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

Authors: Davood Alimohammadi; Taher Ghasemi Honary
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

LetX be a compact Hausdorff space and let τ be a topological involution on X. In 1988, Kulkarni and Arundhathi studied Choquet and Shilov boundaries for real uniform function algebras on (X,τ). Then in 2000, Kulkarni and Limaye studied the concept of boundaries and Choquet sets for uniformly closed real subspaces and subalgebras of C(X,τ) or C(X). In 1971, Dales obtained some properties of peak sets and p-sets for complex Banach function algebras on X. Later in 1990, Arundhathi presented some results on peak sets for real uniform function algebras on (X,τ). 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 C(X,τ) or to real Banach function algebras on (X,τ).


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## Body

## 1. Introduction and Preliminaries

Let𝔽 denote either ℝ or ℂ. We always assume that X is a compact Hausdorff space. We denote by C𝔽(X) the Banach algebra of all continuous functions from X into 𝔽, with the uniform norm
(1)∥f∥X=sup{|f(x)|:x∈X}(f∈C𝔽(X)).
However, we always write C(X) instead of Cℂ(X) and we denote the uniform closure of A by A¯, whenever A is a subset of C(X).LetA be a real or complex subspace of C(X). A nonempty subset S of X is called a boundary (Choquet set, resp.) for A (with respect to X), if for each f∈A the function |f| (Ref, resp.) assumes its maximum on X at some x∈S. Note that every closed boundary for A is a closed Choquet set for A [1, Lemma 1.1] and if A is closed under the complex scalar multiplication, then every Choquet set for A is a boundary for A [1, Section 1]. We denote by Γ(A,X) the intersection of all closed boundaries for A. If Γ(A,X) is a boundary for A, it is called the Shilov boundary for A.LetA be a subspace of C(X) over 𝔽 containing 1. We denote by K𝔽(A) the set of all φ∈A* for which ∥φ∥=φ(1)=1, where A* is the dual space of the normed space (A,∥·∥X) over 𝔽. In fact, the elements of A* are 𝔽-valued linear functionals on A over 𝔽. For each x∈X the map ex:A→ℂ, defined by ex(f)=f(x), is a linear mapping over 𝔽, which is called the evaluation map on A at x. Clearly ex∈Kℂ(A) whenever 𝔽=ℂ and Reex∈Kℝ(A) whenever 𝔽=ℝ.LetB be a complex subspace of C(X) containing 1. A representing measure for φ∈B* is a complex regular Borel measure μ on X such that φ(f)=∫X‍fdμ for all f∈B. It is known that every φ∈Kℂ(B) has a representing measure and every representing measure for such φ is a probability measure [2, Section 2.1]. If x∈X, then δx, the point mass measure at x, is a representing measure for ex. We denote by Ch(B,X) the set of all x∈X for which δx is the only representing measure for ex. If Ch(B,X) is a boundary for B, it is called the Choquet boundary for B.LetA be a real subspace of C(X) containing 1. A real part representing measure for ψ∈A* is a regular Borel measure μ on X such that ψ(f)=∫X‍Refdμ for all f∈A. It is wellknown that every ψ∈Kℝ(A) has a real part representing measure [1, Theorem 1.5]. If x∈X, then δx is a real part representing measure for Reex. We denote by Ch(A,X) the set of all x∈X for which δx is the only real part representing measure for Reex. If Ch(A,X) is a boundary for A, it is called the Choquet boundary for A.LetE be a nonempty set. A self-map τ:E→E is called an involution on E if τ(τ(x))=x for all x∈E. We denote by Fix(τ) the set of all x∈X for which τ(x)=x. A subset S of E is called τ-invariant if τ(S)=S. A τ-invariant measure on E is a measure μ on E such that μ∘τ=μ.An involutionτ on X is called a topological involution on X if τ is continuous. The map σ:C(X)→C(X) defined by σ(f)=f¯∘τ is an algebra involution on C(X), which is called the algebra involution induced by τ on C(X). We now define
(2)C(X,τ)={f∈C(X):σ(f)=f}.
