International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 735139 | https://doi.org/10.5402/2012/735139

Davood Alimohammadi, Hadis Pazandeh, "Extreme Points of the Unit Ball in the Dual Space of Some Real Subspaces of Banach Spaces of Lipschitz Functions", International Scholarly Research Notices, vol. 2012, Article ID 735139, 13 pages, 2012. https://doi.org/10.5402/2012/735139

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

Academic Editor: C. Zhu
Received20 Oct 2011
Accepted16 Nov 2011
Published11 Jan 2012

Abstract

Let ๐‘‹ be a compact Hausdorff space, ๐œ be a continuous involution on ๐‘‹ and ๐ถ(๐‘‹,๐œ) denote the uniformly closed real subalgebra of ๐ถ(๐‘‹) consisting of all ๐‘“โˆˆ๐ถ(๐‘‹) for which ๐‘“โˆ˜๐œ=๐‘“. Let (๐‘‹,๐‘‘) be a compact metric space and let Lip(๐‘‹,๐‘‘๐›ผ) denote the complex Banach space of complex-valued Lipschitz functions of order ๐›ผ on (๐‘‹,๐‘‘) under the norm โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ=max{โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ(๐‘“)}, where ๐›ผโˆˆ(0,1]. For ๐›ผโˆˆ(0,1), the closed subalgebra of Lip(๐‘‹,๐›ผ) consisting of all ๐‘“โˆˆLip(๐‘‹,๐‘‘๐›ผ) for which |๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)|/๐‘‘๐›ผ(๐‘ฅ,๐‘ฆ)โ†’0 as ๐‘‘(๐‘ฅ,๐‘ฆ)โ†’0, denotes by lip(๐‘‹,๐‘‘๐›ผ). Let ๐œ be a Lipschitz involution on (๐‘‹,๐‘‘) and define Lip(๐‘‹,๐œ,๐‘‘๐›ผ)=Lip(๐‘‹,๐‘‘๐›ผ)โˆฉ๐ถ(๐‘‹,๐œ) for ๐›ผโˆˆ(0,1] and lip(๐‘‹,๐œ,๐‘‘๐›ผ)=lip(๐‘‹,๐‘‘๐›ผ)โˆฉ๐ถ(๐‘‹,๐œ) for ๐›ผโˆˆ(0,1). In this paper, we give a characterization of extreme points of ๐ต๐ดโˆ—, where ๐ด is a real linear subspace of Lip(๐‘‹,๐‘‘๐›ผ) or lip(๐‘‹,๐‘‘๐›ผ) which contains 1, in particular, Lip(๐‘‹,๐œ,๐‘‘๐›ผ) or lip(๐‘‹,๐œ,๐‘‘๐›ผ).

1. Introduction and Preliminaries

We let โ„,โ„‚, and ๐•‹={๐‘งโˆˆโ„‚โˆถ|๐‘ง|=1} denote the field of real numbers, complex numbers, and the unit circle, respectively. The symbol ๐•‚ denotes a field that can be either โ„ or โ„‚. The elements of ๐•‚ are called scalars.

Let ๐”› be a normed space over ๐•‚. We denote by ๐”›โˆ— and ๐ต๐”› the dual space ๐”› and the closed unit ball of ๐”›, respectively. If ๐‘† is a subset ๐”›, let Ext(๐‘†) denote the set of all extreme points of ๐‘†. Let ๐ด be a subspace of ๐”› and ๐œ‘โˆˆ๐ดโˆ—. A Hahn-Banach extension of ๐œ‘ to ๐”› is a continuous linear functional ๐œ“โˆˆ๐”›โˆ— such that ๐œ“|๐ด=๐œ‘ and ||๐œ“||=||๐œ‘||. The set of all Hahn-Banach extensions of ๐œ‘ to ๐”› will be denoted by ๐ป๐œ‘.

It is easy to see that if ๐”› and ๐’ด are normed spaces over ๐•‚ and ๐‘‡โˆถ๐”›โ†’๐’ด is a linear isometry from ๐”› onto ๐’ด, then ๐‘‡ is a bijection mapping between Ext(๐ต๐”›) and Ext(๐ต๐’ด).

For a complex normed space ๐”›, we assume that ๐”›๐‘Ÿ denotes ๐”›, regarded as a real normed space by restricting the scalar multiplication to real numbers.

Kulkarni and Limaye gave some conditions for ๐œ‘โˆˆ๐ต๐ดโˆ— to be an extreme point of ๐ต๐ดโˆ— in terms of the Hahn-Banach extension of ๐œ‘ to ๐”› and the extreme points of ๐ต๐”›โˆ— as the following.

Theorem 1.1 (see [1, Theoremโ€‰โ€‰2]). Let ๐”› be a normed space over ๐•‚,๐ด be a nonzero linear subspace of ๐”› and ๐œ‘โˆˆ๐ต๐ดโˆ—. (a)Let ๐œ‘โˆˆExt(๐ต๐ดโˆ—). Then, ๐ป๐œ‘๎€ท๐ตโˆฉExt๐”›โˆ—๎€ธ๎€ท๐ป=Ext๐œ‘๎€ธโ‰ โˆ….(1.1)In particular, ๐œ‘ has an extension to some ๐œ“โˆˆExt(๐ต๐”›โˆ—). Further, if such an extension is unique, then ๐œ‘ has a unique Hahn-Banach extension to ๐”›.(b)Assume that whenever ๐œ“โˆˆExt(๐ต๐”›โˆ—) and ๐œ“(๐‘“)=1 for all ๐‘“โˆˆ๐ด with ๐œ‘(๐‘“)=1=โ€–๐‘“โ€–, one has ๐œ“|๐ด=๐œ‘, then ๐œ‘โˆˆExt(๐ต๐ดโˆ—).(c)If ๐œ‘ has a unique Hahn-Banach extension ๐œ“ to ๐”› and if ๐œ“โˆˆExt(๐ต๐”›โˆ—), then ๐œ‘โˆˆExt(๐ต๐ดโˆ—).

Let ๐‘‹ be a compact Hausdorff space. We denote by ๐ถ(๐‘‹) the complex Banach algebra of all continuous complex-valued functions on ๐‘‹ under the uniform norm โ€–๐‘“โ€–๐‘‹=sup{|๐‘“(๐‘ฅ)|โˆถ๐‘ฅโˆˆ๐‘‹}. For ๐‘ฅโˆˆ๐‘‹, consider the evaluation functional ๐‘’๐‘ฅ given by ๐‘’๐‘ฅ(๐‘“)=๐‘“(๐‘ฅ),๐‘“โˆˆ๐ถ(๐‘‹). Clearly, ๐œ†๐‘’๐‘ฅโˆˆ๐ต๐ถ(๐‘‹)โˆ— for all (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹. It is well known [2, page 441] that๎€ท๐ตExt๐ถ(๐‘‹)โˆ—๎€ธ=๎€ฝ๐œ†๐‘’๐‘ฅโˆถ๎€พ.(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹(1.2)

For ๐‘ฅโˆˆ๐‘‹ and ๐œ†โˆˆ๐•‹, we define the map ๐œ“๐‘ฅ,๐œ†โˆถ๐ถ(๐‘‹)๐‘Ÿโ†’โ„ by ๐œ“๐‘ฅ,๐œ†(๐‘“)=Re(๐œ†๐‘“(๐‘ฅ)) in fact, ๐œ“๐‘ฅ,๐œ†=Re(๐œ†๐‘’๐‘ฅ). Clearly, ๐œ“๐‘ฅ,๐œ†โˆˆ๐ต(๐ถ(๐‘‹)๐‘Ÿ)โˆ— for all (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹. Kulkarni and Limaye showed [1, Propositionโ€‰โ€‰3] that๎€ท๐ตExt(๐ถ(๐‘‹)๐‘Ÿ)โˆ—๎€ธ=๎€ฝ๐œ“๐‘ฅ,๐œ†โˆถ๎€พ,(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹(1.3) and ๐œ“๐‘ฆ,๐œ‡=๐œ“๐‘ฅ,๐œ† if and only if ๐‘ฆ=๐‘ฅ and ๐œ‡=๐œ†.

