# Extreme Points of the Unit Ball in the Dual Space of Some Real Subspaces of Banach Spaces of Lipschitz Functions

Authors: Davood Alimohammadi; Hadis Pazandeh
Journal: ISRN Mathematical Analysis
 (2012)

---

## Abstract

LetX be a compact Hausdorff space, τ be a continuous involution on X and C(X,τ) denote the uniformly closed real subalgebra of C(X) consisting of all f∈C(X) for which f∘τ=f̅. Let (X,d) be a compact metric space and let Lip(X,dα) denote the complex Banach space of complex-valued Lipschitz functions of order α on (X,d) under the norm ∥f∥X,pα=max{∥f∥X,pα(f)}, where α∈(0,1]. For α∈(0,1), the closed subalgebra of Lip(X,α) consisting of all f∈Lip(X,dα) for which |f(x)-f(y)|/dα(x,y)→0 as d(x,y)→0, denotes by lip(X,dα). Let τ be a Lipschitz involution on (X,d) and define Lip(X,τ,dα)=Lip(X,dα)∩C(X,τ) for α∈(0,1] and lip(X,τ,dα)=lip(X,dα)∩C(X,τ) for α∈(0,1). In this paper, we give a characterization of extreme points of BA*, where A is a real linear subspace of Lip(X,dα) or lip(X,dα) which contains 1, in particular, Lip(X,τ,dα) or lip(X,τ,dα).


---

## Body

## 1. Introduction and Preliminaries

We letℝ,ℂ, and 𝕋={z∈ℂ:|z|=1} denote the field of real numbers, complex numbers, and the unit circle, respectively. The symbol 𝕂 denotes a field that can be either ℝ or ℂ. The elements of 𝕂 are called scalars.Let𝔛 be a normed space over 𝕂. We denote by 𝔛* and B𝔛 the dual space 𝔛 and the closed unit ball of 𝔛, respectively. If S is a subset 𝔛, let Ext(S) denote the set of all extreme points of S. Let A be a subspace of 𝔛 and φ∈A*. A Hahn-Banach extension of φ to 𝔛 is a continuous linear functional ψ∈𝔛* such that ψ|A=φ and ||ψ||=||φ||. The set of all Hahn-Banach extensions of φ to 𝔛 will be denoted by Hφ.It is easy to see that if𝔛 and 𝒴 are normed spaces over 𝕂 and T:𝔛→𝒴 is a linear isometry from 𝔛 onto 𝒴, then T is a bijection mapping between Ext(B𝔛) and Ext(B𝒴).For a complex normed space𝔛, we assume that 𝔛r denotes 𝔛, regarded as a real normed space by restricting the scalar multiplication to real numbers.Kulkarni and Limaye gave some conditions forφ∈BA* to be an extreme point of BA* in terms of the Hahn-Banach extension of φ to 𝔛 and the extreme points of B𝔛* as the following.Theorem 1.1 (see [1, Theorem  2]).
Let𝔛 be a normed space over  𝕂,A be a nonzero linear subspace of  𝔛 and  φ∈BA*. (a)
Letφ∈Ext(BA*). Then,
(1.1)Hφ∩Ext⁡(BX*)=Ext⁡(Hφ)≠∅.In particular,φ has an extension to some ψ∈Ext(B𝔛*). Further, if such an extension is unique, then φ has a unique Hahn-Banach extension to 𝔛.(b)
Assume that wheneverψ∈Ext(B𝔛*) and ψ(f)=1 for all f∈A with φ(f)=1=∥f∥, one has ψ|A=φ, then φ∈Ext(BA*).(c)
Ifφ has a unique Hahn-Banach extension ψ to 𝔛 and if ψ∈Ext(B𝔛*), then φ∈Ext(BA*).LetX be a compact Hausdorff space. We denote by C(X) the complex Banach algebra of all continuous complex-valued functions on X under the uniform norm ∥f∥X=sup{|f(x)|:x∈X}. For x∈X, consider the evaluation functional ex given by ex(f)=f(x),f∈C(X). Clearly, λex∈BC(X)* for all (x,λ)∈X×𝕋. It is well known [2, page 441] that(1.2)Ext⁡(BC(X)*)={λex:(x,λ)∈X×T}.Forx∈X and λ∈𝕋, we define the map ψx,λ:C(X)r→ℝ by ψx,λ(f)=Re(λf(x)) in fact, ψx,λ=Re(λex). Clearly, ψx,λ∈B(C(X)r)* for all (x,λ)∈X×𝕋. Kulkarni and Limaye showed [1, Proposition  3] that(1.3)Ext⁡(B(C(X)r)*)={ψx,λ:(x,λ)∈X×T},
