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Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 519893, 9 pages

http://dx.doi.org/10.1155/2013/519893

## Choquet and Shilov Boundaries, Peak Sets, and Peak Points for Real Banach Function Algebras

^{1}Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran^{2}Faculty of Mathematical Sciences and Computer, Kharazmi University (Tarbiat Moallem University), 50 Taleghani Avenue, Tehran 15618-36314, Iran

Received 5 December 2012; Accepted 1 March 2013

Academic Editor: Dachun Yang

Copyright © 2013 Davood Alimohammadi and Taher Ghasemi Honary. 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 be a compact Hausdorff space and let be a topological involution on . In 1988, Kulkarni and Arundhathi studied Choquet and Shilov boundaries for real uniform function algebras on . Then in 2000, Kulkarni and Limaye studied the concept of boundaries and Choquet sets for uniformly closed real subspaces and subalgebras of or . In 1971, Dales obtained some properties of peak sets and *p*-sets for complex Banach function algebras on . Later in 1990, Arundhathi presented some results on peak sets for real uniform function algebras on . In this paper, while we present a brief account of the work of others, we extend some of their results, either to real subspaces of or to real Banach function algebras on .

#### 1. Introduction and Preliminaries

Let denote either or . We always assume that is a compact Hausdorff space. We denote by the Banach algebra of all continuous functions from into , with the uniform norm However, we always write instead of and we denote the uniform closure of by , whenever is a subset of .

Let be a real or complex subspace of . A nonempty subset of is called a *boundary* (*Choquet set*, resp.) for (with respect to ), if for each the function (, resp.) assumes its maximum on at some . Note that every closed boundary for is a closed Choquet set for [1, Lemma 1.1] and if is closed under the complex scalar multiplication, then every Choquet set for is a boundary for [1, Section 1]. We denote by the intersection of all closed boundaries for . If is a boundary for , it is called the *Shilov boundary* for .

Let be a subspace of over containing 1. We denote by the set of all for which , where is the dual space of the normed space over . In fact, the elements of are -valued linear functionals on over . For each the map , defined by , is a linear mapping over , which is called the *evaluation map* on at . Clearly whenever and whenever .

Let be a complex subspace of containing 1. A *representing measure* for is a complex regular Borel measure on such that for all . It is known that every has a representing measure and every representing measure for such is a probability measure [2, Section 2.1]. If , then , the point mass measure 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 .

Let be a real subspace of containing 1. A *real part representing measure* for is a regular Borel measure on such that for all . It is wellknown that every has a real part representing measure [1, Theorem 1.5]. If , then is a real part representing measure for . We denote by the set of all for which is the only real part representing measure for . If is a boundary for , it is called the *Choquet boundary* for .

Let be a nonempty set. A self-map is called an *involution* on if for all . We denote by the set of all for which . A subset of is called *-invariant* if . A *-invariant measure* on is a measure on such that .

An involution on is called a topological involution on if is continuous. The map defined by is an *algebra involution* on , which is called the algebra involution induced by on . We now define
Then is a unital self-adjoint uniformly closed real subalgebra of , which separates the points of and does not contain the constant function . 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 map on . Hence the class real Banach algebras of continuous complex-valued functions are, in fact, larger than the class of real Banach algebras of continuous real-valued functions . This class of continuous functions was defined explicitly by Kulkarni and Limaye in [3].

Let be a real subspace of containing 1. We denote by the intersection of all -invariant closed boundaries for . If is a boundary for , we call it the *Shilov boundary* for with respect to . For each , the measure is a positive regular Borel measure on , which is denoted by . If , then is a -invariant real part representing measure for . We denote by the set of all for which is the only -invariant real part representing measure for . If is a boundary for , we call it the *Choquet boundary* for with respect to .

For a topological involution on let . For a subset of we say that separates the points of if for each and , there exists a function in such that . It is interesting to note that if is a real subalgebra of which separates the points of and contains 1, then separates the points of [4, Lemma 1.3.9].

Let be a real or complex subalgebra of and let be a nonempty subset of . Then is called a *peak set* for if there exists such that and for all . We say that is a *-set* (a *peak set in the weak sense* or a *weak peak set*) for if is the intersection of some collection of peak sets for . If the peak set or -set for is a singleton , then we call a *peak point* or *p-point* for . The set of all peak points, or -points, for is denoted by , or , respectively. Note that is the intersection of all boundaries for .