Let ๐œ be a continuous involution on ๐‘‹; that is, ๐œโˆถ๐‘‹โ†’๐‘‹ is continuous and ๐œโˆ˜๐œ is the identity map on ๐‘‹. The map ๐œŽโˆถ๐ถ(๐‘‹)โ†’๐ถ(๐‘‹) defined by ๐œŽ(๐‘“)=๐‘“โˆ˜๐œ, is an algebra involution on ๐ถ(๐‘‹) which is called the algebra involution induced by ๐œ on ๐ถ(๐‘‹). Define ๐ถ(๐‘‹,๐œ)={๐‘“โˆˆ๐ถ(๐‘‹)โˆถ๐œŽ(๐‘“)=๐‘“}. Then, ๐ถ(๐‘‹,๐œ) is a uniformly closed real subalgebra of ๐ถ(๐‘‹) which contains 1. The real algebras ๐ถ(๐‘‹,๐œ) were first considered in [3]. For a detailed account of several properties of ๐ถ(๐‘‹,๐œ), we refer to [4].

Let ๐‘ƒ={(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹โˆถ๐œ(๐‘ฅ)โ‰ ๐‘ฅ}โˆช{(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹โˆถ๐œ(๐‘ฅ)=๐‘ฅ,๐œ†โˆˆ{โˆ’1,1}}. For each (๐‘ฅ,๐œ†)โˆˆ๐‘ƒ, let ๐œ‘๐‘ฅ,๐œ† denote the restriction of ๐œ“๐‘ฅ,๐œ† to ๐ถ(๐‘‹,๐œ). Grzesiak obtained a characterization of the extreme points of ๐ต(๐ถ(๐‘‹,๐œ))โˆ— in [5] and showed that ๐œ‘โˆˆExt(๐ต(๐ถ(๐‘‹,๐œ))โˆ—) if and only if ๐œ‘=๐œ‘๐‘ฅ,๐œ† for some (๐‘ฅ,๐œ†)โˆˆ๐‘ƒ. Further, if (๐‘ฅ,๐œ†),(๐‘ฆ,๐œ‡)โˆˆ๐‘ƒ, then ๐œ‘๐‘ฆ,๐œ‡=๐œ‘๐‘ฅ,๐œ† if and only if (๐‘ฆ,๐œ‡)=(๐‘ฅ,๐œ†) or (๐‘ฆ,๐œ‡)=(๐œ(๐‘ฅ),๐œ†).

Kulkarni and Limaye obtained [1, Theoremโ€‰โ€‰4] a characterization of Ext(๐ต๐ดโˆ—), where ๐ด is a nonzero real linear subspace of ๐ถ(๐‘‹,๐œ).

Let (๐‘‹,๐‘‘) be a compact metric space. For ๐›ผโˆˆ(0,1], we denote by Lip(๐‘‹,๐‘‘๐›ผ) the set of all complex-valued functions ๐‘“ on ๐‘‹ for which๐‘๐›ผ๎‚ป||||(๐‘“)=sup๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)๐‘‘๐›ผ๎‚ผ(๐‘ฅ,๐‘ฆ)โˆถ๐‘ฅ,๐‘ฆโˆˆ๐‘‹,๐‘ฅโ‰ ๐‘ฆ(1.4) is finite. Then, Lip(๐‘‹,๐›ผ) is a complex subalgebra of ๐ถ(๐‘‹) containing 1 and complex Banach space under the normโ€–๐‘“โ€–๐‘‹,๐‘๐›ผ๎€ฝ=maxโ€–๐‘“โ€–๐‘‹,๐‘๐›ผ๎€พ(๐‘“)(๐‘“โˆˆLip(๐‘‹,๐‘‘๐›ผ)).(1.5) For ๐›ผโˆˆ(0,1), the complex subalgebra of Lip(๐‘‹,๐‘‘๐›ผ) consisting of all ๐‘“โˆˆLip(๐‘‹,๐‘‘๐›ผ) for which||||๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)๐‘‘๐›ผ(๐‘ฅ,๐‘ฆ)โŸถ0as๐‘‘(๐‘ฅ,๐‘ฆ)โŸถ0,(1.6) is denoted by lip(๐‘‹,๐‘‘๐›ผ). Clearly, lip(๐‘‹,๐‘‘๐›ผ) is a closed linear subspace of (Lip(๐‘‹,๐‘‘๐›ผ),โ€–โ‹…โ€–๐‘‹,๐‘๐›ผ) and 1โˆˆlip(๐‘‹,๐‘‘๐›ผ). These Banach spaces were first studied by Leeuw in [6].

Given a compact metric space (๐‘‹,๐‘‘), let ๎‚๐‘‹={(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘‹โˆถ๐‘ฅโ‰ ๐‘ฆ}, and let the compact Hausdorff space ๐‘Š be the disjoint union of ๐‘‹ with ๐›ฝ๎‚๐‘‹, where ๐›ฝ๎‚๐‘‹ is the Stone-Cech compactification of ๎‚๐‘‹. For ๐›ผโˆˆ(0,1], consider the mapping ฮจ๐›ผโˆถLip(๐‘‹,๐‘‘๐›ผ)โ†’๐ถ(๐‘Š) defined for each ๐‘“โˆˆLip(๐‘‹,๐‘‘๐›ผ) byฮจ๐›ผ๎ƒฏ๎‚€๐›ฝ๎‚๐‘“๎‚(๎‚(๐‘“)(๐‘ค)=๐‘“(๐‘ค)if๐‘คโˆˆ๐‘‹,๐‘ค)if๐‘คโˆˆ๐›ฝ๐‘‹,(1.7) where๎‚๐‘“(๐‘ฅ,๐‘ฆ)=๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)๐‘‘๐›ผ๎‚(๐‘ฅ,๐‘ฆ),โˆ€(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹,(1.8) and ๐›ฝ๎‚๐‘“ is the norm-preserving extension of ๎‚๐‘“ to ๐›ฝ๎‚๐‘‹. Clearly, ฮจ๐›ผ is a linear isometry from (Lip(๐‘‹,๐‘‘๐›ผ),โ€–โ‹…โ€–๐‘‹,๐‘๐›ผ) into (๐ถ(๐‘Š),โ€–โ‹…โ€–๐‘Š), which is called the Leeuwโ€™s linear isometry. Therefore, ฮจ๐›ผ(Lip(๐‘‹,๐‘‘๐›ผ)) is a uniformly closed linear subspace of ๐ถ(๐‘Š). It is well known (see [2, page 441]) that๎€ท๐ตExtฮจ๐›ผ(Lip(๐‘‹,๐‘‘๐›ผ))โˆ—๎€ธโІ๎€ฝ๐œ†๐‘’๐‘คโˆฃฮจ๐›ผ(Lip(๐‘‹,๐‘‘๐›ผ))โˆถ๎€พ,(๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹(1.9) where ๐‘’๐‘ค is the evaluation functional at ๐‘ค on ๐ถ(๐‘Š).

