#### Abstract

Let be a compact Hausdorff space, be a continuous involution on and denote the uniformly closed real subalgebra of consisting of all for which . Let be a compact metric space and let denote the complex Banach space of complex-valued Lipschitz functions of order on under the norm , where . For , the closed subalgebra of consisting of all for which as , denotes by . Let be a Lipschitz involution on and define for and for . In this paper, we give a characterization of extreme points of , where is a real linear subspace of or which contains 1, in particular, or .

#### 1. Introduction and Preliminaries

We let , and denote the field of real numbers, complex numbers, and the unit circle, respectively. The symbol denotes a field that can be either or . The elements of are called scalars.

Let be a normed space over . We denote by and the dual space and the closed unit ball of , respectively. If is a subset , let denote the set of all extreme points of . Let be a subspace of and . A Hahn-Banach extension of to is a continuous linear functional such that and . The set of all Hahn-Banach extensions of to will be denoted by .

It is easy to see that if and are normed spaces over and is a linear isometry from onto , then is a bijection mapping between and .

For a complex normed space , we assume that denotes , regarded as a real normed space by restricting the scalar multiplication to real numbers.

Kulkarni and Limaye gave some conditions for to be an extreme point of in terms of the Hahn-Banach extension of to and the extreme points of as the following.

Theorem 1.1 (see [1, Theoremโโ2]). *Let be a normed space over , be a nonzero linear subspace of and . *(a)*Let . Then,
**In particular, has an extension to some . Further, if such an extension is unique, then has a unique Hahn-Banach extension to .*(b)*Assume that whenever and for all with , one has , then .*(c)*If has a unique Hahn-Banach extension to and if , then .*

Let be a compact Hausdorff space. We denote by the complex Banach algebra of all continuous complex-valued functions on under the uniform norm . For , consider the evaluation functional given by . Clearly, for all . It is well known [2, page 441] that

For and , we define the map by in fact, . Clearly, for all . Kulkarni and Limaye showed [1, Propositionโโ3] that and if and only if and .

Let be a continuous involution on ; that is, is continuous and is the identity map on . The map defined by , is an algebra involution on which is called the algebra involution induced by on . Define . Then, is a uniformly closed real subalgebra of which contains 1. The real algebras were first considered in [3]. For a detailed account of several properties of , we refer to [4].

Let . For each , let denote the restriction of to . Grzesiak obtained a characterization of the extreme points of in [5] and showed that if and only if for some . Further, if , then if and only if or .

Kulkarni and Limaye obtained [1, Theoremโโ4] a characterization of , where is a nonzero real linear subspace of .

Let be a compact metric space. For , we denote by the set of all complex-valued functions on for which is finite. Then, is a complex subalgebra of containing 1 and complex Banach space under the norm For , the complex subalgebra of consisting of all for which is denoted by . Clearly, is a closed linear subspace of and . These Banach spaces were first studied by Leeuw in [6].

Given a compact metric space , let , and let the compact Hausdorff space be the disjoint union of with , where is the Stone-Cech compactification of . For , consider the mapping defined for each by where and is the norm-preserving extension of to . Clearly, is a linear isometry from into , which is called the Leeuwโs linear isometry. Therefore, is a uniformly closed linear subspace of . It is well known (see [2, page 441]) that where is the evaluation functional at on .

For each and , define the linear functionals and in by and , respectively. Clearly, and for all . Therefore, . Moreover, for all โโand for all . Thus, we have the following result.

Theorem 1.2. *For , every extreme point of must be either of the form with or of the form with .*

Roy proved the following result by using a result of Leeuw [6, Lemmaโโ1.2].

Theorem 1.3 (see [7, Lemmaโโ1.2]). *For each , is an extreme point of .*

Jimenez-Vargas and Villegas-Vallecillos used above results and obtained a characterization of linear isometries between and in [8].

A map is said to be Lipschitz map from the metric space to the metric space if there exists a constant such that for all .

Let be a compact metric space. The mapping is called a Lipschitz involution on , if is a Lipschitz map from to itself and an involution on . Clearly, every Lipschitz involution on is a continuous involution.

Let be a Lipschitz involution on the compact metric space and let be the algebra involution induced by on . Clearly, We define Then, the following statements hold.(i) (, resp.) is a real subalgebra of (, resp.).(ii) and .(iii) and .(iv) (, resp.) is a real subalgebra of which contains 1 and separates the points of .(v) (, resp.) is uniformly dense in (use (iv) and the Stone-Weierstrass theorem for real subalgebra of [3, Propositionโโ1.1].(vi)For , (vii)There exists a constant such that for all .(viii) is a real Banach space and is its closed real subspace.