and ψy,μ=ψx,λ if and only if y=x and μ=λ.Letτ be a continuous involution on X; that is, τ:X→X is continuous and τ∘τ is the identity map on X. The map σ:C(X)→C(X) defined by σ(f)=f¯∘τ, is an algebra involution on C(X) which is called the algebra involution induced by τ on C(X). Define C(X,τ)={f∈C(X):σ(f)=f}. Then, C(X,τ) is a uniformly closed real subalgebra of C(X) which contains 1. The real algebras C(X,τ) were first considered in [3]. For a detailed account of several properties of C(X,τ), we refer to [4].LetP={(x,λ)∈X×𝕋:τ(x)≠x}∪{(x,λ)∈X×𝕋:τ(x)=x,λ∈{-1,1}}. For each (x,λ)∈P, let φx,λ denote the restriction of ψx,λ to C(X,τ). Grzesiak obtained a characterization of the extreme points of B(C(X,τ))* in [5] and showed that φ∈Ext(B(C(X,τ))*) if and only if φ=φx,λ for some (x,λ)∈P. Further, if (x,λ),(y,μ)∈P, then φy,μ=φx,λ if and only if (y,μ)=(x,λ) or (y,μ)=(τ(x),λ¯).Kulkarni and Limaye obtained [1, Theorem  4] a characterization of Ext(BA*), where A is a nonzero real linear subspace of C(X,τ).Let(X,d) be a compact metric space. For α∈(0,1], we denote by Lip(X,dα) the set of all complex-valued functions f on X for which(1.4)pα(f)=sup⁡{|f(x)-f(y)|dα(x,y):x,y∈X,x≠y}
is finite. Then, Lip(X,α) is a complex subalgebra of C(X) containing 1 and complex Banach space under the norm(1.5)‖f‖X,pα=max⁡{‖f‖X,pα(f)}(f∈Lip(X,dα)).
For α∈(0,1), the complex subalgebra of Lip(X,dα) consisting of all f∈Lip(X,dα) for which(1.6)|f(x)-f(y)|dα(x,y)⟶0asd(x,y)⟶0,
is denoted by lip(X,dα). Clearly, lip(X,dα) is a closed linear subspace of (Lip(X,dα),∥·∥X,pα) and 1∈lip(X,dα). These Banach spaces were first studied by Leeuw in [6].Given a compact metric space(X,d), let X̃={(x,y)∈X×X:x≠y}, and let the compact Hausdorff space W be the disjoint union of X with βX̃, where βX̃ is the Stone-Cech compactification of X̃. For α∈(0,1], consider the mapping Ψα:Lip(X,dα)→C(W) defined for each f∈Lip(X,dα) by(1.7)Ψα(f)(w)={f(w)ifw∈X,(βf̃)(w)ifw∈βX̃,
where(1.8)f̃(x,y)=f(x)-f(y)dα(x,y),∀(x,y)∈X̃,
and βf̃ is the norm-preserving extension of f̃ to βX̃. Clearly, Ψα is a linear isometry from (Lip(X,dα),∥·∥X,pα) into (C(W),∥·∥W), which is called the Leeuw’s linear isometry. Therefore, Ψα(Lip(X,dα)) is a uniformly closed linear subspace of C(W). It is well known (see [2, page 441]) that(1.9)Ext⁡(BΨα(Lip(X,dα))*)⊆{λew∣Ψα(Lip(X,dα)):(w,λ)∈W×T},
where ew is the evaluation functional at w on C(W).For eachx∈X and w∈W, define the linear functionals Δx and Δ̃w in Lip(X,dα)* by Δx(f)=f(x) and Δ̃w(f)=Ψα(f)(w), respectively. Clearly, |Δx(f)|≤∥f∥X,pα and |Δ̃w(f)|≤∥f∥X,pα for all f∈Lip(X,dα). Therefore, Δx,Δ̃w∈BLip(X,dα)*. Moreover, Δ̃x=Δx for all x∈X  and Δ̃w=ewoΨα for all w∈W. Thus, we have the following result.Theorem 1.2.
Forα∈(0,1], every extreme point of  BLip(X,dα)* must be either of the form  λΔx with  (x,λ)∈X×𝕋 or of the form  λΔ̃w with  (w,λ)∈βX̃×𝕋.Roy proved the following result by using a result of Leeuw [6, Lemma  1.2].Theorem 1.3 (see [7, Lemma  1.2]).