Then C(X,τ) is a unital self-adjoint uniformly closed real subalgebra of C(X), which separates the points of X and does not contain the constant function i. Moreover, C(X)=C(X,τ)⊕iC(X,τ) and
(3)max{∥f∥X,∥g∥X}≤∥f+ig∥X≤2max{∥f∥X,∥g∥X}
for all f,g∈C(X,τ). In fact, the complex Banach algebra (C(X),∥·∥X) can be regarded as the complexification of the real Banach algebra (C(X,τ),∥·∥X). Note that C(X,τ)=Cℝ(X) if and only if τ is the identity map on X. Hence the class real Banach algebras of continuous complex-valued functions C(X,τ) are, in fact, larger than the class of real Banach algebras of continuous real-valued functions Cℝ(X). This class of continuous functions was defined explicitly by Kulkarni and Limaye in [3].LetA be a real subspace of C(X,τ) containing 1. We denote by Γ(A,X,τ) the intersection of all τ-invariant closed boundaries for A. If Γ(A,X,τ) is a boundary for A, we call it the Shilov boundary for A with respect to (X,τ). For each x∈X, the measure (1/2)(δx+δτ(x)) is a positive regular Borel measure on X, which is denoted by mx. If x∈X, then mx is a τ-invariant real part representing measure for Reex. We denote by Ch(A,X,τ) the set of all x∈X for which mx is the only τ-invariant real part representing measure for Reex. If Ch(A,X,τ) is a boundary for A, we call it the Choquet boundary for A with respect to (X,τ).For a topological involutionτ on X let X/τ={{x,τ(x)}:x∈X}. For a subset A of C(X,τ) we say that ReA separates the points of X/τ if for each x∈X and y∈X∖{x,τ(x)}, there exists a function f in A such that Ref(x)≠Ref(y). It is interesting to note that if A is a real subalgebra of C(X,τ) which separates the points of X and contains 1, then ReA separates the points of X/τ [4, Lemma 1.3.9].LetA be a real or complex subalgebra of C(X) and let S be a nonempty subset of X. Then S is called a peak set for A if there exists f∈A such that S={x∈X:f(x)=1} and |f(y)|<1 for all y∈X∖S. We say that S is a p-set (a peak set in the weak sense or a weak peak set) for A if S is the intersection of some collection of peak sets for A. If the peak set or p-set S for A is a singleton {x}, then we call x a peak point or p-point for A. The set of all peak points, or p-points, for A is denoted by S0(A,X), or S(A,X), respectively. Note that S0(A,X) is the intersection of all boundaries for A.IfA is a real subalgebra of C(X,τ), then every p-set and hence every peak set for A are τ-invariant. Hence S0(A,X)⊆S(A,X)⊆Fix(τ). Moreover, S0(A,X) is contained in every τ-invariant boundary for A.Definition 1.
LetA be a real subspace of C(X,τ). The point x∈X is a τ-peak point (τ-p-point, resp.) for A if the set {x,τ(x)} is a peak set (a p-set, resp.) for A. We define
(4)T0(A,X,τ)={x∈X:xisaτ-peakpoint  forA},T(A,X,τ)={x∈X:xisaτ-p-point  forA}.
It is easy to see thatT0(A,X,τ) is the intersection of all τ-invariant boundaries for A.Definition 2.
A complex Banach function algebra onX is a complex subalgebra of C(X) which contains 1, separates the points of X, and it is a unital Banach algebra under an algebra norm ∥·∥. If the norm of a complex Banach function algebra on X is the uniform norm on X, then it is called a complex uniform function algebra on X.
Clearly,C(X) is a complex uniform function algebra on X.
LetB be a unital commutative Banach algebra over 𝔽. A character of B is a nonzero homomorphism φ:B→ℂ, where ℂ is regarded as an algebra over 𝔽. We denote by MB the set of all characters of B. It is known that MB, with the Gelfand topology, is a compact Hausdorff space (see [5, Chapter 11] and [4, Chapter 1]).Definition 3.
Areal Banach function algebra on (X,τ) is a real subalgebra of C(X,τ) which contains 1, separates the points of X, and it is a real unital Banach algebra under an algebra norm ∥·∥. If the norm of a real Banach function algebra on (X,τ) is the uniform norm on X, then it is called a real uniform function algebra on (X,τ).
Clearly,C(X,τ) is a real uniform function algebra on (X,τ).
Since the class of real uniform (or Banach) function algebras on(X,τ) 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.
LetA be a real (complex, resp.) Banach function algebra on (X,τ) (on X, resp.). The evaluation character ex is an element of MA for all x∈X. We call A to be natural if every φ∈MA is given by an evaluation character ex at some x∈X.
