Let 𝑋 be a compact Hausdorff space, 𝜏 be a continuous involution on 𝑋 and 𝐢(𝑋,𝜏) denote the uniformly closed real subalgebra of 𝐢(𝑋) consisting of all π‘“βˆˆπΆ(𝑋) for which π‘“βˆ˜πœ=𝑓. 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 π›Όβˆˆ(0,1]. For π›Όβˆˆ(0,1), the closed subalgebra of Lip(𝑋,𝛼) consisting of all π‘“βˆˆLip(𝑋,𝑑𝛼) for which |𝑓(π‘₯)βˆ’π‘“(𝑦)|/𝑑𝛼(π‘₯,𝑦)β†’0 as 𝑑(π‘₯,𝑦)β†’0, denotes by lip(𝑋,𝑑𝛼). Let 𝜏 be a Lipschitz involution on (𝑋,𝑑) and define Lip(𝑋,𝜏,𝑑𝛼)=Lip(𝑋,𝑑𝛼)∩𝐢(𝑋,𝜏) for π›Όβˆˆ(0,1] and lip(𝑋,𝜏,𝑑𝛼)=lip(𝑋,𝑑𝛼)∩𝐢(𝑋,𝜏) for π›Όβˆˆ(0,1). In this paper, we give a characterization of extreme points of π΅π΄βˆ—, 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).

1. Introduction and Preliminaries

We let ℝ,β„‚, and 𝕋={π‘§βˆˆβ„‚βˆΆ|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 β„‚. The elements of 𝕂 are called scalars.

Let 𝔛 be a normed space over 𝕂. We denote by π”›βˆ— and 𝐡𝔛 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 πœ‘βˆˆπ΄βˆ—. A Hahn-Banach extension of πœ‘ to 𝔛 is a continuous linear functional πœ“βˆˆπ”›βˆ— such that πœ“|𝐴=πœ‘ and ||πœ“||=||πœ‘||. The set of all Hahn-Banach extensions of πœ‘ to 𝔛 will be denoted by π»πœ‘.

It is easy to see that if 𝔛 and 𝒴 are normed spaces over 𝕂 and π‘‡βˆΆπ”›β†’π’΄ is a linear isometry from 𝔛 onto 𝒴, then 𝑇 is a bijection mapping between Ext(𝐡𝔛) and Ext(𝐡𝒴).

For a complex normed space 𝔛, we assume that π”›π‘Ÿ denotes 𝔛, regarded as a real normed space by restricting the scalar multiplication to real numbers.

Kulkarni and Limaye gave some conditions for πœ‘βˆˆπ΅π΄βˆ— to be an extreme point of π΅π΄βˆ— in terms of the Hahn-Banach extension of πœ‘ to 𝔛 and the extreme points of π΅π”›βˆ— as the following.

Theorem 1.1 (see [1, Theorem  2]). Let 𝔛 be a normed space over 𝕂,𝐴 be a nonzero linear subspace of 𝔛 and πœ‘βˆˆπ΅π΄βˆ—. (a)Let πœ‘βˆˆExt(π΅π΄βˆ—). Then, π»πœ‘ξ€·π΅βˆ©Extπ”›βˆ—ξ€Έξ€·π»=Extπœ‘ξ€Έβ‰ βˆ….(1.1)In particular, πœ‘ has an extension to some πœ“βˆˆExt(π΅π”›βˆ—). Further, if such an extension is unique, then πœ‘ has a unique Hahn-Banach extension to 𝔛.(b)Assume that whenever πœ“βˆˆExt(π΅π”›βˆ—) and πœ“(𝑓)=1 for all π‘“βˆˆπ΄ with πœ‘(𝑓)=1=‖𝑓‖, one has πœ“|𝐴=πœ‘, then πœ‘βˆˆExt(π΅π΄βˆ—).(c)If πœ‘ has a unique Hahn-Banach extension πœ“ to 𝔛 and if πœ“βˆˆExt(π΅π”›βˆ—), then πœ‘βˆˆExt(π΅π΄βˆ—).

Let 𝑋 be a compact Hausdorff space. We denote by 𝐢(𝑋) the complex Banach algebra of all continuous complex-valued functions on 𝑋 under the uniform norm ‖𝑓‖𝑋=sup{|𝑓(π‘₯)|∢π‘₯βˆˆπ‘‹}. For π‘₯βˆˆπ‘‹, consider the evaluation functional 𝑒π‘₯ given by 𝑒π‘₯(𝑓)=𝑓(π‘₯),π‘“βˆˆπΆ(𝑋). Clearly, πœ†π‘’π‘₯∈𝐡𝐢(𝑋)βˆ— for all (π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹. It is well known [2, page 441] that𝐡Ext𝐢(𝑋)βˆ—ξ€Έ=ξ€½πœ†π‘’π‘₯βˆΆξ€Ύ.(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹(1.2)

For π‘₯βˆˆπ‘‹ and πœ†βˆˆπ•‹, we define the map πœ“π‘₯,πœ†βˆΆπΆ(𝑋)π‘Ÿβ†’β„ by πœ“π‘₯,πœ†(𝑓)=Re(πœ†π‘“(π‘₯)) in fact, πœ“π‘₯,πœ†=Re(πœ†π‘’π‘₯). Clearly, πœ“π‘₯,πœ†βˆˆπ΅(𝐢(𝑋)π‘Ÿ)βˆ— for all (π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹. Kulkarni and Limaye showed [1, Proposition  3] that𝐡Ext(𝐢(𝑋)π‘Ÿ)βˆ—ξ€Έ=ξ€½πœ“π‘₯,πœ†βˆΆξ€Ύ,(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹(1.3) and πœ“π‘¦,πœ‡=πœ“π‘₯,πœ† if and only if 𝑦=π‘₯ and πœ‡=πœ†.

Let 𝜏 be a continuous involution on 𝑋; that is, πœβˆΆπ‘‹β†’π‘‹ is continuous and 𝜏∘𝜏 is the identity map on 𝑋. The map 𝜎∢𝐢(𝑋)→𝐢(𝑋) defined by 𝜎(𝑓)=π‘“βˆ˜πœ, is an algebra involution on 𝐢(𝑋) which is called the algebra involution induced by 𝜏 on 𝐢(𝑋). Define 𝐢(𝑋,𝜏)={π‘“βˆˆπΆ(𝑋)∢𝜎(𝑓)=𝑓}. Then, 𝐢(𝑋,𝜏) is a uniformly closed real subalgebra of 𝐢(𝑋) which contains 1. The real algebras 𝐢(𝑋,𝜏) were first considered in [3]. For a detailed account of several properties of 𝐢(𝑋,𝜏), we refer to [4].