For each(x,λ)∈X×𝕋,  λΔx is an extreme point of  BLip(X,dα)*.Jimenez-Vargas and Villegas-Vallecillos used above results and obtained a characterization of linear isometries betweenLip(X,dX) and Lip(Y,dY) in [8].A mapf:X→Y is said to be Lipschitz map from the metric space (X,dX) to the metric space (Y,dY) if there exists a constant C>0 such that dY(f(x),f(y))≤CdX(x,y) for all x,y∈X.Let(X,d) be a compact metric space. The mapping τ:X→X is called a Lipschitz involution on (X,d), if τ is a Lipschitz map from (X,d) to itself and an involution on X. Clearly, every Lipschitz involution on (X,d) is a continuous involution.Letτ be a Lipschitz involution on the compact metric space (X,d) and let σ be the algebra involution induced by τ on C(X). Clearly,(1.10)σ(Lip(X,dα))=Lip(X,dα),σ(lip(X,dα))=lip(X,dα).
We define(1.11)Lip(X,τ,dα)={h∈Lip(X,dα):σ(h)=h},lip(X,τ,dα)={h∈lip(X,dα):σ(h)=h}.
Then, the following statements hold.(i)
Lip(X,τ,dα) (lip(X,τ,dα), resp.) is a real subalgebra of Lip(X,dα) (lip(X,dα), resp.).(ii)
Lip(X,τ,dα)=Lip(X,dα)∩C(X,τ) and lip(X,τ,dα)=lip(X,dα)∩C(X,τ).(iii)
Lip(X,dα)=Lip(X,τ,dα)⊕iLip(X,τ,dα) and lip(X,dα)=lip(X,τ,dα)⊕ilip(X,τ,dα).(iv)
Lip(X,τ,dα) (lip(X,τ,dα), resp.) is a real subalgebra of C(X,τ) which contains 1 and separates the points of X.(v)
Lip(X,τ,dα) (lip(X,τ,dα), resp.) is uniformly dense in C(X,τ) (use (iv) and the Stone-Weierstrass theorem for real subalgebra of C(X,τ) [3, Proposition  1.1].(vi)
For0<α<β≤1,
(1.12)Lip(X,τ,dβ)⊆lip(X,τ,dα)⊆Lip(X,τ,dα).(vii)
There exists a constantC≥1 such that
(1.13)max⁡{‖f‖X,pα,‖g‖X,pα}≤Cα‖f+g‖X,pα,
for all f,g∈Lip(X,τ,dα).(viii)
Lip(X,τ,dα),∥·∥X,pα) is a real Banach space and lip(X,τ,dα) is its closed real subspace.The real Banach spacesLip(X,τ,dα) and lip(X,τ,dα) are called real Banach spaces of complex Lipschitz functions and first studied in [9].We give a characterization of extreme points of the unit ball in the dual space ofLip(X,dα)r, Lip(X,τ,dα) and some its real linear subspaces for α∈(0,1] in Section 2. Next, we give a characterization of extreme points of the unit ball in the dual spaces of lip(X,dα),lip(X,dα)r,lip(X,τ,dα) and some its real linear subspaces for α∈(0,1) in Section 3.


## 2. Real Linear Subspaces ofLip(X,dα) Containing 1

In the remainder of this paper, we assume thatα∈(0,1], (X,d) is a compact metric space, X̃={(x,y)∈X×X,x≠y}, βX̃ is the Stone-Cech compactification of X̃, W is the compact Hausdorff space X∪βX̃, Ψα is the Leeuw’s linear isometry from (Lip(X,dα),∥·∥X,pα) into (C(W),∥·∥W), and τ is a Lipschitz involution on (X,d).For each(w,λ)∈W×𝕋, we define the map ψ̃w,λ:Lip(X,dα)r→ℝ by ψ̃w,λ(f)=Re(λΔ̃w(f)) in fact, ψ̃w,λ=Re(λ(ewoΨα)). Clearly, ψ̃w,λ∈B(Lip(X,dα)r)* for all (w,λ)∈W×𝕋. Moreover, ψ̃x,λ=Re(λΔx) for all (x,λ)∈X×𝕋.We first give a characterization of the extreme points of the unit ball in the(Lip(X,dα)r)* as the following.Proposition 2.1.
By above notations,(2.1){ψ̃x,λ:(x,λ)∈X×T}⊆Ext⁡(B(Lip(X,dα)r)*)⊆{ψ̃w,λ:(w,λ)∈W×T}.