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 C(X) and C(X,τ) in [1] and obtained interesting results on Choquet and Shilov boundaries for uniformly closed real subalgebras of C(X) or C(X,τ). They also obtained relations between Ch(A,X,τ) and Ch(B,X) (Γ(A,X,τ) and Γ(B,X), resp.), when A is a real uniform function algebra on (X,τ) and B={f+ig:f,g∈A} is the complexification of A.
In Section2, we first show that Ch(A,X)=Ch(A¯,X) and Γ(A,X)=Γ(A¯,X) and by applying these equalities we extend some results due to Kulkarni, Limaye, and Arundhathi, by omitting the uniformly closed condition on A.
The concept of peak points andp-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]).
LetB be a complex subspace of C(X) containing 1 and separating the points of X. Then  (i)
φ∈Kℂ(B) is an extreme point of Kℂ(B) if and only if there exists x∈Ch(B,X) such that φ=ex. In particular, Ch(B,X) is nonempty.(ii)
Ch(B,X) is the set of all x∈X such that ex is an extreme point of Kℂ(B).(iii)
Ch(B,X) is a Choquet set for B and, in particular, a boundary for B.(iv)
The closure ofCh(B,X) is equal to Γ(B,X).Theorem 5 (see [1, Corollary 1.8]).
LetA be a real subspace of C(X). Then the closure of Ch(A,X) is contained in every closed Choquet set as well as every closed boundary for A. In particular, Ch(A,X)⊆Γ(A,X).Theorem 6 (see [1, Lemma 2.1 and Theorem 2.2]).
LetA be a real subspace of C(X) containing 1 and ReA separates the points of X. Then  (i)
The elementφ∈Kℝ(A) is an extreme point of Kℝ(A) if and only if φ=Reex for some x∈Ch(A,X). In particular, Ch(A,X) is nonempty.(ii)
Ch(A,X) is the set of all x∈X such that Reex is an extreme point of Kℝ(A).(iii)
Ch(A,X) is a Choquet set for A.(iv)
The closure ofCh(A,X) is the smallest closed Choquet set for A.Theorem 7 (see [4, Theorem 4.2.2((a), (b))]).
LetA be a real subspace of C(X,τ) containing 1.  (i)
IfS is a τ-invariant boundary for A, then the closure of S is a τ-invariant Choquet set for A.(ii)
Ch(A,X,τ)¯⊆Γ(A,X,τ).Theorem 8 (see [4, Theorem 4.1.10 and Corollary 4.1.11] and [1, Section 4]).
LetA be a real subspace of C(X,τ) containing 1 and let ReA separate the points of X/τ. Then  (i)
φ∈Kℝ(A) is an extreme point of Kℝ(A) if and only if there exists x∈Ch(A,X,τ) such that φ=Reex. In particular, Ch(A,X,τ) is nonempty.(ii)
Ch(A,X,τ) is the set of all x∈X such that Reex is an extreme point of Kℝ(A).(iii)
Ch(A,X,τ) is a τ-invariant Choquet set for A.(iv)
The closure ofCh(A,X,τ) is the smallest τ-invariant Choquet set for A.We now show thatCh(A,X)=Ch(A¯,X) and Γ(A,X)=Γ(A¯,X), for a real or complex subspace A of C(X), and we then extend some results which have been obtained by Kulkarni, Arundhathi, and Limaye in [1, 9].Theorem 9.
(i) LetA be a real subspace of C(X) containing 1. If ReA separates the points of X, then Ch(A,X)=Ch(A¯,X).
(ii)  LetA be a real subspace of C(X,τ) containing 1. If ReA separates the points of X/τ, then Ch(A,X,τ)=Ch(A¯,X,τ).
(iii)  LetB be a complex subspace of C(X) containing 1. If B separates the points of X, then Ch(B,X)=Ch(B¯,X) and Γ(B,X)=Γ(B¯,X).Proof.
(i) For everyx∈X, let ex and εx be the evaluation maps at x on A and A¯, respectively. By Theorem 6(ii), we have
(5)Ch(A,X)={x∈X:Reexisan  extreme  point  ofKℝ(A)},Ch(A¯,X)={x∈X:Reεxis  an  extreme  point  ofKℝ(A-)}.