The real Banach spaces and 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 , and some its real linear subspaces for in Section 2. Next, we give a characterization of extreme points of the unit ball in the dual spaces of and some its real linear subspaces for in Section 3.

#### 2. Real Linear Subspaces of Containing 1

In the remainder of this paper, we assume that , is a compact metric space, , is the Stone-Cech compactification of , is the compact Hausdorff space , is the Leeuwโs linear isometry from into , and is a Lipschitz involution on .

For each , we define the map by in fact, . Clearly, for all . Moreover, for all .

We first give a characterization of the extreme points of the unit ball in the as the following.

Proposition 2.1. *By above notations,
**
Further, for one has if and only if .*

*Proof. *We define the map by . Clearly, is a real-linear mapping. For each , defining the map by . Clearly, and . It follows that and . Thus, is onto.

We claim that is an isometric. Let . Since
for each , we have

Let be an arbitrary positive number. There exists with such that . Choose if and if . Then, , and . If , then , and so,
It follows that

Thus, our claim is justified. The above arguments show that is a real-linear isometry from onto . Therefore,
Since
we conclude that
by Theorems 1.2 and 1.3.

Clearly, for all . Therefore,
It is obvious that if and , then . We now assume that , where . Letting and , we see that and ; that is, . If , there exists such that , but (define by , ); so that
But this is not possible since . Thus, .

The next purpose is giving conditions for to be an extreme point of , where is a real subspace of Lip.

Theorem 2.2. *Let be a real linear subspace of Lip containing 1. For , let . Let .**Let denote the set of such that*(i)*there is with and , *(ii)*for every , there is some with , and .**Then,
**
Further, if and , then if and only if either or .*

*Proof. *Let . Letting in part (a) of Theorem 1.1, and using Proposition 2.1, we see that for some . To prove that , we consider with and show that . For every , we have that is, is a real number. Hence,
This shows that . But since is an extreme point of , we must have . Thus, so that .

Next, let . We claim that the following statement hold.

For with and , there is such that , , but .

By condition (i), there is such that and .*Case 1 (). *Let . Then, and so that . Also,
If , then implies that that is, . Since , this shows that
If , then implies that , that is, . Since , this shows that
*Case 2 (). *By condition (ii), there is such that , and . Let . Now, and , so that . Also,
Thus, our claim is justified. Let and in part (b) of Theorem 1.1. Consider such that for all with . By Proposition 2.1, for some . Thus, for all with . Since , and Re, we have . If , then
Thus, it must be . Choose . Then, and for all with . By our claim, we must have or .

If , then clearly
Now, let . Since , implies that
Therefore, , again. Hence, we see that . This also shows that for all .

Conversely, our claim implies that if , and , then or . Thus, for and , we have if and only if or .

By using the above theorem, we give a characterization of extreme points of unit ball in the dual space of .

Corollary 2.3. *Let be as in Theorem 2.2. For , letโโ denote the restriction of to . Then,
**
Further, if and , then if and only if or .*

*Proof. *Let . By Theorem 2.2, we have
To prove , it is enough to show that for every ,(i)there is with and ,(ii)for every , there is with
Let . We first define the function by
where . Clearly, , , and . Let with . Then,

Therefore, and . Consequently,
Hence, (i) holds.

We now assume that and define the function by
Clearly, , , and there exists such that so that . We define the complex-valued function on by . Then, , , , and . Thus, we have , and
so that . Since , . Thus, (ii) holds.

#### 3. Real Subspaces of Containing 1

Throughout this section, we assume that . Consider the mapping by . Then, is a linear isometric from into .

For each and , define the functionals and in lip by and , respectively. Clearly, for all and , and therefore, . Moreover, for all and for all .

We give a characterization of extreme points of the unit ball in the dual space as the following.

Theorem 3.1. *Every extreme point of must be either the form with or of the form with . Moreover, is an extreme point of for all .*

*Proof. *Since is a Banach space and is a linear isometry from into , we conclude that is a uniformly closed subspace of . It is well known [2, page 441] that
Let and define by . Then, is a linear isometry from onto , so , the adjoint of , is a linear isometry from onto . It is easily to show that for all . Let . Then, . Thus, there exists such that so that . It follows that

Now, let . Clearly, . Assume that and for all with . Since is a nonzero linear subspace of , we conclude that for some or for some by Theorem 1.1. Clearly, and . If for some , then we have
Thus, it must be for some . Since , we have
It follows that . We claim that . Let . We define the function by
where . It is easy to show that , , and . Now, we define the function by
and the function by . It is easy to see that , , , and . Therefore, we have , , , and
so that . It must be that ; that is, . But