Further, for (x,λ),(y,μ)∈X×𝕋 one has ψ̃x,λ=ψ̃y,μ if and only if (x,λ)=(y,μ).Proof.
We define the mapT:(Lip(X,dα)*)r→(Lip(X,dα)r)* by T(φ)=Reφ. Clearly, T is a real-linear mapping. For each u∈(Lip(X,dα)r)*, defining the map φ:Lip(X,dα)*→ℂ by φ(f)=u(f)-iu(if). Clearly, φ∈Lip(X,dα)* and u=Reφ. It follows that φ∈(Lip(X,dα)*)r and T(φ)=u. Thus, T is onto.
We claim thatT is an isometric. Let φ∈(Lip(X,dα)*)r. Since
(2.2)|T(φ)(f)|=|(Reφ)(f)|=|Re(φ(f))|≤|φ(f)|≤‖φ‖‖f‖X,pα,
for each f∈Lip(X,dα)r, we have
(2.3)‖T(φ)‖≤‖φ‖.
Letε be an arbitrary positive number. There exists f∈Lip(X,dα)r with ∥f∥X,pα≤1 such that ∥φ∥<|ψ(f)|+ε. Choose γ=1 if ψ(f)=0 and γ=(1/ψ(f))|φ(f)| if ψ(f)≠0. Then, γ∈ℂ, |γ|=1 and ψ(f)=γ|ψ(f)|. If g=(1/γ)f, then g∈Lip(X,dα)r, ∥g∥X,pα=∥f∥X,pα≤1 and so,
(2.4)‖ψ‖<|Re(φ(g))|+ε=|T(φ)(g)|+ε≤‖T(φ)‖+ε.
It follows that
(2.5)‖φ‖≤‖T(φ)‖.
Thus, our claim is justified. The above arguments show thatT is a real-linear isometry from (Lip(X,dα)*)r onto (Lip(X,dα)r)*. Therefore,
(2.6)Ext⁡(B(Lip(X,dα)r)*)=T(Ext⁡(BLip(x,dα)*)).
Since
(2.7){λΔ̃x:(x,λ)∈X×T}⊆Ext⁡(BLip(X,dα)*)⊆{λΔ̃w:(w,λ)∈W×T},
we conclude that
(2.8){T(λΔ̃x):(x,λ)∈X×T}⊆Ext⁡(B(Lip(X,dα)r)*)⊆{T(λΔ̃w):(w,λ)∈W×T},
by Theorems 1.2 and 1.3.
Clearly,T(λΔ̃w)=Ψ̃λ,w for all (w,λ)∈W×𝕋. Therefore,
(2.9){Ψ̃λ,x:(x,λ)∈X×T}⊆Ext⁡(B(Lip(X,dα)r)*)⊆{Ψ̃λ,w:(w,λ)∈W×T}.
It is obvious that if (x,λ),(y,μ)∈X×𝕋 and (x,λ)=(y,μ), then Ψ̃λ,x=Ψ̃μ,y. We now assume that  Ψ̃λ,x=Ψ̃μ,y, where (x,λ),(y,μ)∈X×𝕋. Letting f=1 and f=i, we see that Reλ=Reμ and Re(λi)=Re(μi); that is, λ=μ. If x≠y, there exists f∈Lip(X,dα)r such that f(y)=μ¯, but f(x)=0 (define f:X→ℂ by f(z)=(μ¯/d(y,x))d(z,x), z∈X); so that
(2.10)|μ|2=Re(μμ¯)=Re(μf(y))=Ψ̃y,μ(f)=Ψ̃x,λ(f)=Re(λf(x))=0.
But this is not possible since |μ|=1. Thus, x=y.The next purpose is giving conditions  forφ∈BA* to be an extreme point of BA*, where A is a real subspace of Lip(X,τ,dα).Theorem 2.2.
LetA be a real linear subspace of  Lip(X,τ,dα) containing 1. For  (w,λ)∈W×𝕋, let η̃w,λ=ψ̃w,λ|A. Let  P={(x,λ)∈X×𝕋:τ(x)≠x}∪{(x,λ)∈X×{-1,1}:τ(x)=x}∪{(w,λ):w∈βX̃,λ∈𝕋}.
LetQA denote the set of (x,λ)∈X×{-1,1} such that(i)
there isg∈A with ∥g∥X,pα≤1 and g(x)=λ,(ii)
for everyy∈X∖{x,τ(x)}, there is some h∈A with ∥h∥X,pα≤1, h(x)=1 and |h(y)|<1.Then,
(2.11){η̃x,λ:(x,λ)∈QA}⊆Ext⁡(BA*)⊆{η̃w,λ:(w,λ)∈P}.
