Theorem 2.1 there exists a closed and uncountable subset T of [0, 1] and a continuous strictly monotonic map ϕ : T → Γ such that Θ(Φ(δ_ys^τ)) = ys^{ϕ(τ )}|_[c,d]if ϕ is increasing and Θ(Φ(δ_y^τ

s)) = y^{ϕ(τ )}_1−s|_[c,d]if ϕ is decreasing for every τ ∈ T and s ∈ {0, 1}. Let U = ϕ(T ). Let z^{ϕ(τ )}s = Φ(δ_y^τ

s) if ϕ is increasing and z^{ϕ(τ )}_1−s = Φ(δ_y^τ

s) if ϕ is decreasing for every τ ∈ T and s ∈ {0, 1}. Then points z₁^γ and z^γ₀ are members of Z ⊂ G_f and

|z₀^γ(α) − a| 6 ₃₂^ε if c 6 α 6 γ, |z₀^γ(α) − b| 6 ₃₂^ε if γ < α 6 d,

|z₁^γ(α) − a| 6 ₃₂^ε if c 6 α < γ, |z₁^γ(α) − b| 6 ₃₂^ε if γ 6 α 6 d, o_[c,d]r{γ}(z₀^γ) 6 ₆₄^ε, o_[c,d]r{γ}(z₁^γ) 6 ₆₄^ε

for every γ ∈ U . The last inequalities are consequences of the following facts. For every γ ∈ U , s ∈ {0, 1} and η > 0 such that γ + 2η, γ − 2η ∈ [c, d], z_s^γis a member of V = Z ∩ ((π_γ+η− πγ−η) ◦ Θ)⁻¹(1 − η, 1 + η). The set V is open in Z. Consequently, z^γ_s is a member of the closure of {x^γ_s : γ ∈ Γ, s ∈ {0, 1}} ∩ V . But for every member x^τ_r of the last set (then τ ∈ [γ − η, γ + η]) (var_[c,γ−2η]+ var_[γ+2η,d])(x^τ_r) 6 ₆₄^ε. Hence, (var_[c,γ−2η]+ var_[γ+2η,d])(z^γ_s) 6 ₆₄^ε. Then

J (π_α)(y^τ_s) = δ_y^τ

s(f (α)) = Φ(δ_y^τ

s)(α) = (

π_α(zs^{ϕ(τ )}) = π_ψ(α)(z^τ_s) if ϕ is increasing π_α(z_1−s^{ϕ(τ )}) if ϕ is decreasing for every α ∈ U and every τ ∈ T and s ∈ {0, 1} where ψ is related to ϕ as in Theorem 3. Hence,

R_T(J (E_U(π_α)))(y^τ_s) = (

π_α(z^{ϕ(τ )}s ) if ϕ is increasing π_α(z^{ϕ(τ )}_1−s) if ϕ is decreasing for every α ∈ U , y_s^τ ∈ L^r_T. Observe that

|πα(z_s^{ϕ(τ )}) − a| 6 ε

32 if ψ(α) < τ and s ∈ {0, 1} or ψ(α) = τ and s = 0

|π_α(z^{ϕ(τ )}_s ) − b| 6 ε

32 if ψ(α) > τ and s ∈ {0, 1} or ψ(α) = τ and s = 1 if ϕ is increasing, and

π_α(z_1−s^{ϕ(τ )}) − a    6

ε

32 if ψ(α) > τ and s ∈ {0, 1} or ψ(α) = τ and s = 1 

π_α(z^{ϕ(τ )}_1−s) − b    6

ε

32 if ψ(α) < τ and s ∈ {0, 1} or ψ(α) = τ and s = 0 if ϕ is decreasing. Let L : C(L^rU) → C(L^rT) be given by

L(π_α) =

(a + (b − a)π_ψ(α) if ϕ is increasing b + (a − b)π_ψ(α) if ϕ is decreasing