If is a real subalgebra of , then every -set and hence every peak set for are -invariant. Hence . Moreover, is contained in every -invariant boundary for .

*Definition 1. *Let be a real subspace of . The point is a -peak point (-*p-point*, resp.) for if the set is a peak set (a -set, resp.) for . We define

It is easy to see that is the intersection of all -invariant boundaries for .

*Definition 2. *A complex Banach function algebra on is a complex subalgebra of which contains 1, separates the points of , and it is a unital Banach algebra under an algebra norm . If the norm of a complex Banach function algebra on is the uniform norm on , then it is called a complex uniform function algebra on .

Clearly, is a complex uniform function algebra on .

Let be a unital commutative Banach algebra over . A *character* of is a nonzero homomorphism , where is regarded as an algebra over . We denote by the set of all characters of . It is known that , with the Gelfand topology, is a compact Hausdorff space (see [5, Chapter 11] and [4, Chapter 1]).

*Definition 3. *A *real Banach function algebra* on is a real subalgebra of which contains 1, separates the points of , and it is a real unital Banach algebra under an algebra norm . If the norm of a real Banach function algebra on is the uniform norm on , then it is called a real uniform function algebra on .

Clearly, is a real uniform function algebra on .

Since the class of real uniform (or Banach) function algebras on is larger than the class of complex uniform (or Banach) function algebras, it is quite natural to ask which properties of a complex algebra can be extended to the corresponding real algebra.

Let be a real (complex, resp.) Banach function algebra on (on , resp.). The evaluation character is an element of for all . We call to be *natural* if every is given by an evaluation character at some .

The concept of the Shilov boundary for a real Banach algebra was studied by Ingelstam, Limaye, and Simha in [6–8]. The concept of Choquet and Shilov boundaries for real uniform function algebras was first studied by Kulkarni and Arundhathi in [9]. Later Kulkarni and Limaye introduced the notions of Choquet sets and boundaries for real subspaces of and in [1] and obtained interesting results on Choquet and Shilov boundaries for uniformly closed real subalgebras of or . They also obtained relations between and ( and , resp.), when is a real uniform function algebra on and is the complexification of .

In Section 2, we first show that and and by applying these equalities we extend some results due to Kulkarni, Limaye, and Arundhathi, by omitting the uniformly closed condition on .

The concept of peak points and -points for real uniform function algebras was studied by Kulkarni and Arundhathi in [9, 10]. Also Dales has obtained many results on the boundaries and peak sets for complex Banach function algebras in [11, 12]. In Sections 3 and 4, we extend some of their results to real Banach function algebras.

#### 2. Choquet and Shilov Boundaries

We begin this section by recalling some well-known results.

Theorem 4 (see [4, Theorem 4.3.3 and Corollary 4.3.4]). *Let be a complex subspace of containing 1 and separating the points of . Then *(i)* is an extreme point of if and only if there exists such that . In particular, is nonempty.*(ii)* is the set of all such that is an extreme point of .*(iii)* is a Choquet set for and, in particular, a boundary for .*(iv)*The closure of is equal to . *

Theorem 5 (see [1, Corollary 1.8]). *Let be a real subspace of . Then the closure of is contained in every closed Choquet set as well as every closed boundary for . In particular, . *

Theorem 6 (see [1, Lemma 2.1 and Theorem 2.2]). *Let be a real subspace of containing 1 and separates the points of . Then *(i)*The element is an extreme point of if and only if for some . In particular, is nonempty.*(ii)* is the set of all such that is an extreme point of .*(iii)* is a Choquet set for .*(iv)*The closure of is the smallest closed Choquet set for . *

Theorem 7 (see [4, Theorem 4.2.2((a), (b))]). *Let be a real subspace of containing 1. *(i)*If is a -invariant boundary for , then the closure of is a -invariant Choquet set for .*(ii)*. *

Theorem 8 (see [4, Theorem 4.1.10 and Corollary 4.1.11] and [1, Section 4]). *Let be a real subspace of containing 1 and let separate the points of . Then *(i)* is an extreme point of if and only if there exists such that . In particular, is nonempty. *(ii)* is the set of all such that is an extreme point of .*(iii)* is a -invariant Choquet set for . *(iv)*The closure of is the smallest -invariant Choquet set for . *

We now show that and , for a real or complex subspace of , and we then extend some results which have been obtained by Kulkarni, Arundhathi, and Limaye in [1, 9].