Further, if (x,λ)∈QA and (y,μ)∈P∩(X×𝕋), then η̃y,μ=η̃x,λ if and only if either (y,μ)=(x,λ) or (y,μ)=(τ(x),λ).Proof.
Letφ∈Ext(BA*). Letting 𝔛=Lip(X,dα)r in part (a) of Theorem 1.1, and using Proposition 2.1, we see that φ=η̃w,λ for some (w,λ)∈W×𝕋. To prove that (w,λ)∈P, we consider w=x∈X with τ(x)=x and show that λ∈{-1,1}. For every f∈A, we have f(x)¯=f(τ(x))=f(x) that is, f(x) is a real number. Hence,
(2.12)|φ(f)|=|η̃x,λ(f)|=|ψ̃x,λ(f)|=|Re(λf(x))|=|Reλ||f(x)|≤|Reλ|‖f‖X,pα.
This shows that ∥φ∥≤|Reλ|. But since φ is an extreme point of BA*, we must have ∥φ∥=1. Thus, 1≤|Reλ|≤|λ|=1 so that λ∈{-1,1}.
Next, let(x,λ)∈QA. We claim that the following statement hold.
For(y,μ)∈X×𝕋 with (y,μ)≠(x,λ) and (y,μ)≠(τ(x),λ), there is f∈A such that ∥f∥X,pα=1, Re(λf(x))=1, but Re(μf(y))≠1.
By condition (i), there isg∈A such that ∥g∥X,pα≤1 and g(x)=λ.
Case 1 (y∈{x,τ(x)}).
Letf=g. Then, ∥f∥X,pα≤1 and |f(x)|=|λ¯|=1 so that ∥f∥X,pα=1. Also,
(2.13)Re(λf(x))=Re(λg(x))=Re(λ2)=1.
If y=x, then (y,μ)≠(x,λ) implies that μ≠λ that is, μλ≠1. Since |μλ|=1, this shows that
(2.14)Re(μf(x))=Re(μg(x))=Re(μλ)≠1.
If y=τ(x), then (y,μ)≠(τ(x),λ) implies that μ≠λ, that is, μλ≠1. Since |μλ|=1, this shows that
(2.15)Re(μf(y))=Re(μg(τ(x)))=Re(μλ¯)=Re(μλ)≠1.Case 2 (y∉{x,τ(x)}).
By condition (ii), there ish∈A such that ∥h∥X,pα≤1, h(x)=1 and |h(y)|<1. Let f=gh. Now, ∥f∥X,pα≤1 and |f(x)|=|f(x)||g(x)|=|g(x)|=|λ|=1, so that ∥f∥X,pα=1. Also,
(2.16)Re(λf(x))=Re(λg(x)h(x))=Re(λ2)=1,Re(μf(y))=Re(μg(y)h(y))≤|μg(y)h(y)|=|g(y)||h(y)|≤|h(y)|<1.
Thus, our claim is justified. Let 𝔛=Lip(X,dα)r and φ=η̃x,λ in part (b) of Theorem 1.1. Consider ψ∈Ext(B𝔛*) such that ψ(f)=1 for all f∈A with φ(f)=1=∥f∥X,pα. By Proposition 2.1, ψ=ψ̃w,μ for some (w,μ)∈W×𝕋. Thus, Re(μΔ̃w(f))=1 for all f∈A with Re(λf(x))=1=∥f∥X,pα. Since λ1∈A, ∥λ1∥X,pα=1 and Re(λ(λ1)(x))=Re(λ2)=1, we have Re(μψα(λ1)(w))=1. If w′∈βX̃, then
(2.17)Re(μψα(λ1)(w′))=Re(μλψα(1)(w′))=Re(μλ(β1̃)(w′))≤|(μλ(β1̃)(w′))|=|(β1̃)(w′)|≤‖β1̃‖βX̃=‖1̃‖X̃=pα(1)=0.
Thus, it must be w∈X. Choose w=y. Then, (y,μ)∈X×𝕋 and Re(μf(y))=1 for all f∈A with ∥f∥X,pα=1=Re(λf(x)). By our claim, we must have (y,μ)=(x,λ) or (y,μ)=(τ(x),λ).
If(y,μ)=(x,λ), then clearly
(2.18)ψ|A=ψ̃y,μ|A=ψ̃x,λ|A=η̃x,λ=φ.