Let 𝑃={(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹βˆΆπœ(π‘₯)β‰ π‘₯}βˆͺ{(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹βˆΆπœ(π‘₯)=π‘₯,πœ†βˆˆ{βˆ’1,1}}. For each (π‘₯,πœ†)βˆˆπ‘ƒ, let πœ‘π‘₯,πœ† denote the restriction of πœ“π‘₯,πœ† to 𝐢(𝑋,𝜏). Grzesiak obtained a characterization of the extreme points of 𝐡(𝐢(𝑋,𝜏))βˆ— in [5] and showed that πœ‘βˆˆExt(𝐡(𝐢(𝑋,𝜏))βˆ—) if and only if πœ‘=πœ‘π‘₯,πœ† for some (π‘₯,πœ†)βˆˆπ‘ƒ. Further, if (π‘₯,πœ†),(𝑦,πœ‡)βˆˆπ‘ƒ, then πœ‘π‘¦,πœ‡=πœ‘π‘₯,πœ† if and only if (𝑦,πœ‡)=(π‘₯,πœ†) or (𝑦,πœ‡)=(𝜏(π‘₯),πœ†).

Kulkarni and Limaye obtained [1, Theorem  4] a characterization of Ext(π΅π΄βˆ—), where 𝐴 is a nonzero real linear subspace of 𝐢(𝑋,𝜏).

Let (𝑋,𝑑) be a compact metric space. For π›Όβˆˆ(0,1], we denote by Lip(𝑋,𝑑𝛼) the set of all complex-valued functions 𝑓 on 𝑋 for which𝑝𝛼||||(𝑓)=sup𝑓(π‘₯)βˆ’π‘“(𝑦)𝑑𝛼(π‘₯,𝑦)∢π‘₯,π‘¦βˆˆπ‘‹,π‘₯≠𝑦(1.4) is finite. Then, Lip(𝑋,𝛼) is a complex subalgebra of 𝐢(𝑋) containing 1 and complex Banach space under the norm‖𝑓‖𝑋,𝑝𝛼=max‖𝑓‖𝑋,𝑝𝛼(𝑓)(π‘“βˆˆLip(𝑋,𝑑𝛼)).(1.5) For π›Όβˆˆ(0,1), the complex subalgebra of Lip(𝑋,𝑑𝛼) consisting of all π‘“βˆˆLip(𝑋,𝑑𝛼) for which||||𝑓(π‘₯)βˆ’π‘“(𝑦)𝑑𝛼(π‘₯,𝑦)⟢0as𝑑(π‘₯,𝑦)⟢0,(1.6) 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 𝑋={(π‘₯,𝑦)βˆˆπ‘‹Γ—π‘‹βˆΆπ‘₯≠𝑦}, and let the compact Hausdorff space π‘Š be the disjoint union of 𝑋 with 𝛽𝑋, where 𝛽𝑋 is the Stone-Cech compactification of 𝑋. For π›Όβˆˆ(0,1], consider the mapping Ξ¨π›ΌβˆΆLip(𝑋,𝑑𝛼)→𝐢(π‘Š) defined for each π‘“βˆˆLip(𝑋,𝑑𝛼) byΨ𝛼𝛽𝑓((𝑓)(𝑀)=𝑓(𝑀)ifπ‘€βˆˆπ‘‹,𝑀)ifπ‘€βˆˆπ›½π‘‹,(1.7) where𝑓(π‘₯,𝑦)=𝑓(π‘₯)βˆ’π‘“(𝑦)𝑑𝛼(π‘₯,𝑦),βˆ€(π‘₯,𝑦)βˆˆπ‘‹,(1.8) and 𝛽𝑓 is the norm-preserving extension of 𝑓 to 𝛽𝑋. Clearly, Ψ𝛼 is a linear isometry from (Lip(𝑋,𝑑𝛼),‖⋅‖𝑋,𝑝𝛼) into (𝐢(π‘Š),β€–β‹…β€–π‘Š), which is called the Leeuw’s linear isometry. Therefore, Ψ𝛼(Lip(𝑋,𝑑𝛼)) is a uniformly closed linear subspace of 𝐢(π‘Š). It is well known (see [2, page 441]) that𝐡ExtΨ𝛼(Lip(𝑋,𝑑𝛼))βˆ—ξ€ΈβŠ†ξ€½πœ†π‘’π‘€βˆ£Ξ¨π›Ό(Lip(𝑋,𝑑𝛼))βˆΆξ€Ύ,(𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹(1.9) where 𝑒𝑀 is the evaluation functional at 𝑀 on 𝐢(π‘Š).

For each π‘₯βˆˆπ‘‹ and π‘€βˆˆπ‘Š, define the linear functionals Ξ”π‘₯ and Δ𝑀 in Lip(𝑋,𝑑𝛼)βˆ— by Ξ”π‘₯(𝑓)=𝑓(π‘₯) and Δ𝑀(𝑓)=Ψ𝛼(𝑓)(𝑀), respectively. Clearly, |Ξ”π‘₯(𝑓)|≀‖𝑓‖𝑋,𝑝𝛼 and |Δ𝑀(𝑓)|≀‖𝑓‖𝑋,𝑝𝛼 for all π‘“βˆˆLip(𝑋,𝑑𝛼). Therefore, Ξ”π‘₯,ξ‚Ξ”π‘€βˆˆπ΅Lip(𝑋,𝑑𝛼)βˆ—. Moreover, Δπ‘₯=Ξ”π‘₯ for all π‘₯βˆˆπ‘‹β€‰β€‰and Δ𝑀=π‘’π‘€π‘œΞ¨π›Ό for all π‘€βˆˆπ‘Š. Thus, we have the following result.

Theorem 1.2. For π›Όβˆˆ(0,1], every extreme point of 𝐡Lip(𝑋,𝑑𝛼)βˆ— must be either of the form πœ†Ξ”π‘₯ with (π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹ or of the form πœ†ξ‚Ξ”π‘€ with (𝑀,πœ†)βˆˆπ›½π‘‹Γ—π•‹.

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 (π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹, πœ†Ξ”π‘₯ is an extreme point of 𝐡Lip(𝑋,𝑑𝛼)βˆ—.

Jimenez-Vargas and Villegas-Vallecillos used above results and obtained a characterization of linear isometries between Lip(𝑋,𝑑𝑋) and Lip(π‘Œ,π‘‘π‘Œ) in [8].