Let x∈Ch(A,X) and Reεx=(1-t)Φ+tΨ, where 0<t<1 and Φ,Ψ∈Kℝ(A¯). Set φ=Φ|A and ψ=Ψ|A. It is easy to show that φ,ψ∈Kℝ(A) and Reex=(1-t)φ+tψ on A. Hence φ=ψ=Reex. Let f be an arbitrary element of A¯. Then there exists a sequence {fn} in A such that limn→∞∥fn-f∥X=0. Since Φ∈(A¯)*, we have
(6)Φ(f)=limn→∞φ(fn)=limn→∞Re((fn(x)))=Ref(x)=(Reεx)(f).
HenceΦ=Reεx. Similarly, Ψ=Reεx. Therefore, εx is an extreme point of Kℝ(A¯) and so x∈Ch(A¯,X).
Conversely, letx∈Ch(A¯,X) and Reex=(1-t)φ+tψ, where 0<t<1 and φ,ψ∈Kℝ(A).
Clearly, ifη∈A* and {hn} is a sequence in A with limn→∞∥hn∥X=0, then limn→∞η(hn)=0.
Now letf∈A¯ and let {fn} be a sequence in A with limn→∞∥fn-f∥X=0. We define φ~ and ψ~ on A¯ by φ~(f)=limn→∞φ(fn) and ψ~(f)=limn→∞ψ(fn). By the previous argument φ~ and ψ~ are well defined. It is easy to see that φ~,ψ~∈Kℝ(A¯) and (1-t)φ~+tψ~=Reεx on A¯. On the other hand, εx is an extreme point of Kℝ(A¯) since x∈Ch(A¯,X). Therefore, φ~=ψ~=εx on A¯ and so φ=ψ=ex on A. It follows that ex is an extreme point Kℝ(A) and so x∈Ch(A,X).
(ii) This is proved exactly with the same argument as in part (i) and applying Theorem8(ii).
(iii) With the same argument as in part (i), by applying Theorem4(ii), we can show that Ch(B,X)=Ch(B¯,X). Therefore, by Theorem 4(iv), we conclude that Γ(B,X)=Γ(B¯,X).We now recall the following known result and then extend it by omitting the uniformly closed condition forA.Theorem 10 (see [1, Lemma 3.1 and Theorem 3.2]).
LetA be a uniformly closed real subalgebra of C(X) containing 1 and let ReA separate the points of X. Then(i)
A subsetS of X is a boundary for A if and only if it is a Choquet set for A.(ii)
Ch(A,X) is a boundary for A.(iii)
The closure ofCh(A,X) is equal to Γ(A,X).Theorem 11.
LetA be a real subalgebra of C(X) containing 1 and let ReA separate the points of X. Then  (i)
Γ(A,X)=Γ(A¯,X).(ii)
Ch(A,X) is a boundary for A¯ and hence for A.(iii)
The closure ofCh(A,X) is equal to Γ(A,X).(iv)
A closed subsetS of X is a boundary for A if and only if it is a Choquet set for A.Proof.
SinceA¯ satisfies the hypotheses of Theorem 10, Ch(A¯,X) is a boundary for A¯ and Ch(A¯,X)¯=Γ(A¯,X). Moreover, we have Ch(A,X)=Ch(A¯,X) by Theorem 9(ii) and Ch(A,X)¯⊆Γ(A,X) by Theorem 5. Therefore, Ch(A,X) is a boundary for A¯ (hence for A) and Γ(A¯,X)⊆Γ(A,X). Since Γ(A,X)⊆Γ(A¯,X) it follows that Γ(A,X)=Γ(A¯,X) and, moreover, Ch(A,X)¯=Γ(A,X).
To prove (iv) letS be a closed boundary for A. By [1, Lemma 1.1], S is a closed Choquet set for A. Conversely, if S is a closed Choquet set for A, then Ch(A,X)¯⊆S by Theorem 6(iv). Therefore, S is a boundary for A by (ii).It is interesting to note that in the previous theorem ifReA does not separate the points of X, a closed Choquet set may not be a boundary for A [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]).
LetA be a uniformly closed real subalgebra of C(X,τ) containing 1 and let ReA separate the points of X/τ. Then  (i)
aτ-invariant subset S of X is a boundary for A if and only if it is a Choquet set for A.(ii)
Ch(A,X,τ) is a τ-invariant boundary for A.(iii)
The closure ofCh(A,X,τ) is equal to Γ(A,X,τ), the smallest τ-invariant closed boundary for A.We now extend the previous theorem by omitting the uniformly closed condition forA.Theorem 13.