Now, let (y,μ)=(τ(x),λ). Since λ∈ℝ, f∈A implies that
(2.19)ψ(f)=Re(μf(y))=Re(λf(τ(x)))=Re(λf̅(x))=λRe(f(x)¯)=λRe(f(x))=Re(λf(x))=ψ̃x,λ(f)=η̃x,λ(f)=φ(f).
Therefore, ψ|A=φ, again. Hence, we see that η̃(x,λ)=φ∈Ext(BA*). This also shows that η̃τ(x),λ=η̃x,λ for all (x,λ)∈X×{-1,1}.
Conversely, our claim implies that if(x,λ)∈QA, (y,μ)∈X×𝕋 and η̃y,μ=η̃x,λ, then (y,μ)=(x,λ) or (y,μ)=(τ(x),λ). Thus, for (x,λ)∈QA and (y,μ)∈P∩(X×𝕋), we have η̃y,μ=η̃x,λ if and only if (y,μ)=(x,λ) or (y,μ)=(τ(x),λ).By using the above theorem,  we give a characterization of extreme points of unit ball in the dual space ofLip(X,τ,dα).Corollary 2.3.
LetP be as in Theorem 2.2. For (w,λ)∈P,  let  η̃w,λ denote the restriction of ψ̃w,λ to Lip(X,τ,dα).  Then,
(2.20){η̃x,λ:(x,λ)∈X×{-1,1}}⊆Ext⁡(BLip(X,τ,dα)*)⊆{η̃w,λ:(w,λ)∈P}.
Further, if (x,λ)∈X×{-1,1} and (y,μ)∈P∩(X×𝕋), then ηy,μ=η̃x,λ if and only if (y,μ)=(x,λ) or (y,μ)=(τ(x),λ).Proof.
LetA=Lip(X,τ,dα). By Theorem 2.2, we have
(2.21)Ext⁡(BA*)⊆{η̃w,λ:(w,λ)∈P}.
To prove {η̃x,λ:(x,λ)∈X×{-1,1}}⊆Ext(BLip(X,τ,dα)*), it is enough to show that for every (x,λ)∈X×{-1,1},(i)
there isg∈Lip(X,τ,dα) with ∥g∥X,pα≤1 and g(x)=λ,(ii)
for everyy∈x∖{x,τ(x)}, there is h∈Lip(X,τ,dα) with(2.22)‖h‖X,pα≤1,h(x)=1,|h(y)|<1.
Let (x,λ)∈X×{-1,1}. We first define the function g:X→ℂ by
(2.23)g(z)=λ(1-d(z,x)d(τ(z),x)1+ρ2+ρ2-2α),
where ρ=diam(X)=sup{d(y,z):y,z∈X}. Clearly, g∈C(X,τ), g(x)=λ, ∥g∥X≤1 and g∈Lip(X,d1). Let y,z∈X with y≠z. Then,
(2.24)|g(y)-g(z)|dα(y,z)=|d(z,x)d(τ(z),x)-d(y,x)d(τ(y),x)|(1+ρ2+ρ2-2α)dα(y,z)≤d1-α(y,z)(1+ρ2+ρ2-2α)(d(z,x)+d(τ(y),x))≤2ρρ1-α1+ρ2+ρ2-2α≤1.
Therefore,g∈Lip(X,dα) and pα(g)≤1. Consequently,
(2.25)g∈Lip(X,τ,dα),‖g‖X,pα≤1,g(x)=λ.
Hence, (i) holds.
We now assume thaty∈X∖{x,τ(x)} and define the function h:X→ℂ by
(2.26)h(z)=1-λg(z)d(x,z)d(x,τ(z))2(1+ρ2+ρ2-2α).
Clearly, h∈C(X,τ), h(x)=1, ∥h∥X≤1 and there exists γ∈(0,1] such that h(y)=1-γ so that |h(y)|<1. We define the complex-valued function g1 on X by g1=(1/2)(1-λg). Then, g1∈C(X,τ), ∥g1∥X≤1/2, g1∈Lip(X,d1), pα(g1)≤1/2 and h=1-gg1. Thus, we have h∈Lip(X,d1), h∈Lip(X,τ,dα) and
(2.27)pα(h)≤pα(g)‖g1‖X+‖g‖Xpα(g1)≤12+12=1,
so that ∥h∥X,pα≤1. Since y∈X∖{x,τ(x)}, |h(y)|=h(y)<1. Thus, (ii) holds.