A map π‘“βˆΆπ‘‹β†’π‘Œ is said to be Lipschitz map from the metric space (𝑋,𝑑𝑋) to the metric space (π‘Œ,π‘‘π‘Œ) if there exists a constant 𝐢>0 such that π‘‘π‘Œ(𝑓(π‘₯),𝑓(𝑦))≀𝐢𝑑𝑋(π‘₯,𝑦) for all π‘₯,π‘¦βˆˆπ‘‹.

Let (𝑋,𝑑) be a compact metric space. The mapping πœβˆΆπ‘‹β†’π‘‹ 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.

Let 𝜏 be a Lipschitz involution on the compact metric space (𝑋,𝑑) and let 𝜎 be the algebra involution induced by 𝜏 on 𝐢(𝑋). Clearly,𝜎(Lip(𝑋,𝑑𝛼))=Lip(𝑋,𝑑𝛼),𝜎(lip(𝑋,𝑑𝛼))=lip(𝑋,𝑑𝛼).(1.10) We defineLip(𝑋,𝜏,𝑑𝛼)={β„ŽβˆˆLip(𝑋,𝑑𝛼)∢𝜎(β„Ž)=β„Ž},lip(𝑋,𝜏,𝑑𝛼)={β„Žβˆˆlip(𝑋,𝑑𝛼)∢𝜎(β„Ž)=β„Ž}.(1.11) 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)∩𝐢(𝑋,𝜏) and lip(𝑋,𝜏,𝑑𝛼)=lip(𝑋,𝑑𝛼)∩𝐢(𝑋,𝜏).(iii)Lip(𝑋,𝑑𝛼)=Lip(𝑋,𝜏,𝑑𝛼)βŠ•π‘–Lip(𝑋,𝜏,𝑑𝛼) and lip(𝑋,𝑑𝛼)=lip(𝑋,𝜏,𝑑𝛼)βŠ•π‘–lip(𝑋,𝜏,𝑑𝛼).(iv)Lip(𝑋,𝜏,𝑑𝛼) (lip(𝑋,𝜏,𝑑𝛼), resp.) is a real subalgebra of 𝐢(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 𝐢(𝑋,𝜏) (use (iv) and the Stone-Weierstrass theorem for real subalgebra of 𝐢(𝑋,𝜏) [3, Proposition  1.1].(vi)For 0<𝛼<𝛽≀1, ξ€·Lip𝑋,𝜏,π‘‘π›½ξ€ΈβŠ†lip(𝑋,𝜏,𝑑𝛼)βŠ†Lip(𝑋,𝜏,𝑑𝛼).(1.12)(vii)There exists a constant 𝐢β‰₯1 such that ξ€½max‖𝑓‖𝑋,𝑝𝛼,‖𝑔‖𝑋,𝑝𝛼≀𝐢𝛼‖𝑓+𝑔‖𝑋,𝑝𝛼,(1.13) for all 𝑓,π‘”βˆˆLip(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 spaces Lip(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 of Lip(𝑋,𝑑𝛼)π‘Ÿ, Lip(𝑋,𝜏,𝑑𝛼) and some its real linear subspaces for π›Όβˆˆ(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)π‘Ÿ,lip(𝑋,𝜏,𝑑𝛼) and some its real linear subspaces for π›Όβˆˆ(0,1) in Section 3.

2. Real Linear Subspaces of Lip(𝑋,𝑑𝛼) Containing 1

In the remainder of this paper, we assume that π›Όβˆˆ(0,1], (𝑋,𝑑) is a compact metric space, 𝑋={(π‘₯,𝑦)βˆˆπ‘‹Γ—π‘‹,π‘₯≠𝑦}, 𝛽𝑋 is the Stone-Cech compactification of 𝑋, π‘Š is the compact Hausdorff space 𝑋𝑋βˆͺ𝛽, Ψ𝛼 is the Leeuw’s linear isometry from (Lip(𝑋,𝑑𝛼),‖⋅‖𝑋,𝑝𝛼) into (𝐢(π‘Š),β€–β‹…β€–π‘Š), and 𝜏 is a Lipschitz involution on (𝑋,𝑑).

For each (𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹, we define the map ξ‚πœ“π‘€,πœ†βˆΆLip(𝑋,𝑑𝛼)π‘Ÿβ†’β„ by ξ‚πœ“π‘€,πœ†ξ‚Ξ”(𝑓)=Re(πœ†π‘€(𝑓)) in fact, ξ‚πœ“π‘€,πœ†=Re(πœ†(π‘’π‘€π‘œΞ¨π›Ό)). Clearly, ξ‚πœ“π‘€,πœ†βˆˆπ΅(Lip(𝑋,𝑑𝛼)π‘Ÿ)βˆ— for all (𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹. Moreover, ξ‚πœ“π‘₯,πœ†=Re(πœ†Ξ”π‘₯) for all (π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹.

We first give a characterization of the extreme points of the unit ball in the (Lip(𝑋,𝑑𝛼)π‘Ÿ)βˆ— as the following.

Proposition 2.1. By above notations, ξ€½ξ‚πœ“π‘₯,πœ†βˆΆξ€Ύξ€·π΅(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹βŠ†Ext(Lip(𝑋,𝑑𝛼)π‘Ÿ)βˆ—ξ€ΈβŠ†ξ€½ξ‚πœ“π‘€,πœ†βˆΆξ€Ύ.(𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹(2.1) Further, for (π‘₯,πœ†),(𝑦,πœ‡)βˆˆπ‘‹Γ—π•‹ one has ξ‚πœ“π‘₯,πœ†=ξ‚πœ“π‘¦,πœ‡ if and only if (π‘₯,πœ†)=(𝑦,πœ‡).

