#### Abstract

Let and be compact Hausdorff spaces, and let and be topological involutions on and , respectively. In 1991, Kulkarni and Arundhathi characterized linear isometries from a real uniform function algebra on (, ) onto a real uniform function algebra on (, ) applying their Choquet boundaries and showed that these mappings are weighted composition operators. In this paper, we characterize all onto linear isometries and certain into linear isometries between and applying the extreme points in the unit balls of and .

#### 1. Introduction and Preliminaries

Let and denote the field of real and complex numbers, respectively. The symbol denotes a field that can be either or . The elements of are called scalars. We also denote by the set of all with .

Let be a normed space over . We denote by and the dual space of and the closed unit ball of , respectively. For a subset of , let denote the set of all extreme points of . Kulkarni and Limaye showed [1, Theorem 2] that if is a nonzero linear subspace of and , then has an extension to some .

We know that if and are normed spaces over and is a linear isometry from onto over , then is a bijection mapping between and .

Let be a compact Hausdorff space. We denote by the unital commutative Banach algebra of all continuous functions from into , with the uniform norm , . We write instead as . For , we consider the linear functional on defined by (), which is called the evaluation functional on at . Clearly, for all . It is known [2, page 441] that

Let be a real or complex linear subspace of . A nonempty subset of is called a *boundary* for (with respect to ), if for each the function assumes its maximum on at some . We denote by the intersection of all closed boundaries for . If is a boundary for , it is called the *Shilov boundary* for (with respect to ).

Let be linear subspace of containing , the constant function with value on . A *representing measure* for is an -valued regular Borel measure on such that for all . Let . If , then and , the point mass measure on at , is a representing measure for . We denote by the set of all for which is the only representing measure for . If is a boundary for , it is called the *Choquet boundary* for (with respect to ). We know that

Let be a real linear subspace of , and let be nonempty subset of . We say that is *extremely regular* at if for every open neighborhood of and for each there is a function with such that for all and for all .

Let be a nonempty set. A self-map is called an *involution* on if for all . A subset of is called *-invariant* if . Clearly, if is -invariant, then . A *-invariant measure* on is a measure on such that .

Let be a topological space. An involution on is called a *topological involution* on , if is continuous.

Let be a compact Hausdorff space, and let be a topological involution on . We define Then, is a unital uniformly closed self-adjoint real subalgebra of which separates the points of and does not contain , the constant function with value on . Note that for all . Moreover, and for all . In fact, the complex Banach algebra can be regarded as the complexification of the real Banach algebra . Note that if and only if is the identity self-map on . Hence, the class of real Banach algebras is, in fact, larger than the class of real Banach algebras .

Let be a compact Hausdorff space, a topological involution on , and a subset of . We say that separates the points of if for each with there exists a function in such that . It is known [3, Lemma ] that if , then separates the points of .

Let be a compact Hausdorff space. We assume that denotes , regarded as a real normed space by restricting the scalar multiplication to real scalars. Let be the topological involution on defined by () and (). Then the map defined by is an isometrical isomorphism from onto . Hence, the class of real Banach algebras is, in fact, larger than the class of real Banach algebras .

Let be a compact Hausdorff space, and let be a topological involution on . Let be a real subspace of containing . A *real part representing measure* for is a -invariant regular real Borel measure on such that for all . Let . If , then and the measure is a real part representing measure for . We denote by the set of all for which is the only real part representing measure for . Clearly, is a -invariant subset of . If is a boundary for , it is called the *Choquet boundary* for with respect to . We know that .

The real Banach algebra and its real linear subspaces were first considered by Kulkarni and Limaye in [4]. For a detailed account of several properties of , we refer to [3].

Let be a compact Hausdorff space. For each , define the map by in fact, . Clearly, .

Let be a topological involution on . For each define the map by . Kulkarni and Limaye showed [1, Proposition 3] that Applying this result, they obtained the following theorem.

Theorem 1 (see [1, Corollary 5]). *Let be a compact Hausdorff space, and let be a topological involution on . Suppose that
**
Then
*

The classical Banach-Stone theorem states that if is a linear isometry from onto , then there exists a homeomorphism from onto and a continuous function with such that

This well-known theorem has been generalized in several directions. In 1966, an important generalization was given by HolsztyÅ„ski [5] by considering into linear isometries as the following.