## 3. Real Subspaces oflip(X,dα) Containing 1

Throughout this section, we assume thatα∈(0,1). Consider the mapping Φα:lip(X,dα)→C(W) by Φα=Ψα|lip(X,dα). Then, Φα is a linear isometric from (lip(X,dα),∥·∥X,pα) into C(W).For eachx∈X and w∈W, define the functionals δx and δ̃w in lip(X,dα)* by δx(f)=f(x) and δ̃w(f)=Φα(f)(w), respectively. Clearly, |δx(f)|≤∥f∥X,pα for allf∈lip(X,dα) and |δ̃w(f)|≤∥f∥X,pα, and therefore, δx,δ̃w∈Blip(X,dα)*. Moreover, δ̃x=δx for all x∈X and δ̃w=ewoΦα for all w∈W.We give a characterization of extreme points of the unit ball in the dual spacelip(X,dα) as the following.Theorem 3.1.
Every extreme point ofBlip(X,dα)* must be either the form  λδx with  (x,λ)∈X×𝕋 or of the form  λδ̃w with  (λ,w)∈βX̃×𝕋.  Moreover, λδx is an extreme point of Blip(X,dα)* for all (x,λ)∈X×𝕋.Proof.
Since(lip(X,dα),∥·∥X,pα) is a Banach space and Φα is a linear isometry from (lip(X,dα),∥·∥X,pα) into (C(W),∥·∥W), we conclude that Φα(lip(X,dα)) is a uniformly closed subspace of C(W). It is well known [2, page 441] that
(3.1)Ext⁡(B(Φα(lip(X,dα)))*)⊆{λew|Φα(lip(X,dα)):(w,λ)∈W×T}.
Let A=Φα(lip(X,dα)) and define Λα:lip(X,dα)→A by Λα(f)=Φα(f). Then, Λα is a linear isometry from (lip(X,dα),∥·∥pα) onto (A,∥·∥W), so Λα*, the adjoint of Λα, is a linear isometry from (A,∥·∥W)* onto (lip(X,dα))*. It is easily to show that Λα*(ew|A)=δ̃w for all w∈W. Let φ∈Ext(Blip(X,dα)*). Then, (Λα*)-1(φ)∈Ext(BA*). Thus, there exists (w,λ)∈W×𝕋 such that (Λα*)-1(φ)=λew|A so that φ=λα*(ew|A)=λδ̃w. It follows that
(3.2)Ext⁡(Blip(X,dα)*)⊆{λδ̃w:(w,λ)∈W×T}.
Now, let(x,λ)∈X×𝕋. Clearly, λδx∈Blip(X,dα)*. Assume that ψ∈Ext(BLip(X,dα)*) and ψ(f)=1 for all f∈lip(X,dα) with (λδx)(f)=1=∥f∥X,pα. Since lip(X,dα) is a nonzero linear subspace of Lip(X,dα), we conclude that ψ=μΔy for some (y,μ)∈X×𝕋 or ψ=μΔ̃w for some (w,μ)∈(βX̃)×𝕋 by Theorem 1.1. Clearly, λ¯1∈lip(X,dα) and (λδx)(λ¯1)=1=∥λ¯1∥X,pα. If ψ=μΔ̃w for some (w,μ)∈(βX̃)×𝕋, then we have
(3.3)1=|ψ(λ¯1)|=|μΔ̃w(λ¯1)|=|Δ̃w(1)|=|(β1̃)(w)|≤‖β1̃‖βX̃=pα(1)=0.
Thus, it must be ψ=μΔy for some (y,μ)∈X×𝕋. Since ψ(λ¯1)=1, we have
(3.4)1=(μΔy)(λ¯1)=μλ¯.
It follows that μ=λ. We claim that y=x. Let y∈X∖{x}. We define the function g:X→ℂ by
(3.5)g(z)=1-d(z,x)1+ρ+ρ1-α,
where ρ=diamX. It is easy to show that g∈Lip(X,d1), g(x)=1, ∥g∥X=1 and pα(g)≤1. Now, we define the function f:X→ℂ by
(3.6)f(z)=λ¯(1-g(z)d(z,x)2(1+ρ+ρ1-α)),
and the function g1:X→ℂ by g1=(1/2)(1-g). It is easy to see that g1∈Lip(X,d1), ∥g1∥X≤1/2, pα(g1)≤1/2, and f=λ¯(1-gg1). Therefore, we have f∈Lip(X,d1)⊆lip(X,dα), f(x)=λ¯, ∥f∥X≤1, and
(3.7)pα(f)≤pα(g)‖g1‖X+‖g‖Xpα(g1)≤12+12=1,
so that ∥f∥X,p=1=(λδx)(f). It must be that ψ(f)=1; that is, (λΔy)(f)=1. But
(3.8)|(λΔy)(f)|=|λf(y)|=|f(y)|=1-g(y)d(y,x)2(1+ρ+ρ1-α)<1.