Proof. We define the map π‘‡βˆΆ(Lip(𝑋,𝑑𝛼)βˆ—)π‘Ÿβ†’(Lip(𝑋,𝑑𝛼)π‘Ÿ)βˆ— by 𝑇(πœ‘)=Reπœ‘. Clearly, 𝑇 is a real-linear mapping. For each π‘’βˆˆ(Lip(𝑋,𝑑𝛼)π‘Ÿ)βˆ—, defining the map πœ‘βˆΆLip(𝑋,𝑑𝛼)βˆ—β†’β„‚ by πœ‘(𝑓)=𝑒(𝑓)βˆ’π‘–π‘’(𝑖𝑓). Clearly, πœ‘βˆˆLip(𝑋,𝑑𝛼)βˆ— and 𝑒=Reπœ‘. It follows that πœ‘βˆˆ(Lip(𝑋,𝑑𝛼)βˆ—)π‘Ÿ and 𝑇(πœ‘)=𝑒. Thus, 𝑇 is onto.
We claim that 𝑇 is an isometric. Let πœ‘βˆˆ(Lip(𝑋,𝑑𝛼)βˆ—)π‘Ÿ. Since ||||=||||=||||≀||||𝑇(πœ‘)(𝑓)(Reπœ‘)(𝑓)Re(πœ‘(𝑓))πœ‘(𝑓)β‰€β€–πœ‘β€–β€–π‘“β€–π‘‹,𝑝𝛼,(2.2) for each π‘“βˆˆLip(𝑋,𝑑𝛼)π‘Ÿ, we have ‖𝑇(πœ‘)β€–β‰€β€–πœ‘β€–.(2.3)
Let πœ€ be an arbitrary positive number. There exists π‘“βˆˆLip(𝑋,𝑑𝛼)π‘Ÿ with ‖𝑓‖𝑋,𝑝𝛼≀1 such that β€–πœ‘β€–<|πœ“(𝑓)|+πœ€. Choose 𝛾=1 if πœ“(𝑓)=0 and 𝛾=(1/πœ“(𝑓))|πœ‘(𝑓)| if πœ“(𝑓)β‰ 0. Then, π›Ύβˆˆβ„‚, |𝛾|=1 and πœ“(𝑓)=𝛾|πœ“(𝑓)|. If 𝑔=(1/𝛾)𝑓, then π‘”βˆˆLip(𝑋,𝑑𝛼)π‘Ÿ, ‖𝑔‖𝑋,𝑝𝛼=‖𝑓‖𝑋,𝑝𝛼≀1 and so, β€–||||||||πœ“β€–<Re(πœ‘(𝑔))+πœ€=𝑇(πœ‘)(𝑔)+πœ€β‰€β€–π‘‡(πœ‘)β€–+πœ€.(2.4) It follows that β€–πœ‘β€–β‰€β€–π‘‡(πœ‘)β€–.(2.5)
Thus, our claim is justified. The above arguments show that 𝑇 is a real-linear isometry from (Lip(𝑋,𝑑𝛼)βˆ—)π‘Ÿ onto (Lip(𝑋,𝑑𝛼)π‘Ÿ)βˆ—. Therefore, 𝐡Ext(Lip(𝑋,𝑑𝛼)π‘Ÿ)βˆ—ξ€Έξ€·ξ€·π΅=𝑇ExtLip(π‘₯,𝑑𝛼)βˆ—.ξ€Έξ€Έ(2.6) Since ξ‚†πœ†ξ‚Ξ”π‘₯ξ‚‡ξ€·π΅βˆΆ(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹βŠ†ExtLip(𝑋,𝑑𝛼)βˆ—ξ€ΈβŠ†ξ‚†πœ†ξ‚Ξ”π‘€ξ‚‡βˆΆ(𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹,(2.7) we conclude that ξ‚†π‘‡ξ‚€πœ†ξ‚Ξ”π‘₯ξ‚ξ‚‡ξ€·π΅βˆΆ(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹βŠ†Ext(Lip(𝑋,𝑑𝛼)π‘Ÿ)βˆ—ξ€ΈβŠ†ξ‚†π‘‡ξ‚€πœ†ξ‚Ξ”π‘€ξ‚ξ‚‡βˆΆ(𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹,(2.8) by Theorems 1.2 and 1.3.
Clearly, Δ𝑇(πœ†π‘€ξ‚Ξ¨)=πœ†,𝑀 for all (𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹. Therefore, ξ‚†ξ‚Ξ¨πœ†,π‘₯ξ‚‡ξ€·π΅βˆΆ(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹βŠ†Ext(Lip(𝑋,𝑑𝛼)π‘Ÿ)βˆ—ξ€ΈβŠ†ξ‚†ξ‚Ξ¨πœ†,π‘€ξ‚‡βˆΆ(𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹.(2.9) It is obvious that if (π‘₯,πœ†),(𝑦,πœ‡)βˆˆπ‘‹Γ—π•‹ and (π‘₯,πœ†)=(𝑦,πœ‡), then ξ‚Ξ¨πœ†,π‘₯=ξ‚Ξ¨πœ‡,𝑦. We now assume that ξ‚Ξ¨πœ†,π‘₯=ξ‚Ξ¨πœ‡,𝑦, where (π‘₯,πœ†),(𝑦,πœ‡)βˆˆπ‘‹Γ—π•‹. Letting 𝑓=1 and 𝑓=𝑖, we see that Reπœ†=Reπœ‡ and Re(πœ†π‘–)=Re(πœ‡π‘–); that is, πœ†=πœ‡. If π‘₯≠𝑦, there exists π‘“βˆˆLip(𝑋,𝑑𝛼)π‘Ÿ such that 𝑓(𝑦)=πœ‡, but 𝑓(π‘₯)=0 (define π‘“βˆΆπ‘‹β†’β„‚ by 𝑓(𝑧)=(πœ‡/𝑑(𝑦,π‘₯))𝑑(𝑧,π‘₯), π‘§βˆˆπ‘‹); so that ||πœ‡||2ξ€·πœ‡=Reπœ‡ξ€Έξ‚Ξ¨=Re(πœ‡π‘“(𝑦))=𝑦,πœ‡ξ‚Ξ¨(𝑓)=π‘₯,πœ†(𝑓)=Re(πœ†π‘“(π‘₯))=0.(2.10) But this is not possible since |πœ‡|=1. Thus, π‘₯=𝑦.

The next purpose is giving conditions for πœ‘βˆˆπ΅π΄βˆ— to be an extreme point of π΅π΄βˆ—, where 𝐴 is a real subspace of Lip(𝑋,𝜏,𝑑𝛼).