LetA be a real subalgebra of C(X,τ) containing 1 and let ReA separate the points of X/τ. Then(i)
Γ(A,X,τ)=Γ(A¯,X,τ).(ii)
Ch(A,X,τ) is a τ-invariant boundary for A.(iii)
The closure ofCh(A,X,τ) is equal to Γ(A,X,τ), the smallest τ-invariant closed boundary for A.(iv)
Aτ-invariant closed subset S of X is a boundary for A if and only if it is a Choquet set for A.Proof.
The proof is similar to that of Theorem11 by replacing entries in the first column by the corresponding entries in the second column of Table 1.Table 1
C
(
X
)
C
(
X
,
τ
)
points ofX
points ofX/τ
Ch
(
A
-
,
X
)
Ch
(
A
-
,
X
,
τ
)
(closed) boundary
τ
-invariant (closed) boundary
Γ
(
A
-
,
X
)
Γ
(
A
-
,
X
,
τ
)
Theorem10
Theorem12
Ch
(
A
,
X
)
Ch
(
A
,
X
,
τ
)
Γ
(
A
,
X
)
Γ
(
A
,
X
,
τ
)
Theorem5
Theorem7(ii)
Choquet set
τ-invariant Choquet set
[1, Lemma 1.1]
Theorem7(i).Corollary 14.
IfA is a real Banach function algebra on (X,τ), then  (i)
Ch(A,X,τ)=Ch(A¯,X,τ).(ii)
Ch(A,X,τ) is a τ-invariant boundary for A.(iii)
Γ(A,X,τ)=Γ(A¯,X,τ).(iv)
Γ(A,X,τ)=Ch(A,X,τ)¯.Proof.
SinceA separates the points of X, we conclude that ReA separates the points of X/τ. 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]).
LetA be a real uniform function algebra on (X,τ) and suppose that B={f+ig:f,g∈A}. Then
(7)Ch(A,X,τ)=Ch(B,X),Γ(A,X,τ)=Γ(B,X).We now extend the previous theorem by omitting the uniformly closed condition forA.Theorem 16.
LetA be a real subalgebra of C(X,τ) containing 1 and separating the points of X and let B={f+ig:f,g∈A}. Then Ch(A,X,τ)=Ch(B,X) and Γ(A,X,τ)=Γ(B,X).Proof.
SinceA separates the points of X, we conclude that ReA separates the points of X/τ. Therefore, Ch(A,X,τ)=Ch(A¯,X,τ) and Γ(A,X,τ)=Γ(A¯,X,τ) by Theorem 9(ii) and Theorem 13(i). Clearly, A¯ is real uniform function algebra on (X,τ). Since
(8)B={f+ig:f,g∈A},C(X)=C(X,τ)⊕iC(X,τ),max{∥f∥X,∥g∥X}≤∥f+ig∥X
for all f,g∈C(X,τ), we have B¯={f+ig:f,g∈A¯}. Hence, Ch(A¯,X,τ)=Ch(B¯,X) and Γ(A¯,X,τ)=Γ(B¯,X) by Theorem 15. On the other hand, since B is a complex subspace of C(X) which contains 1 and separates the points of X, we have Ch(B,X)=Ch(B¯,X) and Γ(B,X)=Γ(B¯,X) by Theorem 9(iii). Thus the result follows.Corollary 17.
IfA is a real Banach function algebra on (X,τ) and B={f+ig:f,g∈A}, then Ch(A,X,τ)=Ch(B,X) and Γ(A,X,τ)=Γ(B,X).At the end of this section, we present two examples of real subalgebras ofC(X,τ) which are not uniformly closed and then determine their Choquet and Shilov boundaries by Theorem 9, Theorem 13, and Theorem 16.Example 18.
Let(X,d) be a compact metric space and let 0<α≤1. Let Lip(X,α) be the complex algebra of all complex-valued functions f on X for which
(9)pα(f)=sup{|f(x)-f(y)|dα(x,y):x,y∈X,x≠y}
is finite. For 0<α<1, we take lip(X,α) to be the set of all f∈Lip(X,α) for which
(10)|f(x)-f(y)|dα(x,y)⟶0asd(x,y)⟶0.
The Lipschitz norm ∥·∥α on Lip(X,α) is defined by
(11)∥f∥α=∥f∥X+pα(f).
It is known that (Lip(X,α),∥·∥α) and (lip(X,α),∥·∥α) are complex Banach function algebras on X. These algebras are called Lipschitz algebras of order α and were first studied by Sherbert in [13, 14].