Theorem 2. *Let and be compact Hausdorff spaces. If is a linear isometry from into , then there exists a closed subset ofâ€‰â€‰, a continuous map from onto and a function with and such that
*

In 1975, another important generalization of Banach-Stone theorem was given by Novinger [6] for certain complex subalgebras of and considering the concept of their Choquet boundaries. In 1991, Kulkarni and Arundhathi [7] generalized the given results by Novinger for certain real subalgebras of and considering the concept of their Choquet boundaries. We can cite some other generalizations of Banach-Stone theorem, for example, the generalization obtained by Cambern [8] for space of vector-valued continuous functions, by Jeang and Wong [9] for spaces of scalar-valued continuous functions vanishing at infinity, by Araujo and Font [10] for certain subspaces of scalar-valued continuous functions considering concepts of their Choquet and Shilov boundaries, and by JimÃ©nez-Vargas and Villegas-Vallecillos [11] for spaces of scalar-valued Lipschitz functions.

Throughout the rest of this paper, and are compact Hausdorff spaces and and are topological involutions on and , respectively.

In this paper, we characterize all linear isometries from onto and certain linear isometries from into , applying the extreme points in and . Our results in Section 2 are some of the given results by Kulkarni and Arundhathi in [7] that we obtain by applying the extreme points in and . The main result in Section 3 is a generalization of the given result by HolsztyÅ„ski in [5] for certain into isometries.

#### 2. Onto Linear Isometries

We first determine unit-preserving linear isometries from onto . For this purpose, we need the following lemmas.

Lemma 3 (see [3, Lemma ]). *Let and be real linear subspaces ofâ€‰â€‰ and , respectively. Suppose that , , is extremely regular at and is extremely regular at . Let be a linear isometry from onto and for all . Then, for all or for all . *

Lemma 4. *Let be a nonempty -invariant closed subset of . Then, is extremely regular at . In particular, is extremely regular at for all in . *

*Proof. *Assume that and is an open neighborhood of in . By Urysohnâ€™s lemma, there exists a function with for all , for all , and for all . Define the function by . Then, , , for all and for all . Hence, is extremely regular at .

Theorem 5. *Let be a linear isometry from onto with . Then, there exists a homeomorphism from onto with on such that
*

*Proof. *We claim that
Let . By Theorem 1, . Since is a linear isometry from onto , we conclude that is a linear isometry from onto and so
Therefore, . Hence, our claim is justified.

We now show that for each there exists a unique such that
Let . By (13), we have . Hence, by Theorem 1, there exists such that
Since and , by (16), we have
Since and , we have and so . Thus, by (16), we have
From (18) we deduce that
Since is a linear isometry from onto and by Lemma 4, is extremely regular at , and is extremely regular at , we conclude that for all in or for all by Lemma 3. We assume that whenever for all in and whenever for all in . Then, and we have
This proves the existent of . To show uniqueness, assume that there exists such that
Then, for all in . This implies that since separates the points of .

Now we define the map â€‰â€‰byâ€‰â€‰ whenever for all in . In fact, we have
Now we prove that on . Let . Then, . From (22), for each we have
This implies that since separates the points of . Therefore, on .

Continually, we prove that is bijective. Since is a linear isometry from onto with , by the above argument, we deduce that there exists a map such that
To prove the injectivity of , let . For each , by (22) and (24) we have
This implies that since separates the points of . Therefore, , the identity mapping on , and so is injective.

To prove that is onto, let . For each , by (24) and (22) we have
This implies that since separates the points of . Therefore , the identity mapping on , and so is onto.

We now check that is continuous. Let , and let be a net in such that in . Then we have
By (27) and definition of , we deduce that
Since separates the points of , by (28), we have in . Therefore, is continuous.

Since is a bijective continuous mapping, is a compact space, and is a Hausdorff space, we conclude that is continuous, and so is a homeomorphism. Hence, the proof is complete.

Corollary 6. *Let be an isometry mapping from onto with and . Then is an isomorphism. *

*Proof. *Since and are real Banach algebras and is an isometry from onto with , we conclude that is a linear isometry by Mazur-Ulam theorem (see [12]). Since , applying Theorem 5, we deduce that there exists a homeomorphism from onto such that
It follows that is a homomorphism. Hence, the proof is complete.

We now study the onto case (not necessarily unit preserving).

Lemma 7. *Let with . Then, for all , if for every there exists such that with implies that
*

*Proof. *Assume that there exists such that . Then, since . Let . Then, and . Take . Since , we deduce that is a -invariant open subset of and . By Urysohnâ€™s lemma, there exists such that for all , , and for all . Let . Then, , for all , , and for all . Therefore,
Hence, for each we have
This completes the proof.