This contradiction implies that y=x and so our claim is justified. Therefore, ψ=λΔx. It follows that ψ|lip(X,dα)=λδx. Therefore, λδx∈Ext(Blip(X,dα)*) by part (b) of Theorem 1.1. Consequently,
(3.9){λδx:(x,λ)∈X×T}⊆Ext⁡(Blip(X,dα)*)⊆{λδ̃w:(w,λ)∈W×T}.
Thus, the proof is complete.For each(w,λ)∈W×𝕋, we define the map φ̃w,λ:lip(X,dα)r→ℝ by φ̃w,λ(f)=Re(λδ̃w(f)); in fact, φ̃w,λ=Re(λ(ewoΦα)). Clearly, φ̃w,λ∈B(lip(X,dα)r)* for all (w,λ)∈W×𝕋. Moreover, φ̃x,λ=Re(λδx) for all (x,λ)∈X×𝕋.We now give a characterization of extreme points of the unit ball in the(lip(X,dα)r)* as the following.Proposition 3.2.
By above notations,(3.10){φ̃x,λ:(x,λ)∈X×T}⊆Ext⁡(B(lip(X,dα)r)*)⊆{φ̃w,λ:(w,λ)∈W×T}.
Further, for (x,λ),(y,μ)∈X×𝕋, one has φ̃x,λ=φ̃y,μ if and only if (x,λ)=(y,μ).Proof.
The proof is similar to that of Proposition2.1 by replacing lip(X,dα),δ̃x,δ̃w,φ̃x,λ,φ̃w,λ and Theorems 1.2 and 1.3 by Lip(X,dα),Δx,Δ̃w,ψ̃x,λ,ψ̃w,λ and Theorem 3.1, respectively.Theorem 3.3.
LetA be a real linear subspace of lip(X,τ,dα) containing 1. For  (w,λ)∈W×𝕋, let  θ̃w,λ=φ̃w,λ|A. Let P={(x,λ)∈X×𝕋:τ(x)≠x}∪{(x,λ)∈X×𝕋:τ(x)=x,λ∈{-1,1}}∪{(w,λ)∈W×𝕋:w∈βX̃}. Let QA denote the set of all (x,λ)∈X×{-1,1} such that(i)
there isg∈A with ∥g∥X,pα≤1 and g(x)=λ¯,(ii)
for everyy∈X∖{x,τ(x)}, there is h∈A with ∥h∥X,pα≤1, h(x)=1 and |h(y)|<1. Then,
(3.11){θ̃x,λ:(x,λ)∈QA}⊆Ext⁡(BA*)⊆{θ̃w,λ:(w,λ)∈P}.
Further, if (x,λ)∈QA and (y,μ)∈P∩(X×𝕋), then θ̃y,μ=θ̃x,λ if and only if (y,μ)=(x,λ) or (y,μ)=(τ(x),λ).Proof.
The proof is similar to that of Theorem2.2 by replacing lip(X,dα),θ̃w,λ,θ̃x,λ,φ̃x,λ, Proposition 3.2 and Φα by Lip(X,dα),η̃w,λ,η̃x,λ,ψ̃x,λ, Proposition 2.1 and Ψα, respectively.Corollary 3.4.
LetP be as in Theorem 3.3. For  (w,λ)∈P, let  θ̃w,λ denote the restriction of  φ̃w,λ to  lip(X,τ,dα). Then,
(3.12){θ̃x,λ:(x,λ)∈X×{-1,1}}⊆Ext⁡(Blip(X,τ,α)*)⊆{θ̃w,λ:(w,λ)∈P}.
Further, if (x,λ)∈X×{-1,1} and (y,μ)∈P∩(X×𝕋), then θy,μ=θ̃x,λ if and only if (y,μ)=(x,λ) or (y,μ)=(τ(x),λ).Proof.
The proof is similar to that Corollary2.3 by replacing lip(X,τ,dα),θ̃w,λ,θ̃x,λ and lip(X,dα) by Lip(X,τ,dα),η̃w,λ,η̃x,λ and Lip(X,dα), respectively.



---

*Source: 735139-2012-01-11.xml*