Letτ be a Lipschitz involution on (X,d), that is, an involution τ on X satisfying the Lipschitz condition d(τ(x),τ(y))≤Cd(x,y) for a positive constant C and for all x,y in X. Let σ be the algebra involution induced by τ on C(X). Then σ(Lip(X,α))=Lip(X,α) and σ(lip(X,α))=lip(X,α); that is, these (complex) Lipschitz algebras are σ-invariant. We now define
(12)Lip(X,τ,α)={h∈Lip(X,α):σ(h)=h},lip(X,τ,α)={h∈lip(X,α):σ(h)=h}.
Then Lip(X,τ,α) and lip(X,τ,α) are real Banach function algebras on (X,τ) under the norm ||·||α and we have
(13)Lip(X,α)=Lip(X,τ,α)⊕iLip(X,τ,α),lip(X,α)=lip(X,τ,α)⊕ilip(X,τ,α).
These algebras are called real Lipschitz algebras of complex functions of order α and were first studied in [15].
SinceLip(X,1)⊆lip(X,α)⊆Lip(X,α) and for each x∈X the function fx:X→ℂ, defined by
(14)fx(y)=1-d(x,y)diamX,
belongs to Lip(X,1) and peaks at x, we conclude that
(15)S0(Lip(X,1),X)=S0(lip(X,α),X)=S0(Lip(X,α),X)=X.
If B is the algebra Lip(X,α) for α∈(0,1] or lip(X,α) for α∈(0,1), then by Theorem 16, Ch(B,X)=Ch(A,X,τ) and Γ(A,X,τ)=Γ(B,X), whenever A=Lip(X,τ,α), for α∈(0,1], or A=lip(X,τ,α), for α∈(0,1). Since B¯=C(X) and
(16)Ch(B,X)=Ch(B¯,X)=Γ(B,X)=Γ(B¯,X)=X,
we have Ch(A,X,τ)=Γ(A,X,τ)=X.Example 19.
LetX be the closed unit interval [0,1] with the usual Euclidean topology. Let B=D∞(X) be the complex algebra of all (continuous) complex-valued functions with derivatives of all orders on X. It is known that B 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 B is a self-adjoint complex subalgebra of C(X) which contains 1 and separates the points of X, we have B¯=C(X) by the Stone-Weierstrass theorem for complex subalgebras of C(X). Since Ch(C(X),X)=Γ(C(X),X)=X, by Theorem 9(iii) it follows that Ch(B,X)=Γ(B,X)=X.
We now define the topological involutionτ on X by τ(x)=1-x and let σ be the algebra involution induced by τ on C(X). Since for every h∈B, σ(h) is a differentiable function of all orders on X and (σ(h))(n)=(-1)nσ(h(n)) for each n∈ℕ, it follows that σ(B)=B. If we define A={h∈B:σ(h)=h}, then A is a real subalgebra of C(X,τ) containing 1, separating the points of X, and moreover, the algebra B={f+ig:f,g∈A} is the complexification of A. Then by [15, Theorem 1.2], A is not complete under any algebra norm. Since Ch(B,X)=Γ(B,X)=X, we conclude that Ch(A,X,τ)=Γ(A,X,τ)=X by Theorem 16.
On the other hand,A¯=C(X,τ) by the Stone-Weierstrass theorem for real subalgebras of C(X,τ) [3]. Since Ch(C(X,τ),X)=Γ(C(X,τ),X)=X, we conclude that Ch(A,X,τ)=Γ(A,X,τ)=X 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 andp-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 onX can be regarded as a real uniform function algebra on a compact Hausdorff space with a suitable topological involution τ [3]. We may take Y as a disjoint union of two copies of X and τ as a homeomorphism that sends a point in one copy of X to the corresponding point in the other copy of X [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(B,|||·|||) be a complex Banach function algebra on a compact Hausdorff space X. Then there exists a compact Hausdorff space Y, a topological involution τ on Y, and a real Banach function algebra (A,∥·∥) on (Y,τ) such that (B,|||·|||), regarded as a real Banach algebra, is isometrically isomorphic to (A,∥·∥). Moreover,  (i)
the algebraA is natural if and only if B is natural.(ii)
There exists a one-to-one correspondence between the set of all peak sets (p-sets, resp.) for B and the set of all peak sets (p-sets, resp.) for A, with respect to (Y,τ).Proof.