Lemma 8. *Let be a linear isometry from onto , and let . Then, for all . *

*Proof. *Since and , by Lemma 7, it is sufficient to show that for each there exists such that with implies that

Let , and let with . The surjectivity of implies that there exists such that . Since is an isometry, we have . This implies that there exists such that . Therefore,
It follows that
Hence,
Since and , we have
It is enough that we choose .

In the following result, we show that every linear isometry from onto is a weighted composition operator.

Theorem 9. *Let be a linear isometry from onto . Then, there exist a function with for all and a homeomorphism from onto with on such that
*

*Proof. *Assume that . Then . By Lemma 8, we have for all . Clearly, for all and . We define the map by . Then, is a linear isometry from onto , and . By Theorem 5, there exists a homeomorphism from onto with on such that
This implies that
Hence, the proof is complete.

#### 3. Into Linear Isometries

We formulate our main result in this section which is a version for into linear isometries of -spaces of a known HolsztyÅ„skisâ€™s theorem (Theorem 2) for into linear isometries of -spaces. We first study unit-preserving linear isometries from into .

Theorem 10. *Let be a linear isometry from into satisfying the following conditions:*(i)*,*(ii)*if and there is a point such that for all , then is extremely regular at . **Then, there exist a -invariant closed boundary for and a continuous map from onto with on such that
*

*Proof. *Let . Since is a linear isometry, we deduce that is a uniformly closed linear subspace of . Moreover, implies that . We define the map by . Then, is a linear isometry from the real Banach space onto the real Banach space . Therefore, is a linear isometry from onto and so
Now we define
We first show that is nonempty. To prove this fact, we show that the following statement, namely (I), holds.

(I) For each there exists such that .

Let . Then by Theorem 1. From (42), there exists such that
Since is a linear subspace of , there exists such that
By Theorem 1, there exists such that
From (44), (45), and (46), we have
Since and , by (47) we have
This implies that since . Hence, by (47) we have
and so statement (I) holds.

Since for all in , statement (I) implies that is nonempty.

We next show that the following statement, namely, (II), holds.

(II) For each there exists a unique point such that for all in .

Let . Then, . Hence, by Theorem 1, there exists such that
Since and , by (50) we have
This implies that since . Thus, by (50) we have
From (52) we give
It follows that

According to (54) and condition (ii), is extremely regular at . On the other hand, is extremely regular at by Lemma 4. Hence, by Lemma 3, we deduce that for all in or for all in . We assume that whenever for all in and whenever for all in . Then, and we have
This proves the existent of . To show uniqueness, assume that there exists such that
From (55) and (56) we have for all in . This implies that since separates the points of . Thus, statement (II) holds.

Now we define the map by whenever for all in . The statement (II) implies that is well-defined. By definition of , we have
We prove that in . Let . For each , by (57) we have
This implies that since separates the points of . Hence, on .

Continually, we show that is a -invariant. Let . For each we have
This implies that
Hence, by Theorem 1. Therefore, and so is -invariant.

Now we show that is surjective. Let . According to statement (I), we deduce that there exists a point such that
It follows that and we have
Statement (II) implies that there exists a point such that
From (62) and (63), we have for all in . This implies that since separates the points of . If , then by (63) and definition of we have . If , then by (63) and definition of we have
for all in , and so . Therefore, is surjective.

Continually, we show that is a boundary for . Let . Then, there exists a function such that . Since is a continuous complex-valued function on the compact space , there exists a point such that . The surjectivity of implies that there exists a point such that . Hence, , and so
Therefore, is a boundary for .

We now check that is continuous. Let , and let be a net in such that in . Then, for each we have in . This implies that for each in . Since separates the points of , we have in . Therefore, is continuous.

Finally, we show that is closed in . Let , the closure of in . Then, there exists a net in such that in . Then, for each we have , and so in . Since the net converges for all in , we conclude that the net converges in to a point . Hence, for each we have . Therefore, for all in . This implies that for each we have
Hence,
This implies that , by Theorem 1. Therefore, . So is closed in .

Note that if is a linear isometry from onto , then and so is extremely regular at for all in . Therefore, if is a linear isometry from on with , then conditions (i) and (ii) in Theorem 10 hold.

We now study the into case (not necessarily unit-preserving).

Theorem 11. *Let be a linear isometry from into satisfying the following conditions:*(i)* for all , *(ii)*if and there is a point such that for all in , then is extremely regular at . **Then, there exist a -invariant closed boundary for , a function with for all , and a continuous map from onto with *