Theorem 9. *
(i) Let be a real subspace of containing 1. If separates the points of , then .**
(ii) Let be a real subspace of containing 1. If separates the points of , then .**
(iii) Let be a complex subspace of containing 1. If separates the points of , then and . *

*Proof. * (i) For every , let and be the evaluation maps at on and , respectively. By Theorem 6(ii), we have
Let and , where and . Set and . It is easy to show that and on . Hence . Let be an arbitrary element of . Then there exists a sequence in such that . Since , we have

Hence . Similarly, . Therefore, is an extreme point of and so .

Conversely, let and , where and .

Clearly, if and is a sequence in with , then .

Now let and let be a sequence in with . We define and on by and . By the previous argument and are well defined. It is easy to see that and on . On the other hand, is an extreme point of since . Therefore, on and so on . It follows that is an extreme point and so .

(ii) This is proved exactly with the same argument as in part (i) and applying Theorem 8(ii).

(iii) With the same argument as in part (i), by applying Theorem 4(ii), we can show that . Therefore, by Theorem 4(iv), we conclude that .

We now recall the following known result and then extend it by omitting the uniformly closed condition for .

Theorem 10 (see [1, Lemma 3.1 and Theorem 3.2]). *Let be a uniformly closed real subalgebra of containing 1 and let separate the points of . Then*(i)*A subset of is a boundary for if and only if it is a Choquet set for .*(ii)* is a boundary for .*(iii)*The closure of is equal to . *

Theorem 11. *Let be a real subalgebra of containing 1 and let separate the points of . Then *(i)*.*(ii)* is a boundary for and hence for .*(iii)*The closure of is equal to .*(iv)*A closed subset of is a boundary for if and only if it is a Choquet set for . *

*Proof. *Since satisfies the hypotheses of Theorem 10, is a boundary for and . Moreover, we have by Theorem 9(ii) and by Theorem 5. Therefore, is a boundary for (hence for ) and . Since it follows that and, moreover, .

To prove (iv) let be a closed boundary for . By [1, Lemma 1.1], is a closed Choquet set for . Conversely, if is a closed Choquet set for , then by Theorem 6(iv). Therefore, is a boundary for by (ii).

It is interesting to note that in the previous theorem if does not separate the points of , a closed Choquet set may not be a boundary for [1, Lemma 1.1].

The following result is also known and we bring it here for easy reference.

Theorem 12 (see [4, Theorem 4.2.5] and [9]). *Let be a uniformly closed real subalgebra of containing 1 and let separate the points of . Then *(i)*a -invariant subset of is a boundary for if and only if it is a Choquet set for .*(ii)* is a -invariant boundary for .*(iii)*The closure of is equal to , the smallest -invariant closed boundary for . *

We now extend the previous theorem by omitting the uniformly closed condition for .

Theorem 13. *Let be a real subalgebra of containing 1 and let separate the points of . Then*(i)*.*(ii)* is a -invariant boundary for .*(iii)*The closure of is equal to , the smallest -invariant closed boundary for .*(iv)*A -invariant closed subset of is a boundary for if and only if it is a Choquet set for . *

*Proof. *The proof is similar to that of Theorem 11 by replacing entries in the first column by the corresponding entries in the second column of Table 1.

Corollary 14. *If is a real Banach function algebra on , then *(i)*.*(ii)* is a -invariant boundary for .*(iii)*.*(iv)*. *

* Proof. *Since separates the points of , we conclude that separates the points of . Therefore, the result follows by Theorem 9(ii) and Theorem 13.

The following result is also known.

Theorem 15 (see [9, Theorem 3.7, Corollary 3.8] or [4, Theorem 4.3.7]). *Let be a real uniform function algebra on and suppose that . Then
*

We now extend the previous theorem by omitting the uniformly closed condition for .

Theorem 16. *Let be a real subalgebra of containing 1 and separating the points of and let . Then and . *

*Proof. *Since separates the points of , we conclude that separates the points of . Therefore, and by Theorem 9(ii) and Theorem 13(i). Clearly, is real uniform function algebra on . Since
for all , we have . Hence, and by Theorem 15. On the other hand, since is a complex subspace of which contains 1 and separates the points of , we have and by Theorem 9(iii). Thus the result follows.