Theorem 2.2. Let 𝐴 be a real linear subspace of Lip(𝑋,𝜏,𝑑𝛼) containing 1. For (𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹, let Μƒπœ‚π‘€,πœ†=ξ‚πœ“π‘€,πœ†|𝐴. Let 𝑃={(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹βˆΆπœ(π‘₯)β‰ π‘₯}βˆͺ{(π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1}∢𝜏(π‘₯)=π‘₯}βˆͺ{(𝑀,πœ†)βˆΆπ‘€βˆˆπ›½π‘‹,πœ†βˆˆπ•‹}.
Let 𝑄𝐴 denote the set of (π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1} such that(i)there is π‘”βˆˆπ΄ with ‖𝑔‖𝑋,𝑝𝛼≀1 and 𝑔(π‘₯)=πœ†, (ii)for every π‘¦βˆˆπ‘‹β§΅{π‘₯,𝜏(π‘₯)}, there is some β„Žβˆˆπ΄ with β€–β„Žβ€–π‘‹,𝑝𝛼≀1, β„Ž(π‘₯)=1 and |β„Ž(𝑦)|<1.Then, ξ€½Μƒπœ‚π‘₯,πœ†βˆΆ(π‘₯,πœ†)βˆˆπ‘„π΄ξ€Ύξ€·π΅βŠ†Extπ΄βˆ—ξ€ΈβŠ†ξ€½Μƒπœ‚π‘€,πœ†βˆΆξ€Ύ(𝑀,πœ†)βˆˆπ‘ƒ.(2.11) Further, if (π‘₯,πœ†)βˆˆπ‘„π΄ and (𝑦,πœ‡)βˆˆπ‘ƒβˆ©(𝑋×𝕋), then Μƒπœ‚π‘¦,πœ‡=Μƒπœ‚π‘₯,πœ† if and only if either (𝑦,πœ‡)=(π‘₯,πœ†) or (𝑦,πœ‡)=(𝜏(π‘₯),πœ†).

Proof. Let πœ‘βˆˆExt(π΅π΄βˆ—). Letting 𝔛=Lip(𝑋,𝑑𝛼)π‘Ÿ in part (a) of Theorem 1.1, and using Proposition 2.1, we see that πœ‘=Μƒπœ‚π‘€,πœ† for some (𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹. To prove that (𝑀,πœ†)βˆˆπ‘ƒ, we consider 𝑀=π‘₯βˆˆπ‘‹ with 𝜏(π‘₯)=π‘₯ and show that πœ†βˆˆ{βˆ’1,1}. For every π‘“βˆˆπ΄, we have 𝑓(π‘₯)=𝑓(𝜏(π‘₯))=𝑓(π‘₯) that is, 𝑓(π‘₯) is a real number. Hence, ||||=||πœ‘(𝑓)Μƒπœ‚π‘₯,πœ†||=||(𝑓)ξ‚πœ“π‘₯,πœ†||=||||=||||||𝑓||≀||||(𝑓)Re(πœ†π‘“(π‘₯))Reπœ†(π‘₯)Reπœ†β€–π‘“β€–π‘‹,𝑝𝛼.(2.12) This shows that β€–πœ‘β€–β‰€|Reπœ†|. But since πœ‘ is an extreme point of π΅π΄βˆ—, we must have β€–πœ‘β€–=1. Thus, 1≀|Reπœ†|≀|πœ†|=1 so that πœ†βˆˆ{βˆ’1,1}.
Next, let (π‘₯,πœ†)βˆˆπ‘„π΄. We claim that the following statement hold.
For (𝑦,πœ‡)βˆˆπ‘‹Γ—π•‹ with (𝑦,πœ‡)β‰ (π‘₯,πœ†) and (𝑦,πœ‡)β‰ (𝜏(π‘₯),πœ†), there is π‘“βˆˆπ΄ such that ‖𝑓‖𝑋,𝑝𝛼=1, Re(πœ†π‘“(π‘₯))=1, but Re(πœ‡π‘“(𝑦))β‰ 1.
By condition (i), there is π‘”βˆˆπ΄ such that ‖𝑔‖𝑋,𝑝𝛼≀1 and 𝑔(π‘₯)=πœ†.
Case 1 (π‘¦βˆˆ{π‘₯,𝜏(π‘₯)}). Let 𝑓=𝑔. Then, ‖𝑓‖𝑋,𝑝𝛼≀1 and |𝑓(π‘₯)|=|πœ†|=1 so that ‖𝑓‖𝑋,𝑝𝛼=1. Also, ξ€·πœ†Re(πœ†π‘“(π‘₯))=Re(πœ†π‘”(π‘₯))=Re2ξ€Έ=1.(2.13) If 𝑦=π‘₯, then (𝑦,πœ‡)β‰ (π‘₯,πœ†) implies that πœ‡β‰ πœ† that is, πœ‡πœ†β‰ 1. Since |πœ‡πœ†|=1, this shows that Re(πœ‡π‘“(π‘₯))=Re(πœ‡π‘”(π‘₯))=Re(πœ‡πœ†)β‰ 1.(2.14) If 𝑦=𝜏(π‘₯), then (𝑦,πœ‡)β‰ (𝜏(π‘₯),πœ†) implies that πœ‡β‰ πœ†, that is, πœ‡πœ†β‰ 1. Since |πœ‡πœ†|=1, this shows that ξ‚€πœ‡Re(πœ‡π‘“(𝑦))=Re(πœ‡π‘”(𝜏(π‘₯)))=Reπœ†ξ‚=Re(πœ‡πœ†)β‰ 1.(2.15)Case 2 (π‘¦βˆ‰{π‘₯,𝜏(π‘₯)}). By condition (ii), there is β„Žβˆˆπ΄ such that β€–β„Žβ€–π‘‹,𝑝𝛼≀1, β„Ž(π‘₯)=1 and |β„Ž(𝑦)|<1. Let 𝑓=π‘”β„Ž. Now, ‖𝑓‖𝑋,𝑝𝛼≀1 and |𝑓(π‘₯)|=|𝑓(π‘₯)||𝑔(π‘₯)|=|𝑔(π‘₯)|=|πœ†|=1, so that ‖𝑓‖𝑋,𝑝𝛼=1. Also, ξ€·πœ†Re(πœ†π‘“(π‘₯))=Re(πœ†π‘”(π‘₯)β„Ž(π‘₯))=Re2ξ€Έ||||=||||||||≀||||=1,Re(πœ‡π‘“(𝑦))=Re(πœ‡π‘”(𝑦)β„Ž(𝑦))β‰€πœ‡π‘”(𝑦)β„Ž(𝑦)𝑔(𝑦)β„Ž(𝑦)β„Ž(𝑦)<1.(2.16) Thus, our claim is justified. Let 𝔛=Lip(𝑋,𝑑𝛼)π‘Ÿ and πœ‘=Μƒπœ‚π‘₯,πœ† in part (b) of Theorem 1.1. Consider πœ“βˆˆExt(π΅π”›βˆ—) such that πœ“(𝑓)=1 for all π‘“βˆˆπ΄ with πœ‘(𝑓)=1=‖𝑓‖𝑋,𝑝𝛼. By Proposition 2.1, πœ“=ξ‚πœ“π‘€,πœ‡ for some (𝑀,πœ‡)βˆˆπ‘ŠΓ—π•‹. Thus, ΔRe(πœ‡π‘€(𝑓))=1 for all π‘“βˆˆπ΄ with Re(πœ†π‘“(π‘₯))=1=‖𝑓‖𝑋,𝑝𝛼. Since πœ†1∈𝐴, β€–πœ†1‖𝑋,𝑝𝛼=1 and Re(πœ†(πœ†1)(π‘₯))=Re(πœ†2)=1, we have Re(πœ‡πœ“π›Ό(πœ†1)(𝑀))=1. If ξ‚π‘‹π‘€β€²βˆˆπ›½, then ξ€·Reπœ‡πœ“π›Όξ€·π‘€(πœ†1)ξ…žξ€·ξ€Έξ€Έ=Reπœ‡πœ†πœ“π›Όξ€·π‘€(1)ξ…žξ€·ξ€·π›½Μƒ1𝑀=Reπœ‡πœ†ξ€Έξ€·ξ…žβ‰€||𝛽̃1ξ€Έ(ξ€Έ||=||𝛽̃1π‘€ξ€Έξ€Έπœ‡πœ†π‘€β€²)ξ€Έξ€·ξ…žξ€Έ||≀‖‖𝛽̃1‖‖𝛽𝑋=β€–β€–Μƒ1‖‖𝑋=𝑝𝛼(1)=0.(2.17) Thus, it must be π‘€βˆˆπ‘‹. Choose 𝑀=𝑦. Then, (𝑦,πœ‡)βˆˆπ‘‹Γ—π•‹ and Re(πœ‡π‘“(𝑦))=1 for all π‘“βˆˆπ΄ with ‖𝑓‖𝑋,𝑝𝛼=1=Re(πœ†π‘“(π‘₯)). By our claim, we must have (𝑦,πœ‡)=(π‘₯,πœ†) or (𝑦,πœ‡)=(𝜏(π‘₯),πœ†).
If (𝑦,πœ‡)=(π‘₯,πœ†), then clearly πœ“||𝐴=ξ‚πœ“π‘¦,πœ‡||𝐴=ξ‚πœ“π‘₯,πœ†||𝐴=Μƒπœ‚π‘₯,πœ†=πœ‘.(2.18) Now, let (𝑦,πœ‡)=(𝜏(π‘₯),πœ†). Since πœ†βˆˆβ„, π‘“βˆˆπ΄ implies that ξ‚€πœ†πœ“(𝑓)=Re(πœ‡π‘“(𝑦))=Re(πœ†π‘“(𝜏(π‘₯)))=Re𝑓(π‘₯)=πœ†Re𝑓(π‘₯)=πœ†Re(𝑓(π‘₯))=Re(πœ†π‘“(π‘₯))=ξ‚πœ“π‘₯,πœ†(𝑓)=Μƒπœ‚π‘₯,πœ†(𝑓)=πœ‘(𝑓).(2.19) Therefore, πœ“|𝐴=πœ‘, again. Hence, we see that Μƒπœ‚(π‘₯,πœ†)=πœ‘βˆˆExt(π΅π΄βˆ—). This also shows that Μƒπœ‚πœ(π‘₯),πœ†=Μƒπœ‚π‘₯,πœ† for all (π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1}.