For each ๐‘ฅโˆˆ๐‘‹ and ๐‘คโˆˆ๐‘Š, define the linear functionals ฮ”๐‘ฅ and ๎‚ฮ”๐‘ค in Lip(๐‘‹,๐‘‘๐›ผ)โˆ— by ฮ”๐‘ฅ(๐‘“)=๐‘“(๐‘ฅ) and ๎‚ฮ”๐‘ค(๐‘“)=ฮจ๐›ผ(๐‘“)(๐‘ค), respectively. Clearly, |ฮ”๐‘ฅ(๐‘“)|โ‰คโ€–๐‘“โ€–๐‘‹,๐‘๐›ผ and |๎‚ฮ”๐‘ค(๐‘“)|โ‰คโ€–๐‘“โ€–๐‘‹,๐‘๐›ผ for all ๐‘“โˆˆLip(๐‘‹,๐‘‘๐›ผ). Therefore, ฮ”๐‘ฅ,๎‚ฮ”๐‘คโˆˆ๐ตLip(๐‘‹,๐‘‘๐›ผ)โˆ—. Moreover, ๎‚ฮ”๐‘ฅ=ฮ”๐‘ฅ for all ๐‘ฅโˆˆ๐‘‹โ€‰โ€‰and ๎‚ฮ”๐‘ค=๐‘’๐‘ค๐‘œฮจ๐›ผ for all ๐‘คโˆˆ๐‘Š. Thus, we have the following result.

Theorem 1.2. For ๐›ผโˆˆ(0,1], every extreme point of ๐ตLip(๐‘‹,๐‘‘๐›ผ)โˆ— must be either of the form ๐œ†ฮ”๐‘ฅ with (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹ or of the form ๐œ†๎‚ฮ”๐‘ค with ๎‚(๐‘ค,๐œ†)โˆˆ๐›ฝ๐‘‹ร—๐•‹.

Roy proved the following result by using a result of Leeuw [6, Lemmaโ€‰โ€‰1.2].

Theorem 1.3 (see [7, Lemmaโ€‰โ€‰1.2]). For each (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹, ๐œ†ฮ”๐‘ฅ is an extreme point of ๐ตLip(๐‘‹,๐‘‘๐›ผ)โˆ—.

Jimenez-Vargas and Villegas-Vallecillos used above results and obtained a characterization of linear isometries between Lip(๐‘‹,๐‘‘๐‘‹) and Lip(๐‘Œ,๐‘‘๐‘Œ) in [8].

A map ๐‘“โˆถ๐‘‹โ†’๐‘Œ is said to be Lipschitz map from the metric space (๐‘‹,๐‘‘๐‘‹) to the metric space (๐‘Œ,๐‘‘๐‘Œ) if there exists a constant ๐ถ>0 such that ๐‘‘๐‘Œ(๐‘“(๐‘ฅ),๐‘“(๐‘ฆ))โ‰ค๐ถ๐‘‘๐‘‹(๐‘ฅ,๐‘ฆ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹.

Let (๐‘‹,๐‘‘) be a compact metric space. The mapping ๐œโˆถ๐‘‹โ†’๐‘‹ is called a Lipschitz involution on (๐‘‹,๐‘‘), if ๐œ is a Lipschitz map from (๐‘‹,๐‘‘) to itself and an involution on ๐‘‹. Clearly, every Lipschitz involution on (๐‘‹,๐‘‘) is a continuous involution.

Let ๐œ be a Lipschitz involution on the compact metric space (๐‘‹,๐‘‘) and let ๐œŽ be the algebra involution induced by ๐œ on ๐ถ(๐‘‹). Clearly,๐œŽ(Lip(๐‘‹,๐‘‘๐›ผ))=Lip(๐‘‹,๐‘‘๐›ผ),๐œŽ(lip(๐‘‹,๐‘‘๐›ผ))=lip(๐‘‹,๐‘‘๐›ผ).(1.10) We defineLip(๐‘‹,๐œ,๐‘‘๐›ผ)={โ„ŽโˆˆLip(๐‘‹,๐‘‘๐›ผ)โˆถ๐œŽ(โ„Ž)=โ„Ž},lip(๐‘‹,๐œ,๐‘‘๐›ผ)={โ„Žโˆˆlip(๐‘‹,๐‘‘๐›ผ)โˆถ๐œŽ(โ„Ž)=โ„Ž}.(1.11) Then, the following statements hold.(i)Lip(๐‘‹,๐œ,๐‘‘๐›ผ) (lip(๐‘‹,๐œ,๐‘‘๐›ผ), resp.) is a real subalgebra of Lip(๐‘‹,๐‘‘๐›ผ) (lip(๐‘‹,๐‘‘๐›ผ), resp.).(ii)Lip(๐‘‹,๐œ,๐‘‘๐›ผ)=Lip(๐‘‹,๐‘‘๐›ผ)โˆฉ๐ถ(๐‘‹,๐œ) and lip(๐‘‹,๐œ,๐‘‘๐›ผ)=lip(๐‘‹,๐‘‘๐›ผ)โˆฉ๐ถ(๐‘‹,๐œ).(iii)Lip(๐‘‹,๐‘‘๐›ผ)=Lip(๐‘‹,๐œ,๐‘‘๐›ผ)โŠ•๐‘–Lip(๐‘‹,๐œ,๐‘‘๐›ผ) and lip(๐‘‹,๐‘‘๐›ผ)=lip(๐‘‹,๐œ,๐‘‘๐›ผ)โŠ•๐‘–lip(๐‘‹,๐œ,๐‘‘๐›ผ).(iv)Lip(๐‘‹,๐œ,๐‘‘๐›ผ) (lip(๐‘‹,๐œ,๐‘‘๐›ผ), resp.) is a real subalgebra of ๐ถ(๐‘‹,๐œ) which contains 1 and separates the points of ๐‘‹.(v)Lip(๐‘‹,๐œ,๐‘‘๐›ผ) (lip(๐‘‹,๐œ,๐‘‘๐›ผ), resp.) is uniformly dense in ๐ถ(๐‘‹,๐œ) (use (iv) and the Stone-Weierstrass theorem for real subalgebra of ๐ถ(๐‘‹,๐œ) [3, Propositionโ€‰โ€‰1.1].(vi)For 0<๐›ผ<๐›ฝโ‰ค1, ๎€ทLip๐‘‹,๐œ,๐‘‘๐›ฝ๎€ธโІlip(๐‘‹,๐œ,๐‘‘๐›ผ)โІLip(๐‘‹,๐œ,๐‘‘๐›ผ).(1.12)(vii)There exists a constant ๐ถโ‰ฅ1 such that ๎€ฝmaxโ€–๐‘“โ€–๐‘‹,๐‘๐›ผ,โ€–๐‘”โ€–๐‘‹,๐‘๐›ผ๎€พโ‰ค๐ถ๐›ผโ€–๐‘“+๐‘”โ€–๐‘‹,๐‘๐›ผ,(1.13) for all ๐‘“,๐‘”โˆˆLip(๐‘‹,๐œ,๐‘‘๐›ผ).(viii)Lip(๐‘‹,๐œ,๐‘‘๐›ผ),โ€–โ‹…โ€–๐‘‹,๐‘๐›ผ) is a real Banach space and lip(๐‘‹,๐œ,๐‘‘๐›ผ) is its closed real subspace.