Corollary 17. *If is a real Banach function algebra on and , then and . *

At the end of this section, we present two examples of real subalgebras of which are not uniformly closed and then determine their Choquet and Shilov boundaries by Theorem 9, Theorem 13, and Theorem 16.

*Example 18. *Let be a compact metric space and let . Let be the complex algebra of all complex-valued functions on for which
is finite. For , we take to be the set of all for which
The Lipschitz norm on is defined by
It is known that and are complex Banach function algebras on . These algebras are called Lipschitz algebras of order and were first studied by Sherbert in [13, 14].

Let be a *Lipschitz involution* on , that is, an involution on satisfying the Lipschitz condition for a positive constant and for all in . Let be the algebra involution induced by on . Then and ; that is, these (complex) Lipschitz algebras are -invariant. We now define
Then and are real Banach function algebras on under the norm and we have
These algebras are called real Lipschitz algebras of complex functions of order and were first studied in [15].

Since and for each the function , defined by
belongs to and peaks at , we conclude that
If is the algebra for or for , then by Theorem 16, and , whenever , for , or , for . Since and
we have .

*Example 19. *Let be the closed unit interval with the usual Euclidean topology. Let be the complex algebra of all (continuous) complex-valued functions with derivatives of all orders on . It is known that is not complete under any algebra norm. For this property see, for example, Carpenter's theorem [16] or Singer-Wermer theorem [17, 18, Theorem 16]. Since is a self-adjoint complex subalgebra of which contains 1 and separates the points of , we have by the Stone-Weierstrass theorem for complex subalgebras of . Since , by Theorem 9(iii) it follows that .

We now define the topological involution on by and let be the algebra involution induced by on . Since for every , is a differentiable function of all orders on and for each , it follows that . If we define , then is a real subalgebra of containing 1, separating the points of , and moreover, the algebra is the complexification of . Then by [15, Theorem 1.2], is not complete under any algebra norm. Since , we conclude that by Theorem 16.

On the other hand, by the Stone-Weierstrass theorem for real subalgebras of [3]. Since , we conclude that by Theorem 9(ii) and Theorem 13(i).

#### 3. Peak Sets and Peak Points

In this section we extend some of the known results concerning peak sets and -sets for complex Banach (uniform, resp.) function algebras to those for real Banach (uniform, resp.) function algebras.

We know that every complex uniform function algebra on can be regarded as a real uniform function algebra on a compact Hausdorff space with a suitable topological involution [3]. We may take as a disjoint union of two copies of and as a homeomorphism that sends a point in one copy of to the corresponding point in the other copy of [4, page 29]. Now a natural question arises here whether the above fact holds for Banach function algebras. The answer is affirmative as the following result shows. However, its proof is similar to that of Kulkarni and Limaye for uniform function algebras.

Theorem 20. *Let be a complex Banach function algebra on a compact Hausdorff space . Then there exists a compact Hausdorff space , a topological involution on , and a real Banach function algebra on such that , regarded as a real Banach algebra, is isometrically isomorphic to . Moreover, *(i)*the algebra is natural if and only if is natural. *(ii)*There exists a one-to-one correspondence between the set of all peak sets (-sets, resp.) for and the set of all peak sets (-sets, resp.) for , with respect to . *

* Proof. *Let be the compact Hausdorff space with the product topology. We define the map by and for all . Clearly, is a topological involution on . Now, we define the map by
Then is an isometrically isomorphism from , regarded as a real Banach algebra, onto and . Set . It follows that is a real subalgebra of which contains 1 and separates the points of . We define for all . Then is a complete algebra norm on and . Therefore, is a real Banach function algebra on . Now, we define the map by . Clearly, is an isometric isomorphism from , regarded as a real Banach algebra, onto .

Let denote as a real algebra. Then it is easy to show that . To prove the naturality of , let . Then . If is natural, either there exists such that or there exists such that . We can easily show that is the evaluation character at or at , respectively. Therefore, is natural.

Conversely, let be natural and . Hence . Therefore, either there exists such that is the evaluation character at some on or it is the evaluation character at some on . Since and is a complex algebra, the latter case does not occur, and hence . Since , it follows that and thus is natural.

Now let be a peak set (-set, resp.) for . It is easy to see that is a peak set (-set, resp.) for .

Conversely, let be a peak set for . Then there exists a subset of such that . We can easily show that is a peak set for .