Conversely, our claim implies that if (π‘₯,πœ†)βˆˆπ‘„π΄, (𝑦,πœ‡)βˆˆπ‘‹Γ—π•‹ and Μƒπœ‚π‘¦,πœ‡=Μƒπœ‚π‘₯,πœ†, then (𝑦,πœ‡)=(π‘₯,πœ†) or (𝑦,πœ‡)=(𝜏(π‘₯),πœ†). Thus, for (π‘₯,πœ†)βˆˆπ‘„π΄ and (𝑦,πœ‡)βˆˆπ‘ƒβˆ©(𝑋×𝕋), we have Μƒπœ‚π‘¦,πœ‡=Μƒπœ‚π‘₯,πœ† if and only if (𝑦,πœ‡)=(π‘₯,πœ†) or (𝑦,πœ‡)=(𝜏(π‘₯),πœ†).

By using the above theorem, we give a characterization of extreme points of unit ball in the dual space of Lip(𝑋,𝜏,𝑑𝛼).

Corollary 2.3. Let 𝑃 be as in Theorem 2.2. For (𝑀,πœ†)βˆˆπ‘ƒ, letβ€‰β€‰Μƒπœ‚π‘€,πœ† denote the restriction of ξ‚πœ“π‘€,πœ† to Lip(𝑋,𝜏,𝑑𝛼). Then, ξ€½Μƒπœ‚π‘₯,πœ†βˆΆξ€Ύξ€·π΅(π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1}βŠ†ExtLip(𝑋,𝜏,𝑑𝛼)βˆ—ξ€ΈβŠ†ξ€½Μƒπœ‚π‘€,πœ†βˆΆξ€Ύ.(𝑀,πœ†)βˆˆπ‘ƒ(2.20) Further, if (π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1} and (𝑦,πœ‡)βˆˆπ‘ƒβˆ©(𝑋×𝕋), then πœ‚π‘¦,πœ‡=Μƒπœ‚π‘₯,πœ† if and only if (𝑦,πœ‡)=(π‘₯,πœ†) or (𝑦,πœ‡)=(𝜏(π‘₯),πœ†).

