Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2012, Article IDΒ 735139, 13 pages
http://dx.doi.org/10.5402/2012/735139
Research Article

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

Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

Received 20 October 2011; Accepted 16 November 2011

Academic Editor: C.Β Zhu

Copyright Β© 2012 Davood Alimohammadi and Hadis Pazandeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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) This contradiction implies that 𝑦=π‘₯ and so our claim is justified. Therefore, πœ“=πœ†Ξ”π‘₯. It follows that πœ“|lip(𝑋,𝑑𝛼)=πœ†π›Ώπ‘₯. Therefore, πœ†π›Ώπ‘₯∈Ext(𝐡lip(𝑋,𝑑𝛼)βˆ—) by part (b) of Theorem 1.1. Consequently, ξ€½πœ†π›Ώπ‘₯βˆΆξ€Ύξ€·π΅(π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹βŠ†Extlip(𝑋,𝑑𝛼)βˆ—ξ€ΈβŠ†ξ€½πœ†Μƒπ›Ώπ‘€βˆΆξ€Ύ.(𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹(3.9) Thus, the proof is complete.

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†Μƒπ›Ώ(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†π›Ώπ‘₯) for all (π‘₯,πœ†)βˆˆπ‘‹Γ—π•‹.

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

Proposition 3.2. 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•‹(3.10) 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. The proof is similar to that of Proposition 2.1 by replacing lip(𝑋,𝑑𝛼),̃𝛿π‘₯,̃𝛿𝑀,ξ‚πœ‘π‘₯,πœ†,ξ‚πœ‘π‘€,πœ† and Theorems 1.2 and 1.3 by Lip(𝑋,𝑑𝛼),Ξ”π‘₯,Δ𝑀,ξ‚πœ“π‘₯,πœ†,ξ‚πœ“π‘€,πœ† and Theorem 3.1, respectively.

Theorem 3.3. Let 𝐴 be a real linear subspace of lip(𝑋,𝜏,𝑑𝛼) containing 1. For (𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹, let Μƒπœƒπ‘€,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—π•‹βˆΆπœ(π‘₯)=π‘₯,πœ†βˆˆ{βˆ’1,1}}βˆͺ{(𝑀,πœ†)βˆˆπ‘ŠΓ—π•‹βˆΆπ‘€βˆˆπ›½π‘‹}. Let 𝑄𝐴 denote the set of all (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 β„Žβˆˆπ΄ with β€–β„Žβ€–π‘‹,𝑝𝛼≀1, β„Ž(π‘₯)=1 and |β„Ž(𝑦)|<1. Then, ξ€½Μƒπœƒπ‘₯,πœ†βˆΆ(π‘₯,πœ†)βˆˆπ‘„π΄ξ€Ύξ€·π΅βŠ†Extπ΄βˆ—ξ€ΈβŠ†ξ€½Μƒπœƒπ‘€,πœ†βˆΆξ€Ύ(𝑀,πœ†)βˆˆπ‘ƒ.(3.11) Further, if (π‘₯,πœ†)βˆˆπ‘„π΄ and (𝑦,πœ‡)βˆˆπ‘ƒβˆ©(𝑋×𝕋), then Μƒπœƒπ‘¦,πœ‡=Μƒπœƒπ‘₯,πœ† if and only if (𝑦,πœ‡)=(π‘₯,πœ†) or (𝑦,πœ‡)=(𝜏(π‘₯),πœ†).

Proof. The proof is similar to that of Theorem 2.2 by replacing lip(𝑋,π‘‘π›ΌΜƒπœƒ),𝑀,πœ†,Μƒπœƒπ‘₯,πœ†,ξ‚πœ‘π‘₯,πœ†, Proposition 3.2 and Φ𝛼 by Lip(𝑋,𝑑𝛼),Μƒπœ‚π‘€,πœ†,Μƒπœ‚π‘₯,πœ†,ξ‚πœ“π‘₯,πœ†, Proposition 2.1 and Ψ𝛼, respectively.

Corollary 3.4. Let 𝑃 be as in Theorem 3.3. For (𝑀,πœ†)βˆˆπ‘ƒ, let Μƒπœƒπ‘€,πœ† denote the restriction of ξ‚πœ‘π‘€,πœ† to lip(𝑋,𝜏,𝑑𝛼). Then, ξ€½Μƒπœƒπ‘₯,πœ†βˆΆξ€Ύξ€·π΅(π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1}βŠ†Extlip(𝑋,𝜏,𝛼)βˆ—ξ€ΈβŠ†ξ€½Μƒπœƒπ‘€,πœ†βˆΆξ€Ύ.(𝑀,πœ†)βˆˆπ‘ƒ(3.12) Further, if (π‘₯,πœ†)βˆˆπ‘‹Γ—{βˆ’1,1} and (𝑦,πœ‡)βˆˆπ‘ƒβˆ©(𝑋×𝕋), then πœƒπ‘¦,πœ‡=Μƒπœƒπ‘₯,πœ† if and only if (𝑦,πœ‡)=(π‘₯,πœ†) or (𝑦,πœ‡)=(𝜏(π‘₯),πœ†).

Proof. The proof is similar to that Corollary 2.3 by replacing lip(𝑋,𝜏,π‘‘π›ΌΜƒπœƒ),𝑀,πœ†,Μƒπœƒπ‘₯,πœ† and lip(𝑋,𝑑𝛼) by Lip(𝑋,𝜏,𝑑𝛼),Μƒπœ‚π‘€,πœ†,Μƒπœ‚π‘₯,πœ† and Lip(𝑋,𝑑𝛼), respectively.

References

  1. S. H. Kulkarni and B. V. Limaye, β€œExtreme points of the unit ball in the dual spaces of some real subspaces of C(X),” Glasnik Matematički. Serija III, vol. 29(49), no. 2, pp. 333–340, 1994. View at Google Scholar Β· View at Zentralblatt MATH
  2. N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience, New York, NY, USA, 1958.
  3. S. H. Kulkarni and B. V. Limaye, β€œGleason parts of real function algebras,” Canadian Journal of Mathematics, vol. 33, no. 1, pp. 181–200, 1981. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. S. H. Kulkarni and B. V. Limaye, Real Function Algebras, vol. 168 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992.
  5. M. Grzesiak, β€œExtreme points of the unit ball in the dual space of some real Banach algebra of continuous complex functions,” Fasciculi Mathematici, no. 16, pp. 5–10, 1986. View at Google Scholar Β· View at Zentralblatt MATH
  6. K. de Leeuw, β€œBanach spaces of Lipschitz functions,” Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica, vol. 21, pp. 55–66, 1961/1962. View at Google Scholar
  7. A. K. Roy, β€œExtreme points and linear isometries of the Banach space of Lipschitz functions,” Canadian Journal of Mathematics, vol. 20, pp. 1150–1164, 1968. View at Google Scholar Β· View at Zentralblatt MATH
  8. A. Jiménez-Vargas and M. Villegas-Vallecillos, β€œInto linear isometries between spaces of Lipschitz functions,” Houston Journal of Mathematics, vol. 34, no. 4, pp. 1165–1184, 2008. View at Google Scholar Β· View at Zentralblatt MATH
  9. D. Alimohammadi and A. Ebadian, β€œHedberg's theorem in real Lipschitz algebras,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 10, pp. 1479–1493, 2001. View at Google Scholar