If is a -set for , then , where is a peak set for . Since , it follows that is a -set for .

Concerning the union of peak sets and -sets for real or complex subspaces of , the following results are well-known.

Theorem 21 (see [4, Remark 2.2.10]). *Let be a uniformly closed real subalgebra of containing 1. If and are peak sets (-sets, resp.) for , then is a peak set (-set, resp.) for . *

Theorem 22 (see [18, II. Corollary 12.8]). * Let be a complex uniform function algebra on . If is a sequence of -sets for and is closed, then is a -set for . *

Note that the above two theorems do not hold for complex Banach function algebras, in general. In fact, Dales has proved the following result in [12] and has shown, by counterexamples, that the restriction imposed on and the peak sets (-sets, resp.) are necessary.

Theorem 23. *Let be a natural complex Banach function algebra on . Then a finite union of pairwise disjoint peak sets (-sets, resp.) for is a peak set (-set, resp.) for . *

The following two results state the relation between the peak sets (-sets, resp.) for a real subspace of and those of its complexification, which are modifications of [4, Theorem 2.2.11].

Theorem 24. *Let be a real subspace of containing 1 and . *(i)*If is a peak set (-set, resp.) for , then it is a peak set (-set, resp.) for . Moreover, is also a peak set (-set, resp.) for and .*(ii)*If is a peak set (-set, resp.) for , then is also a peak set (-set, resp.) for . In case , is, in fact, a peak set (-sets, resp.) for .*(iii)* and . *

Theorem 25. *Let be a uniformly closed subalgebra of containing 1 and let . *(i)*If is a peak set (-set, resp.) for , then is a peak set (-set, resp.) for .*(ii)* and .*(iii)* and .*(iv)*If is a real uniform function algebra on , then and . *

We now extend Theorem 23 for natural real Banach function algebras.

Theorem 26. *Let be a natural real Banach function algebra on . Then a finite union of pairwise disjoint peak sets (-sets, resp.) for is a peak set (-set, resp.) for . *

* Proof. *It is sufficient to show that if and are two disjoint peak sets (-sets, resp.) for , then is a peak set (-set, resp.) for . Let . Then is a complex subalgebra of and there exists an algebra norm on such that is a natural complex Banach function algebra on by [15, Theorem 1.2]. On the other hand, and are peak sets (-sets, resp.) for by Theorem 24(i). Thus is a peak set (-set, resp.) for by Theorem 23. Since and are -invariant, is also -invariant. Therefore, is a peak set (-set, resp.) for by Theorem 24(ii).

*Remark 27. *By using the counter examples in [7] and considering Theorem 20, we can show that all restrictions imposed on in the above theorem are necessary.

Now, we show that part (i) of Theorem 25 holds for real Banach function algebras with small modification.

Theorem 28. *Let be a real Banach function algebra on and be its complexification. If is a peak set (-set, resp.) for then is a peak set (-set, resp.) for .*

*Proof. *Since is a peak set (-set, resp.) for , we conclude that is also a peak set (-set, resp.) for by Theorem 24(ii). Let be the complexification of . Clearly, and are peak sets (-sets, resp.) for and hence by Theorem 25(i), is a peak set (-set, resp.) for .

Theorem 29. *Let be a natural complex Banach function algebra on and be the algebra involution on , induced by the topological involution . Let and . If and are -invariant peak sets (-sets, resp.) for such that , then is a peak set (-set, resp.) for .*

*Proof. *By [15, Theorem 1.1], is a natural real Banach function algebra on and . On the other hand, and are peak sets (-sets, resp.) for by Theorem 24(ii). Therefore, is a peak set(-set, resp.) for by Theorem 26.

The following example shows that the uniformly closed condition of in Theorem 25(i) and the condition in Theorem 29 are essential.

*Example 30. *Let and let be the complex algebra of all continuous complex-valued functions on which have continuous first-order partial derivatives with respect to and . We define the algebra norm on by
Then is a natural complex Banach function algebra on and although the sets and are peak sets for , the set is not a peak set for , as shown by Dales in [12, Example 1]. Note that is nonempty.