The real Banach spaces Lip(๐‘‹,๐œ,๐‘‘๐›ผ) and lip(๐‘‹,๐œ,๐‘‘๐›ผ) 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(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ, Lip(๐‘‹,๐œ,๐‘‘๐›ผ) and some its real linear subspaces for ๐›ผโˆˆ(0,1] in Section 2. Next, we give a characterization of extreme points of the unit ball in the dual spaces of lip(๐‘‹,๐‘‘๐›ผ),lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ,lip(๐‘‹,๐œ,๐‘‘๐›ผ) and some its real linear subspaces for ๐›ผโˆˆ(0,1) in Section 3.

2. Real Linear Subspaces of Lip(๐‘‹,๐‘‘๐›ผ) Containing 1

In the remainder of this paper, we assume that ๐›ผโˆˆ(0,1], (๐‘‹,๐‘‘) is a compact metric space, ๎‚๐‘‹={(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘‹,๐‘ฅโ‰ ๐‘ฆ}, ๐›ฝ๎‚๐‘‹ is the Stone-Cech compactification of ๎‚๐‘‹, ๐‘Š is the compact Hausdorff space ๎‚๐‘‹๐‘‹โˆช๐›ฝ, ฮจ๐›ผ is the Leeuwโ€™s linear isometry from (Lip(๐‘‹,๐‘‘๐›ผ),โ€–โ‹…โ€–๐‘‹,๐‘๐›ผ) into (๐ถ(๐‘Š),โ€–โ‹…โ€–๐‘Š), and ๐œ is a Lipschitz involution on (๐‘‹,๐‘‘).

For each (๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹, we define the map ๎‚๐œ“๐‘ค,๐œ†โˆถLip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿโ†’โ„ by ๎‚๐œ“๐‘ค,๐œ†๎‚ฮ”(๐‘“)=Re(๐œ†๐‘ค(๐‘“)) in fact, ๎‚๐œ“๐‘ค,๐œ†=Re(๐œ†(๐‘’๐‘ค๐‘œฮจ๐›ผ)). Clearly, ๎‚๐œ“๐‘ค,๐œ†โˆˆ๐ต(Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ)โˆ— for all (๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹. Moreover, ๎‚๐œ“๐‘ฅ,๐œ†=Re(๐œ†ฮ”๐‘ฅ) for all (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹.

We first give a characterization of the extreme points of the unit ball in the (Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ)โˆ— as the following.

Proposition 2.1. By above notations, ๎€ฝ๎‚๐œ“๐‘ฅ,๐œ†โˆถ๎€พ๎€ท๐ต(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹โІExt(Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ)โˆ—๎€ธโІ๎€ฝ๎‚๐œ“๐‘ค,๐œ†โˆถ๎€พ.(๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹(2.1) Further, for (๐‘ฅ,๐œ†),(๐‘ฆ,๐œ‡)โˆˆ๐‘‹ร—๐•‹ one has ๎‚๐œ“๐‘ฅ,๐œ†=๎‚๐œ“๐‘ฆ,๐œ‡ if and only if (๐‘ฅ,๐œ†)=(๐‘ฆ,๐œ‡).

Proof. We define the map ๐‘‡โˆถ(Lip(๐‘‹,๐‘‘๐›ผ)โˆ—)๐‘Ÿโ†’(Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ)โˆ— by ๐‘‡(๐œ‘)=Re๐œ‘. Clearly, ๐‘‡ is a real-linear mapping. For each ๐‘ขโˆˆ(Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ)โˆ—, defining the map ๐œ‘โˆถLip(๐‘‹,๐‘‘๐›ผ)โˆ—โ†’โ„‚ by ๐œ‘(๐‘“)=๐‘ข(๐‘“)โˆ’๐‘–๐‘ข(๐‘–๐‘“). Clearly, ๐œ‘โˆˆLip(๐‘‹,๐‘‘๐›ผ)โˆ— and ๐‘ข=Re๐œ‘. It follows that ๐œ‘โˆˆ(Lip(๐‘‹,๐‘‘๐›ผ)โˆ—)๐‘Ÿ and ๐‘‡(๐œ‘)=๐‘ข. Thus, ๐‘‡ is onto.
We claim that ๐‘‡ is an isometric. Let ๐œ‘โˆˆ(Lip(๐‘‹,๐‘‘๐›ผ)โˆ—)๐‘Ÿ. Since ||||=||||=||||โ‰ค||||๐‘‡(๐œ‘)(๐‘“)(Re๐œ‘)(๐‘“)Re(๐œ‘(๐‘“))๐œ‘(๐‘“)โ‰คโ€–๐œ‘โ€–โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ,(2.2) for each ๐‘“โˆˆLip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ, we have โ€–๐‘‡(๐œ‘)โ€–โ‰คโ€–๐œ‘โ€–.(2.3)
Let ๐œ€ be an arbitrary positive number. There exists ๐‘“โˆˆLip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ with โ€–๐‘“โ€–๐‘‹,๐‘๐›ผโ‰ค1 such that โ€–๐œ‘โ€–<|๐œ“(๐‘“)|+๐œ€. Choose ๐›พ=1 if ๐œ“(๐‘“)=0 and ๐›พ=(1/๐œ“(๐‘“))|๐œ‘(๐‘“)| if ๐œ“(๐‘“)โ‰ 0. Then, ๐›พโˆˆโ„‚, |๐›พ|=1 and ๐œ“(๐‘“)=๐›พ|๐œ“(๐‘“)|. If ๐‘”=(1/๐›พ)๐‘“, then ๐‘”โˆˆLip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ, โ€–๐‘”โ€–๐‘‹,๐‘๐›ผ=โ€–๐‘“โ€–๐‘‹,๐‘๐›ผโ‰ค1 and so, โ€–||||||||๐œ“โ€–<Re(๐œ‘(๐‘”))+๐œ€=๐‘‡(๐œ‘)(๐‘”)+๐œ€โ‰คโ€–๐‘‡(๐œ‘)โ€–+๐œ€.(2.4) It follows that โ€–๐œ‘โ€–โ‰คโ€–๐‘‡(๐œ‘)โ€–.(2.5)
Thus, our claim is justified. The above arguments show that ๐‘‡ is a real-linear isometry from (Lip(๐‘‹,๐‘‘๐›ผ)โˆ—)๐‘Ÿ onto (Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ)โˆ—. Therefore, ๎€ท๐ตExt(Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ)โˆ—๎€ธ๎€ท๎€ท๐ต=๐‘‡ExtLip(๐‘ฅ,๐‘‘๐›ผ)โˆ—.๎€ธ๎€ธ(2.6) Since ๎‚†๐œ†๎‚ฮ”๐‘ฅ๎‚‡๎€ท๐ตโˆถ(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹โІExtLip(๐‘‹,๐‘‘๐›ผ)โˆ—๎€ธโІ๎‚†๐œ†๎‚ฮ”๐‘ค๎‚‡โˆถ(๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹,(2.7) we conclude that ๎‚†๐‘‡๎‚€๐œ†๎‚ฮ”๐‘ฅ๎‚๎‚‡๎€ท๐ตโˆถ(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹โІExt(Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ)โˆ—๎€ธโІ๎‚†๐‘‡๎‚€๐œ†๎‚ฮ”๐‘ค๎‚๎‚‡โˆถ(๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹,(2.8) by Theorems 1.2 and 1.3.
Clearly, ๎‚ฮ”๐‘‡(๐œ†๐‘ค๎‚ฮจ)=๐œ†,๐‘ค for all (๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹. Therefore, ๎‚†๎‚ฮจ๐œ†,๐‘ฅ๎‚‡๎€ท๐ตโˆถ(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹โІExt(Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ)โˆ—๎€ธโІ๎‚†๎‚ฮจ๐œ†,๐‘ค๎‚‡โˆถ(๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹.(2.9) It is obvious that if (๐‘ฅ,๐œ†),(๐‘ฆ,๐œ‡)โˆˆ๐‘‹ร—๐•‹ and (๐‘ฅ,๐œ†)=(๐‘ฆ,๐œ‡), then ๎‚ฮจ๐œ†,๐‘ฅ=๎‚ฮจ๐œ‡,๐‘ฆ. We now assume that ๎‚ฮจ๐œ†,๐‘ฅ=๎‚ฮจ๐œ‡,๐‘ฆ, where (๐‘ฅ,๐œ†),(๐‘ฆ,๐œ‡)โˆˆ๐‘‹ร—๐•‹. Letting ๐‘“=1 and ๐‘“=๐‘–, we see that Re๐œ†=Re๐œ‡ and Re(๐œ†๐‘–)=Re(๐œ‡๐‘–); that is, ๐œ†=๐œ‡. If ๐‘ฅโ‰ ๐‘ฆ, there exists ๐‘“โˆˆLip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ such that ๐‘“(๐‘ฆ)=๐œ‡, but ๐‘“(๐‘ฅ)=0 (define ๐‘“โˆถ๐‘‹โ†’โ„‚ by ๐‘“(๐‘ง)=(๐œ‡/๐‘‘(๐‘ฆ,๐‘ฅ))๐‘‘(๐‘ง,๐‘ฅ), ๐‘งโˆˆ๐‘‹); so that ||๐œ‡||2๎€ท๐œ‡=Re๐œ‡๎€ธ๎‚ฮจ=Re(๐œ‡๐‘“(๐‘ฆ))=๐‘ฆ,๐œ‡๎‚ฮจ(๐‘“)=๐‘ฅ,๐œ†(๐‘“)=Re(๐œ†๐‘“(๐‘ฅ))=0.(2.10) But this is not possible since |๐œ‡|=1. Thus, ๐‘ฅ=๐‘ฆ.