Proof. Let 𝐴=Lip(𝑋,𝜏,𝑑𝛼). By Theorem 2.2, we have 𝐡Extπ΄βˆ—ξ€ΈβŠ†ξ€½Μƒπœ‚π‘€,πœ†βˆΆξ€Ύ(𝑀,πœ†)βˆˆπ‘ƒ.(2.21) To prove {Μƒπœ‚π‘₯,πœ†βˆΆ(π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1}}βŠ†Ext(𝐡Lip(𝑋,𝜏,𝑑𝛼)βˆ—), it is enough to show that for every (π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1},(i)there is π‘”βˆˆLip(𝑋,𝜏,𝑑𝛼) with ‖𝑔‖𝑋,𝑝𝛼≀1 and 𝑔(π‘₯)=πœ†,(ii)for every π‘¦βˆˆπ‘₯⧡{π‘₯,𝜏(π‘₯)}, there is β„ŽβˆˆLip(𝑋,𝜏,𝑑𝛼) withβ€–β„Žβ€–π‘‹,𝑝𝛼||||≀1,β„Ž(π‘₯)=1,β„Ž(𝑦)<1.(2.22) Let (π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1}. We first define the function π‘”βˆΆπ‘‹β†’β„‚ by 𝑔(𝑧)=πœ†1βˆ’π‘‘(𝑧,π‘₯)𝑑(𝜏(𝑧),π‘₯)1+𝜌2+𝜌2βˆ’2𝛼,(2.23) where 𝜌=diam(𝑋)=sup{𝑑(𝑦,𝑧)βˆΆπ‘¦,π‘§βˆˆπ‘‹}. Clearly, π‘”βˆˆπΆ(𝑋,𝜏), 𝑔(π‘₯)=πœ†, ‖𝑔‖𝑋≀1 and π‘”βˆˆLip(𝑋,𝑑1). Let 𝑦,π‘§βˆˆπ‘‹ with 𝑦≠𝑧. Then, ||||𝑔(𝑦)βˆ’π‘”(𝑧)𝑑𝛼=||||(𝑦,𝑧)𝑑(𝑧,π‘₯)𝑑(𝜏(𝑧),π‘₯)βˆ’π‘‘(𝑦,π‘₯)𝑑(𝜏(𝑦),π‘₯)ξ€·1+𝜌2+𝜌2βˆ’2𝛼𝑑𝛼≀𝑑(𝑦,𝑧)1βˆ’π›Ό(𝑦,𝑧)ξ€·1+𝜌2+𝜌2βˆ’2𝛼≀(𝑑(𝑧,π‘₯)+𝑑(𝜏(𝑦),π‘₯))2𝜌𝜌1βˆ’π›Ό1+𝜌2+𝜌2βˆ’2𝛼≀1.(2.24)
Therefore, π‘”βˆˆLip(𝑋,𝑑𝛼) and 𝑝𝛼(𝑔)≀1. Consequently, π‘”βˆˆLip(𝑋,𝜏,𝑑𝛼‖),𝑔‖𝑋,𝑝𝛼≀1,𝑔(π‘₯)=πœ†.(2.25) Hence, (i) holds.
We now assume that π‘¦βˆˆπ‘‹β§΅{π‘₯,𝜏(π‘₯)} and define the function β„ŽβˆΆπ‘‹β†’β„‚ by β„Ž(𝑧)=1βˆ’πœ†π‘”(𝑧)𝑑(π‘₯,𝑧)𝑑(π‘₯,𝜏(𝑧))2ξ€·1+𝜌2+𝜌2βˆ’2𝛼.(2.26) Clearly, β„ŽβˆˆπΆ(𝑋,𝜏), β„Ž(π‘₯)=1, β€–β„Žβ€–π‘‹β‰€1 and there exists π›Ύβˆˆ(0,1] such that β„Ž(𝑦)=1βˆ’π›Ύ so that |β„Ž(𝑦)|<1. We define the complex-valued function 𝑔1 on 𝑋 by 𝑔1=(1/2)(1βˆ’πœ†π‘”). Then, 𝑔1∈𝐢(𝑋,𝜏), ‖𝑔1‖𝑋≀1/2, 𝑔1∈Lip(𝑋,𝑑1), 𝑝𝛼(𝑔1)≀1/2 and β„Ž=1βˆ’π‘”π‘”1. Thus, we have β„ŽβˆˆLip(𝑋,𝑑1), β„ŽβˆˆLip(𝑋,𝜏,𝑑𝛼) and 𝑝𝛼(β„Ž)≀𝑝𝛼‖‖𝑔(𝑔)1‖‖𝑋+‖𝑔‖𝑋𝑝𝛼𝑔1≀12+12=1,(2.27) so that β€–β„Žβ€–π‘‹,𝑝𝛼≀1. Since π‘¦βˆˆπ‘‹β§΅{π‘₯,𝜏(π‘₯)}, |β„Ž(𝑦)|=β„Ž(𝑦)<1. Thus, (ii) holds.

3. Real Subspaces of lip(𝑋,𝑑𝛼) Containing 1

Throughout this section, we assume that π›Όβˆˆ(0,1). Consider the mapping Ξ¦π›ΌβˆΆlip(𝑋,𝑑𝛼)→𝐢(π‘Š) by Φ𝛼=Ψ𝛼|lip(𝑋,𝑑𝛼). Then, Φ𝛼 is a linear isometric from (lip(𝑋,𝑑𝛼),‖⋅‖𝑋,𝑝𝛼) into 𝐢(π‘Š).

For each π‘₯βˆˆπ‘‹ and π‘€βˆˆπ‘Š, define the functionals 𝛿π‘₯ and ̃𝛿𝑀 in lip(𝑋,𝑑𝛼)βˆ— by 𝛿π‘₯(𝑓)=𝑓(π‘₯) and ̃𝛿𝑀(𝑓)=Φ𝛼(𝑓)(𝑀), respectively. Clearly, |𝛿π‘₯(𝑓)|≀‖𝑓‖𝑋,𝑝𝛼 for allπ‘“βˆˆlip(𝑋,𝑑𝛼) and |̃𝛿𝑀(𝑓)|≀‖𝑓‖𝑋,𝑝𝛼, and therefore, 𝛿π‘₯,Μƒπ›Ώπ‘€βˆˆπ΅lip(𝑋,𝑑𝛼)βˆ—. Moreover, ̃𝛿π‘₯=𝛿π‘₯ for all π‘₯βˆˆπ‘‹ and ̃𝛿𝑀=π‘’π‘€π‘œΞ¦π›Ό for all π‘€βˆˆπ‘Š.

We give a characterization of extreme points of the unit ball in the dual space lip(𝑋,𝑑𝛼) as the following.

Theorem 3.1. Every extreme point of 𝐡lip(𝑋,𝑑𝛼)βˆ— must be either the form πœ†π›Ώπ‘₯ with (π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹ or of the form πœ†Μƒπ›Ώπ‘€ with (πœ†,𝑀)βˆˆπ›½π‘‹Γ—π•‹. Moreover, πœ†π›Ώπ‘₯ is an extreme point of 𝐡lip(𝑋,𝑑𝛼)βˆ— for all (π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹.