We first define the topological involution on by . Let be the algebra involution on , induced by . By the definition of , for every and for each fixed in the closed unit disk , has continuous derivative on and hence has also continuous derivative on as a function of . Similarly, for each fixed , has continuous derivative on and hence, as a function of , has also continuous derivative on . Therefore, has continuous partial derivatives on with respect to and . Since , for all , it follows that and hence . If we define , then is a natural real Banach function algebra on and is the complexification of by [15, Theorem 1.1]. Moreover, is not uniformly closed. Since , the set is not a peak set for . Therefore, is not a peak set for by Theorem 24(i).

We now define the topological involution on by . Clearly, and are -invariant. Let be the algebra involution on , induced by . By a similar argument as in the above case, it is easy to see that . If we define , then is a natural real Banach function algebra on and by [15, Theorem 1.1]. Since is not a peak set for , it is not a peak set for by Theorem 24(i).

Theorem 31. *Let be a natural real Banach function algebra on and let . Then and . *

* Proof. * Let . If then by Theorem 24(ii), and hence . If , then is a peak set for by Theorem 24(ii) and Theorem 23. Since is -invariant, it follows that it is a peak set for by Theorem 24(ii) and hence . Therefore, the first inclusion holds.

By applying parts (i) and (iii) in Theorem 24 and Theorem 29 for -sets, the second inclusion holds with the same argument.

Concerning the existence of a -point (a peak point, respectively) in every peak set for a complex uniform function algebra, the following result holds.

Theorem 32 (see [2, Corollary 2.4.6]). *Let a complex uniform function algebra on . Then every peak set for contains a -point for . Moreover, if is first countable, then every peak set for contains a peak point for . *

Note that the above theorem is not true for complex Banach function algebras, in general (see [11, 19, 20]).

The following example shows that the above theorem does not hold even for real uniform function algebras, in general.

*Example 33. *We denote by , , and the open unit disk , the closed unit disc , and the unit circle , respectively. Let , the complex disc algebra. It is known that is a complex uniform algebra on and . We define the topological involution on by . Let , where is the algebra involution induced by on . Since is continuous on and it is analytic on for each , it follows that . In fact, is the real disc algebra and is the complexification of . Moreover,
by Theorem 24(iii). Hence for each with , the set is a peak set for which does not contain any peak point for .

The following is a modification of Theorem 32 for real uniform function algebras.

Theorem 34. *Let be a real uniform function algebra on . Then every peak set for contains a --point for . Moreover, if is first countable, every peak set for contains a -peak point for . *

* Proof. *Let be a peak set for . If we take then is a complex uniform function algebra on by [4, Theorem 1.3.20] and is also a peak set for . Hence and when is first countable, , by Theorem 32. Since and , we have , and if is first countable, then .

#### 4. On the Density of -Peak Points in the Shilov Boundary

We first state some known results on the density of the set of peak points in the Shilov boundary for complex Banach function algebras.

Theorem 35 (see [2, Theorem 2.3.4]). *Let be a complex uniform function algebra on . Then . Moreover, if is first countable, then and so is the closure of . *

Theorem 36 (see [11, Theorem 2.3] and [21]). *Let be a Banach function algebra on a compact metrizable space . Then is the closure of . *

The following example shows that the above theorem does not hold even for real uniform algebras on compact metrizable spaces, in general.

*Example 37. *Let be the complex disc algebra and let be the real disc algebra, considered in Example 33. It is known that . Therefore, by Theorem 16. On the other hand, as shown in Example 33. Hence, the closure of is not equal to .

Theorem 38 (see [10] or [4, Theorem 4.2.4]). *Let be a uniformly closed real subalgebra of containing 1. Then
*

*Remark 39. *Let be a real uniform function algebra on and . Then we have by Theorem 38, by Theorem 15, and by Theorem 35. Therefore, , which has already been obtained in Theorem 25(iv).

Theorem 40. *Let be a uniformly closed real subalgebra of containing 1 and let separate the points of . Then
**
In particular, if is a first countable space, then
*

*Proof. *By Theorem 38 we have and since by Theorem 13(iii), we conclude that
In the case that is first countable, every finite subset of is a -set. Hence and so . Therefore, by the previous argument.

Theorem 41. *Let be a compact metrizable space and let be a topological involution on . If is a natural real Banach function algebra on , then
*

*Proof. *Let . The naturality of implies that by Theorem 31. On the other hand, there exists an algebra norm on such that is a complex Banach function algebra on by [15, Theorem 1.2]. Since is a metrizable compact space, we have by Theorem 36. On the other hand, by Theorem 16. Therefore, .

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