The next purpose is giving conditions for ๐œ‘โˆˆ๐ต๐ดโˆ— to be an extreme point of ๐ต๐ดโˆ—, where ๐ด is a real subspace of Lip(๐‘‹,๐œ,๐‘‘๐›ผ).

Theorem 2.2. Let ๐ด be a real linear subspace of Lip(๐‘‹,๐œ,๐‘‘๐›ผ) containing 1. For (๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹, let ฬƒ๐œ‚๐‘ค,๐œ†=๎‚๐œ“๐‘ค,๐œ†|๐ด. Let ๐‘ƒ={(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹โˆถ๐œ(๐‘ฅ)โ‰ ๐‘ฅ}โˆช{(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—{โˆ’1,1}โˆถ๐œ(๐‘ฅ)=๐‘ฅ}๎‚โˆช{(๐‘ค,๐œ†)โˆถ๐‘คโˆˆ๐›ฝ๐‘‹,๐œ†โˆˆ๐•‹}.
Let ๐‘„๐ด denote the set of (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—{โˆ’1,1} such that(i)there is ๐‘”โˆˆ๐ด with โ€–๐‘”โ€–๐‘‹,๐‘๐›ผโ‰ค1 and ๐‘”(๐‘ฅ)=๐œ†, (ii)for every ๐‘ฆโˆˆ๐‘‹โงต{๐‘ฅ,๐œ(๐‘ฅ)}, there is some โ„Žโˆˆ๐ด with โ€–โ„Žโ€–๐‘‹,๐‘๐›ผโ‰ค1, โ„Ž(๐‘ฅ)=1 and |โ„Ž(๐‘ฆ)|<1.Then, ๎€ฝฬƒ๐œ‚๐‘ฅ,๐œ†โˆถ(๐‘ฅ,๐œ†)โˆˆ๐‘„๐ด๎€พ๎€ท๐ตโІExt๐ดโˆ—๎€ธโІ๎€ฝฬƒ๐œ‚๐‘ค,๐œ†โˆถ๎€พ(๐‘ค,๐œ†)โˆˆ๐‘ƒ.(2.11) Further, if (๐‘ฅ,๐œ†)โˆˆ๐‘„๐ด and (๐‘ฆ,๐œ‡)โˆˆ๐‘ƒโˆฉ(๐‘‹ร—๐•‹), then ฬƒ๐œ‚๐‘ฆ,๐œ‡=ฬƒ๐œ‚๐‘ฅ,๐œ† if and only if either (๐‘ฆ,๐œ‡)=(๐‘ฅ,๐œ†) or (๐‘ฆ,๐œ‡)=(๐œ(๐‘ฅ),๐œ†).