Proof. Since (lip(𝑋,𝑑𝛼),‖⋅‖𝑋,𝑝𝛼) is a Banach space and Φ𝛼 is a linear isometry from (lip(𝑋,𝑑𝛼),‖⋅‖𝑋,𝑝𝛼) into (𝐢(π‘Š),β€–β‹…β€–π‘Š), we conclude that Φ𝛼(lip(𝑋,𝑑𝛼)) is a uniformly closed subspace of 𝐢(π‘Š). It is well known [2, page 441] that 𝐡Ext(Φ𝛼(lip(𝑋,𝑑𝛼)))βˆ—ξ€ΈβŠ†ξ‚†πœ†π‘’π‘€||Φ𝛼(lip(𝑋,𝑑𝛼))ξ‚‡βˆΆ(𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹.(3.1) Let 𝐴=Φ𝛼(lip(𝑋,𝑑𝛼)) and define Ξ›π›ΌβˆΆlip(𝑋,𝑑𝛼)→𝐴 by Λ𝛼(𝑓)=Φ𝛼(𝑓). Then, Λ𝛼 is a linear isometry from (lip(𝑋,𝑑𝛼),‖⋅‖𝑝𝛼) onto (𝐴,β€–β‹…β€–π‘Š), so Ξ›βˆ—π›Ό, the adjoint of Λ𝛼, is a linear isometry from (𝐴,β€–β‹…β€–π‘Š)βˆ— onto (lip(𝑋,𝑑𝛼))βˆ—. It is easily to show that Ξ›βˆ—π›Ό(𝑒𝑀|𝐴̃𝛿)=𝑀 for all π‘€βˆˆπ‘Š. Let πœ‘βˆˆExt(𝐡lip(𝑋,𝑑𝛼)βˆ—). Then, (Ξ›βˆ—π›Ό)βˆ’1(πœ‘)∈Ext(π΅π΄βˆ—). Thus, there exists (𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹ such that (Ξ›βˆ—π›Ό)βˆ’1(πœ‘)=πœ†π‘’π‘€|𝐴 so that πœ‘=πœ†βˆ—π›Ό(𝑒𝑀|𝐴̃𝛿)=πœ†π‘€. It follows that 𝐡Extlip(𝑋,𝑑𝛼)βˆ—ξ€ΈβŠ†ξ€½πœ†Μƒπ›Ώπ‘€βˆΆξ€Ύ.(𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹(3.2)
Now, let (π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹. Clearly, πœ†π›Ώπ‘₯∈𝐡lip(𝑋,𝑑𝛼)βˆ—. Assume that πœ“βˆˆExt(𝐡Lip(𝑋,𝑑𝛼)βˆ—) and πœ“(𝑓)=1 for all π‘“βˆˆlip(𝑋,𝑑𝛼) with (πœ†π›Ώπ‘₯)(𝑓)=1=‖𝑓‖𝑋,𝑝𝛼. Since lip(𝑋,𝑑𝛼) is a nonzero linear subspace of Lip(𝑋,𝑑𝛼), we conclude that πœ“=πœ‡Ξ”π‘¦ for some (𝑦,πœ‡)βˆˆπ‘‹Γ—π•‹ or ξ‚Ξ”πœ“=πœ‡π‘€ for some (𝑀,πœ‡)∈(𝛽𝑋)×𝕋 by Theorem 1.1. Clearly, πœ†1∈lip(𝑋,𝑑𝛼) and (πœ†π›Ώπ‘₯)(πœ†1)=1=β€–πœ†1‖𝑋,𝑝𝛼. If ξ‚Ξ”πœ“=πœ‡π‘€ for some (𝑀,πœ‡)∈(𝛽𝑋)×𝕋, then we have |||πœ“ξ‚€1=|||=|||πœ‡ξ‚Ξ”πœ†1𝑀|||=||ξ‚Ξ”πœ†1𝑀||=||𝛽̃1ξ€Έ||≀‖‖𝛽̃1β€–β€–(1)(𝑀)𝛽𝑋=𝑝𝛼(1)=0.(3.3) Thus, it must be πœ“=πœ‡Ξ”π‘¦ for some (𝑦,πœ‡)βˆˆπ‘‹Γ—π•‹. Since πœ“(πœ†1)=1, we have ξ€·1=πœ‡Ξ”π‘¦ξ€Έξ‚€ξ‚πœ†1=πœ‡πœ†.(3.4) It follows that πœ‡=πœ†. We claim that 𝑦=π‘₯. Let π‘¦βˆˆπ‘‹β§΅{π‘₯}. We define the function π‘”βˆΆπ‘‹β†’β„‚ by 𝑔(𝑧)=1βˆ’π‘‘(𝑧,π‘₯)1+𝜌+𝜌1βˆ’π›Ό,(3.5) where 𝜌=diam𝑋. It is easy to show that π‘”βˆˆLip(𝑋,𝑑1), 𝑔(π‘₯)=1, ‖𝑔‖𝑋=1 and 𝑝𝛼(𝑔)≀1. Now, we define the function π‘“βˆΆπ‘‹β†’β„‚ by 𝑓(𝑧)=πœ†ξƒ©1βˆ’π‘”(𝑧)𝑑(𝑧,π‘₯)2ξ€·1+𝜌+𝜌1βˆ’π›Όξ€Έξƒͺ,(3.6) and the function 𝑔1βˆΆπ‘‹β†’β„‚ by 𝑔1=(1/2)(1βˆ’π‘”). It is easy to see that 𝑔1∈Lip(𝑋,𝑑1), ‖𝑔1‖𝑋≀1/2, 𝑝𝛼(𝑔1)≀1/2, and 𝑓=πœ†(1βˆ’π‘”π‘”1). Therefore, we have π‘“βˆˆLip(𝑋,𝑑1)βŠ†lip(𝑋,𝑑𝛼), 𝑓(π‘₯)=πœ†, ‖𝑓‖𝑋≀1, and 𝑝𝛼(𝑓)≀𝑝𝛼‖‖𝑔(𝑔)1‖‖𝑋+‖𝑔‖𝑋𝑝𝛼𝑔1≀12+12=1,(3.7) so that ‖𝑓‖𝑋,𝑝=1=(πœ†π›Ώπ‘₯)(𝑓). It must be that πœ“(𝑓)=1; that is, (πœ†Ξ”π‘¦)(𝑓)=1. But ||ξ€·πœ†Ξ”π‘¦ξ€Έ||=||||=||𝑓||(𝑓)πœ†π‘“(𝑦)(𝑦)=1βˆ’π‘”(𝑦)𝑑(𝑦,π‘₯)2ξ€·1+𝜌+𝜌1βˆ’π›Όξ€Έ<1.(3.8