Proof. Let ๐œ‘โˆˆExt(๐ต๐ดโˆ—). Letting ๐”›=Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ in part (a) of Theorem 1.1, and using Proposition 2.1, we see that ๐œ‘=ฬƒ๐œ‚๐‘ค,๐œ† for some (๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹. To prove that (๐‘ค,๐œ†)โˆˆ๐‘ƒ, we consider ๐‘ค=๐‘ฅโˆˆ๐‘‹ with ๐œ(๐‘ฅ)=๐‘ฅ and show that ๐œ†โˆˆ{โˆ’1,1}. For every ๐‘“โˆˆ๐ด, we have ๐‘“(๐‘ฅ)=๐‘“(๐œ(๐‘ฅ))=๐‘“(๐‘ฅ) that is, ๐‘“(๐‘ฅ) is a real number. Hence, ||||=||๐œ‘(๐‘“)ฬƒ๐œ‚๐‘ฅ,๐œ†||=||(๐‘“)๎‚๐œ“๐‘ฅ,๐œ†||=||||=||||||๐‘“||โ‰ค||||(๐‘“)Re(๐œ†๐‘“(๐‘ฅ))Re๐œ†(๐‘ฅ)Re๐œ†โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ.(2.12) This shows that โ€–๐œ‘โ€–โ‰ค|Re๐œ†|. But since ๐œ‘ is an extreme point of ๐ต๐ดโˆ—, we must have โ€–๐œ‘โ€–=1. Thus, 1โ‰ค|Re๐œ†|โ‰ค|๐œ†|=1 so that ๐œ†โˆˆ{โˆ’1,1}.
Next, let (๐‘ฅ,๐œ†)โˆˆ๐‘„๐ด. We claim that the following statement hold.
For (๐‘ฆ,๐œ‡)โˆˆ๐‘‹ร—๐•‹ with (๐‘ฆ,๐œ‡)โ‰ (๐‘ฅ,๐œ†) and (๐‘ฆ,๐œ‡)โ‰ (๐œ(๐‘ฅ),๐œ†), there is ๐‘“โˆˆ๐ด such that โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ=1, Re(๐œ†๐‘“(๐‘ฅ))=1, but Re(๐œ‡๐‘“(๐‘ฆ))โ‰ 1.
By condition (i), there is ๐‘”โˆˆ๐ด such that โ€–๐‘”โ€–๐‘‹,๐‘๐›ผโ‰ค1 and ๐‘”(๐‘ฅ)=๐œ†.
Case 1 (๐‘ฆโˆˆ{๐‘ฅ,๐œ(๐‘ฅ)}). Let ๐‘“=๐‘”. Then, โ€–๐‘“โ€–๐‘‹,๐‘๐›ผโ‰ค1 and |๐‘“(๐‘ฅ)|=|๐œ†|=1 so that โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ=1. Also, ๎€ท๐œ†Re(๐œ†๐‘“(๐‘ฅ))=Re(๐œ†๐‘”(๐‘ฅ))=Re2๎€ธ=1.(2.13) If ๐‘ฆ=๐‘ฅ, then (๐‘ฆ,๐œ‡)โ‰ (๐‘ฅ,๐œ†) implies that ๐œ‡โ‰ ๐œ† that is, ๐œ‡๐œ†โ‰ 1. Since |๐œ‡๐œ†|=1, this shows that Re(๐œ‡๐‘“(๐‘ฅ))=Re(๐œ‡๐‘”(๐‘ฅ))=Re(๐œ‡๐œ†)โ‰ 1.(2.14) If ๐‘ฆ=๐œ(๐‘ฅ), then (๐‘ฆ,๐œ‡)โ‰ (๐œ(๐‘ฅ),๐œ†) implies that ๐œ‡โ‰ ๐œ†, that is, ๐œ‡๐œ†โ‰ 1. Since |๐œ‡๐œ†|=1, this shows that ๎‚€๐œ‡Re(๐œ‡๐‘“(๐‘ฆ))=Re(๐œ‡๐‘”(๐œ(๐‘ฅ)))=Re๐œ†๎‚=Re(๐œ‡๐œ†)โ‰ 1.(2.15)Case 2 (๐‘ฆโˆ‰{๐‘ฅ,๐œ(๐‘ฅ)}). By condition (ii), there is โ„Žโˆˆ๐ด such that โ€–โ„Žโ€–๐‘‹,๐‘๐›ผโ‰ค1, โ„Ž(๐‘ฅ)=1 and |โ„Ž(๐‘ฆ)|<1. Let ๐‘“=๐‘”โ„Ž. Now, โ€–๐‘“โ€–๐‘‹,๐‘๐›ผโ‰ค1 and |๐‘“(๐‘ฅ)|=|๐‘“(๐‘ฅ)||๐‘”(๐‘ฅ)|=|๐‘”(๐‘ฅ)|=|๐œ†|=1, so that โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ=1. Also, ๎€ท๐œ†Re(๐œ†๐‘“(๐‘ฅ))=Re(๐œ†๐‘”(๐‘ฅ)โ„Ž(๐‘ฅ))=Re2๎€ธ||||=||||||||โ‰ค||||=1,Re(๐œ‡๐‘“(๐‘ฆ))=Re(๐œ‡๐‘”(๐‘ฆ)โ„Ž(๐‘ฆ))โ‰ค๐œ‡๐‘”(๐‘ฆ)โ„Ž(๐‘ฆ)๐‘”(๐‘ฆ)โ„Ž(๐‘ฆ)โ„Ž(๐‘ฆ)<1.(2.16) Thus, our claim is justified. Let ๐”›=Lip(๐‘‹,๐‘‘๐›ผ)๐‘Ÿ and ๐œ‘=ฬƒ๐œ‚๐‘ฅ,๐œ† in part (b) of Theorem 1.1. Consider ๐œ“โˆˆExt(๐ต๐”›โˆ—) such that ๐œ“(๐‘“)=1 for all ๐‘“โˆˆ๐ด with ๐œ‘(๐‘“)=1=โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ. By Proposition 2.1, ๐œ“=๎‚๐œ“๐‘ค,๐œ‡ for some (๐‘ค,๐œ‡)โˆˆ๐‘Šร—๐•‹. Thus, ๎‚ฮ”Re(๐œ‡๐‘ค(๐‘“))=1 for all ๐‘“โˆˆ๐ด with Re(๐œ†๐‘“(๐‘ฅ))=1=โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ. Since ๐œ†1โˆˆ๐ด, โ€–๐œ†1โ€–๐‘‹,๐‘๐›ผ=1 and Re(๐œ†(๐œ†1)(๐‘ฅ))=Re(๐œ†2)=1, we have Re(๐œ‡๐œ“๐›ผ(๐œ†1)(๐‘ค))=1. If ๎‚๐‘‹๐‘คโ€ฒโˆˆ๐›ฝ, then ๎€ทRe๐œ‡๐œ“๐›ผ๎€ท๐‘ค(๐œ†1)๎…ž๎€ท๎€ธ๎€ธ=Re๐œ‡๐œ†๐œ“๐›ผ๎€ท๐‘ค(1)๎…ž๎€ท๎€ท๐›ฝฬƒ1๐‘ค๎€ธ๎€ธ=Re๐œ‡๐œ†๎€ธ๎€ท๎…žโ‰ค||๎€ท๎€ท๐›ฝฬƒ1๎€ธ(๎€ธ||=||๎€ท๐›ฝฬƒ1๐‘ค๎€ธ๎€ธ๐œ‡๐œ†๐‘คโ€ฒ)๎€ธ๎€ท๎…ž๎€ธ||โ‰คโ€–โ€–๐›ฝฬƒ1โ€–โ€–๐›ฝ๎‚๐‘‹=โ€–โ€–ฬƒ1โ€–โ€–๎‚๐‘‹=๐‘๐›ผ(1)=0.(2.17) Thus, it must be ๐‘คโˆˆ๐‘‹. Choose ๐‘ค=๐‘ฆ. Then, (๐‘ฆ,๐œ‡)โˆˆ๐‘‹ร—๐•‹ and Re(๐œ‡๐‘“(๐‘ฆ))=1 for all ๐‘“โˆˆ๐ด with โ€–๐‘“โ€–๐‘‹,๐‘๐›ผ=1=Re(๐œ†๐‘“(๐‘ฅ)). By our claim, we must have (๐‘ฆ,๐œ‡)=(๐‘ฅ,๐œ†) or (๐‘ฆ,๐œ‡)=(๐œ(๐‘ฅ),๐œ†).
If (๐‘ฆ,๐œ‡)=(๐‘ฅ,๐œ†), then clearly ๐œ“||๐ด=๎‚๐œ“๐‘ฆ,๐œ‡||๐ด=๎‚๐œ“๐‘ฅ,๐œ†||๐ด=ฬƒ๐œ‚๐‘ฅ,๐œ†=๐œ‘.(2.18) Now, let (๐‘ฆ,๐œ‡)=(๐œ(๐‘ฅ),๐œ†). Since ๐œ†โˆˆโ„, ๐‘“โˆˆ๐ด implies that ๎‚€๐œ†๐œ“(๐‘“)=Re(๐œ‡๐‘“(๐‘ฆ))=Re(๐œ†๐‘“(๐œ(๐‘ฅ)))=Re๎‚๎‚€๐‘“(๐‘ฅ)=๐œ†Re๎‚๐‘“(๐‘ฅ)=๐œ†Re(๐‘“(๐‘ฅ))=Re(๐œ†๐‘“(๐‘ฅ))=๎‚๐œ“๐‘ฅ,๐œ†(๐‘“)=ฬƒ๐œ‚๐‘ฅ,๐œ†(๐‘“)=๐œ‘(๐‘“).(2.19) Therefore, ๐œ“|๐ด=๐œ‘, again. Hence, we see that ฬƒ๐œ‚(๐‘ฅ,๐œ†)=๐œ‘โˆˆExt(๐ต๐ดโˆ—). This also shows that ฬƒ๐œ‚๐œ(๐‘ฅ),๐œ†=ฬƒ๐œ‚๐‘ฅ,๐œ† for all (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—{โˆ’1,1}.

Conversely, our claim implies that if (๐‘ฅ,๐œ†)โˆˆ๐‘„๐ด, (๐‘ฆ,๐œ‡)โˆˆ๐‘‹ร—๐•‹ and ฬƒ๐œ‚๐‘ฆ,๐œ‡=ฬƒ๐œ‚๐‘ฅ,๐œ†, then (๐‘ฆ,๐œ‡)=(๐‘ฅ,๐œ†) or (๐‘ฆ,๐œ‡)=(๐œ(๐‘ฅ),๐œ†). Thus, for (๐‘ฅ,๐œ†)โˆˆ๐‘„๐ด and (๐‘ฆ,๐œ‡)โˆˆ๐‘ƒโˆฉ(๐‘‹ร—๐•‹), we have ฬƒ๐œ‚๐‘ฆ,๐œ‡=ฬƒ๐œ‚๐‘ฅ,๐œ† if and only if (๐‘ฆ,๐œ‡)=(๐‘ฅ,๐œ†) or (๐‘ฆ,๐œ‡)=(๐œ(๐‘ฅ),๐œ†).

By using the above theorem, we give a characterization of extreme points of unit ball in the dual space of Lip(๐‘‹,๐œ,๐‘‘๐›ผ).

Corollary 2.3. Let ๐‘ƒ be as in Theorem 2.2. For (๐‘ค,๐œ†)โˆˆ๐‘ƒ, letโ€‰โ€‰ฬƒ๐œ‚๐‘ค,๐œ† denote the restriction of ๎‚๐œ“๐‘ค,๐œ† to Lip(๐‘‹,๐œ,๐‘‘๐›ผ). Then, ๎€ฝฬƒ๐œ‚๐‘ฅ,๐œ†โˆถ๎€พ๎€ท๐ต(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—{โˆ’1,1}โІExtLip(๐‘‹,๐œ,๐‘‘๐›ผ)โˆ—๎€ธโІ๎€ฝฬƒ๐œ‚๐‘ค,๐œ†โˆถ๎€พ.(๐‘ค,๐œ†)โˆˆ๐‘ƒ(2.20) Further, if (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—{โˆ’1,1} and (๐‘ฆ,๐œ‡)โˆˆ๐‘ƒโˆฉ(๐‘‹ร—๐•‹), then ๐œ‚๐‘ฆ,๐œ‡=ฬƒ๐œ‚๐‘ฅ,๐œ† if and only if (๐‘ฆ,๐œ‡)=(๐‘ฅ,๐œ†) or (๐‘ฆ,๐œ‡)=(๐œ(๐‘ฅ),๐œ†).

Proof. Let ๐ด=Lip(๐‘‹,๐œ,๐‘‘๐›ผ). By Theorem 2.2, we have ๎€ท๐ตExt๐ดโˆ—๎€ธโІ๎€ฝฬƒ๐œ‚๐‘ค,๐œ†โˆถ๎€พ(๐‘ค,๐œ†)โˆˆ๐‘ƒ.(2.21) To prove {ฬƒ๐œ‚๐‘ฅ,๐œ†โˆถ(๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—{โˆ’1,1}}โІExt(๐ตLip(๐‘‹,๐œ,๐‘‘๐›ผ)โˆ—), it is enough to show that for every (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—{โˆ’1,1},(i)there is ๐‘”โˆˆLip(๐‘‹,๐œ,๐‘‘๐›ผ) with โ€–๐‘”โ€–๐‘‹,๐‘๐›ผโ‰ค1 and ๐‘”(๐‘ฅ)=๐œ†,(ii)for every ๐‘ฆโˆˆ๐‘ฅโงต{๐‘ฅ,๐œ(๐‘ฅ)}, there is โ„ŽโˆˆLip(๐‘‹,๐œ,๐‘‘๐›ผ) withโ€–โ„Žโ€–๐‘‹,๐‘๐›ผ||||โ‰ค1,โ„Ž(๐‘ฅ)=1,โ„Ž(๐‘ฆ)<1.(2.22) Let (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—{โˆ’1,1}. We first define the function ๐‘”โˆถ๐‘‹โ†’โ„‚ by ๎‚ต๐‘”(๐‘ง)=๐œ†1โˆ’๐‘‘(๐‘ง,๐‘ฅ)๐‘‘(๐œ(๐‘ง),๐‘ฅ)1+๐œŒ2+๐œŒ2โˆ’2๐›ผ๎‚ถ,(2.23) where ๐œŒ=diam(๐‘‹)=sup{๐‘‘(๐‘ฆ,๐‘ง)โˆถ๐‘ฆ,๐‘งโˆˆ๐‘‹}. Clearly, ๐‘”โˆˆ๐ถ(๐‘‹,๐œ), ๐‘”(๐‘ฅ)=๐œ†, โ€–๐‘”โ€–๐‘‹โ‰ค1 and ๐‘”โˆˆLip(๐‘‹,๐‘‘1). Let ๐‘ฆ,๐‘งโˆˆ๐‘‹ with ๐‘ฆโ‰ ๐‘ง. Then, ||||๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ง)๐‘‘๐›ผ=||||(๐‘ฆ,๐‘ง)๐‘‘(๐‘ง,๐‘ฅ)๐‘‘(๐œ(๐‘ง),๐‘ฅ)โˆ’๐‘‘(๐‘ฆ,๐‘ฅ)๐‘‘(๐œ(๐‘ฆ),๐‘ฅ)๎€ท1+๐œŒ2+๐œŒ2โˆ’2๐›ผ๎€ธ๐‘‘๐›ผโ‰ค๐‘‘(๐‘ฆ,๐‘ง)1โˆ’๐›ผ(๐‘ฆ,๐‘ง)๎€ท1+๐œŒ2+๐œŒ2โˆ’2๐›ผ๎€ธโ‰ค(๐‘‘(๐‘ง,๐‘ฅ)+๐‘‘(๐œ(๐‘ฆ),๐‘ฅ))2๐œŒ๐œŒ1โˆ’๐›ผ1+๐œŒ2+๐œŒ2โˆ’2๐›ผโ‰ค1.(2.24)
Therefore, ๐‘”โˆˆLip(๐‘‹,๐‘‘๐›ผ) and ๐‘๐›ผ(๐‘”)โ‰ค1. Consequently, ๐‘”โˆˆLip(๐‘‹,๐œ,๐‘‘๐›ผโ€–),๐‘”โ€–๐‘‹,๐‘๐›ผโ‰ค1,๐‘”(๐‘ฅ)=๐œ†.(2.25) Hence, (i) holds.
We now assume that ๐‘ฆโˆˆ๐‘‹โงต{๐‘ฅ,๐œ(๐‘ฅ)} and define the function โ„Žโˆถ๐‘‹โ†’โ„‚ by โ„Ž(๐‘ง)=1โˆ’๐œ†๐‘”(๐‘ง)๐‘‘(๐‘ฅ,๐‘ง)๐‘‘(๐‘ฅ,๐œ(๐‘ง))2๎€ท1+๐œŒ2+๐œŒ2โˆ’2๐›ผ๎€ธ.(2.26) Clearly, โ„Žโˆˆ๐ถ(๐‘‹,๐œ), โ„Ž(๐‘ฅ)=1, โ€–โ„Žโ€–๐‘‹โ‰ค1 and there exists ๐›พโˆˆ(0,1] such that โ„Ž(๐‘ฆ)=1โˆ’๐›พ so that |โ„Ž(๐‘ฆ)|<1. We define the complex-valued function ๐‘”1 on ๐‘‹ by ๐‘”1=(1/2)(1โˆ’๐œ†๐‘”). Then, ๐‘”1โˆˆ๐ถ(๐‘‹,๐œ), โ€–๐‘”1โ€–๐‘‹โ‰ค1/2, ๐‘”1โˆˆLip(๐‘‹,๐‘‘1), ๐‘๐›ผ(๐‘”1)โ‰ค1/2 and โ„Ž=1โˆ’๐‘”๐‘”1. Thus, we have โ„ŽโˆˆLip(๐‘‹,๐‘‘1), โ„ŽโˆˆLip(๐‘‹,๐œ,๐‘‘๐›ผ) and ๐‘๐›ผ(โ„Ž)โ‰ค๐‘๐›ผโ€–โ€–๐‘”(๐‘”)1โ€–โ€–๐‘‹+โ€–๐‘”โ€–๐‘‹๐‘๐›ผ๎€ท๐‘”1๎€ธโ‰ค12+12=1,(2.27) so that โ€–โ„Žโ€–๐‘‹,๐‘๐›ผโ‰ค1. Since ๐‘ฆโˆˆ๐‘‹โงต{๐‘ฅ,๐œ(๐‘ฅ)}, |โ„Ž(๐‘ฆ)|=โ„Ž(๐‘ฆ)<1. Thus, (ii) holds.

3. Real Subspaces of lip(๐‘‹,๐‘‘๐›ผ) Containing 1

Throughout this section, we assume that ๐›ผโˆˆ(0,1). Consider the mapping ฮฆ๐›ผโˆถlip(๐‘‹,๐‘‘๐›ผ)โ†’๐ถ(๐‘Š) by ฮฆ๐›ผ=ฮจ๐›ผ|lip(๐‘‹,๐‘‘๐›ผ). Then, ฮฆ๐›ผ is a linear isometric from (lip(๐‘‹,๐‘‘๐›ผ),โ€–โ‹…โ€–๐‘‹,๐‘๐›ผ) into ๐ถ(๐‘Š).

For each ๐‘ฅโˆˆ๐‘‹ and ๐‘คโˆˆ๐‘Š, define the functionals ๐›ฟ๐‘ฅ and ฬƒ๐›ฟ๐‘ค in lip(๐‘‹,๐‘‘๐›ผ)โˆ— by ๐›ฟ๐‘ฅ(๐‘“)=๐‘“(๐‘ฅ) and ฬƒ๐›ฟ๐‘ค(๐‘“)=ฮฆ๐›ผ(๐‘“)(๐‘ค), respectively. Clearly, |๐›ฟ๐‘ฅ(๐‘“)|โ‰คโ€–๐‘“โ€–๐‘‹,๐‘๐›ผ for all๐‘“โˆˆlip(๐‘‹,๐‘‘๐›ผ) and |ฬƒ๐›ฟ๐‘ค(๐‘“)|โ‰คโ€–๐‘“โ€–๐‘‹,๐‘๐›ผ, and therefore, ๐›ฟ๐‘ฅ,ฬƒ๐›ฟ๐‘คโˆˆ๐ตlip(๐‘‹,๐‘‘๐›ผ)โˆ—. Moreover, ฬƒ๐›ฟ๐‘ฅ=๐›ฟ๐‘ฅ for all ๐‘ฅโˆˆ๐‘‹ and ฬƒ๐›ฟ๐‘ค=๐‘’๐‘ค๐‘œฮฆ๐›ผ for all ๐‘คโˆˆ๐‘Š.

We give a characterization of extreme points of the unit ball in the dual space lip(๐‘‹,๐‘‘๐›ผ) as the following.

Theorem 3.1. Every extreme point of ๐ตlip(๐‘‹,๐‘‘๐›ผ)โˆ— must be either the form ๐œ†๐›ฟ๐‘ฅ with (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹ or of the form ๐œ†ฬƒ๐›ฟ๐‘ค with ๎‚(๐œ†,๐‘ค)โˆˆ๐›ฝ๐‘‹ร—๐•‹. Moreover, ๐œ†๐›ฟ๐‘ฅ is an extreme point of ๐ตlip(๐‘‹,๐‘‘๐›ผ)โˆ— for all (๐‘ฅ,๐œ†)โˆˆ๐‘‹ร—๐•‹.

Proof. Since (lip(๐‘‹,๐‘‘๐›ผ),โ€–โ‹…โ€–๐‘‹,๐‘๐›ผ) is a Banach space and ฮฆ๐›ผ is a linear isometry from (lip(๐‘‹,๐‘‘๐›ผ),โ€–โ‹…โ€–๐‘‹,๐‘๐›ผ) into (๐ถ(๐‘Š),โ€–โ‹…โ€–๐‘Š), we conclude that ฮฆ๐›ผ(lip(๐‘‹,๐‘‘๐›ผ)) is a uniformly closed subspace of ๐ถ(๐‘Š). It is well known [2, page 441] that ๎€ท๐ตExt(ฮฆ๐›ผ(lip(๐‘‹,๐‘‘๐›ผ)))โˆ—๎€ธโІ๎‚†๐œ†๐‘’๐‘ค||ฮฆ๐›ผ(lip(๐‘‹,๐‘‘๐›ผ))๎‚‡โˆถ(๐‘ค,๐œ†)โˆˆ๐‘Šร—๐•‹.(3.1) Let ๐ด=ฮฆ๐›ผ(lip(๐‘‹,๐‘‘๐›ผ)) and define ฮ›๐›ผโˆถlip(๐‘‹,๐‘‘๐›ผ)โ†’๐ด by ฮ›๐›ผ(๐‘“)=ฮฆ๐›ผ(๐‘“). Then, ฮ›๐›ผ is a linear isometry from (lip(๐‘‹,๐‘‘๐›ผ),โ€–โ‹…โ€–๐‘๐›ผ) onto (๐ด,โ€–โ‹…โ€–๐‘Š), so ฮ›โˆ—๐›ผ, the adjoint of ฮ›๐›ผ, is a linear isometry from (๐ด,โ€–โ‹…โ€–๐‘Š)โˆ— onto (lip(๐‘‹,๐‘‘๐›ผ))โˆ—. It is easily to show that ฮ›โˆ—๐›ผ(๐‘’๐‘ค|๐ดฬƒ๐›ฟ)=๐‘ค for all