LetY be the compact Hausdorff space X×{0,1} with the product topology. We define the map τ:Y→Y by τ(x,0)=(x,1) and τ(x,1)=(x,0) for all x∈X. Clearly, τ is a topological involution on Y. Now, we define the map Ψ:C(X)→C(Y,τ) by
(17)Ψ(f)(x,0)=f(x),Ψ(f)(x,1)=f(x)¯(f∈C(X),x∈X).
Then Ψ is an isometrically isomorphism from (C(X),∥·∥X), regarded as a real Banach algebra, onto (C(Y),∥·∥Y) and Ψ(1)=1. Set A=Ψ(B). It follows that A is a real subalgebra of C(Y,τ) which contains 1 and separates the points of Y. We define ∥g∥=|||Ψ-1(g)||| for all g∈A. Then ∥·∥ is a complete algebra norm on A and ∥1∥=1. Therefore, A is a real Banach function algebra on (Y,τ). Now, we define the map Φ:B→A by Φ=Ψ|B. Clearly, Φ is an isometric isomorphism from (B,|||·|||), regarded as a real Banach algebra, onto (A,∥·∥).
LetBr denote B as a real algebra. Then it is easy to show that MBr=MB∪{η¯:η∈MB}. To prove the naturality of A, let φ∈MA. Then φ∘Φ∈MBr. If B is natural, either there exists x∈X such that φ∘Φ=ex or there exists y∈X such that φ∘Φ=ey¯. We can easily show that φ is the evaluation character at (x,0) or at (y,1), respectively. Therefore, A is natural.
Conversely, letA be natural and η∈MB. Hence η∘Φ-1∈MA. Therefore, either there exists x∈X such that η∘Φ-1 is the evaluation character at some (x,0) on A or it is the evaluation character at some (y,1) on A. Since η∈MB and B is a complex algebra, the latter case does not occur, and hence η∘Φ-1=e(x,0). Since Φ(B)=A, it follows that η=ex and thus B is natural.
Now letS be a peak set (p-set, resp.) for B. It is easy to see that S×{0,1} is a peak set (p-set, resp.) for A.
Conversely, letE be a peak set for A. Then there exists a subset S of X such that E=S×{0,1}. We can easily show that S is a peak set for B.
IfP is a p-set for B, then P=∩α∈ISα, where Sα is a peak set for B. Since P×{0,1}=∩α∈I(Sα×{0,1}), it follows that P×{0,1} is a p-set for A.Concerning the union of peak sets andp-sets for real or complex subspaces of C(X), the following results are well-known.Theorem 21 (see [4, Remark 2.2.10]).
LetA be a uniformly closed real subalgebra of C(X) containing 1. If S and T are peak sets (p-sets, resp.) for A, then S∪T is a peak set (p-set, resp.) for A.Theorem 22 (see [18, II.  Corollary 12.8]).
LetB be a complex uniform function algebra on X. If {Sj}j=1∞ is a sequence of p-sets for B and S=⋃j=1∞‍Sj is closed, then S is a p-set for B.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 B and the peak sets (p-sets, resp.) are necessary.Theorem 23.
LetB be a natural complex Banach function algebra on X. Then a finite union of pairwise disjoint peak sets (p-sets, resp.) for B is a peak set (p-set, resp.) for B.The following two results state the relation between the peak sets (p-sets, resp.) for a real subspace of C(X,τ) and those of its complexification, which are modifications of [4, Theorem 2.2.11].Theorem 24.
LetA be a real subspace of C(X,τ) containing 1 and B={f+ig:f,g∈A}.  (i)
IfS is a peak set (p-set, resp.) for A, then it is a peak set (p-set, resp.) for B. Moreover, τ(S) is also a peak set (p-set, resp.) for A and τ(S)=S.(ii)
IfS is a peak set (p-set, resp.) for B, then τ(S) is also a peak set (p-set, resp.) for B. In case τ(S)=S, S is, in fact, a peak set (p-sets, resp.) for A.(iii)
S0(A,X)=S0(B,X)∩Fix(τ) and S(A,X)=S(B,X)∩Fix(τ).Theorem 25.
LetA be a uniformly closed subalgebra of C(X,τ) containing 1 and let B={f+ig:f,g∈A}.  (i)
IfS is a peak set (p-set, resp.) for B, then S∪τ(S) is a peak set (p-set, resp.) for A.(ii)
S0(B,X)⊆T0(A,X,τ) and S(B,X)⊆T(A,X,τ).(iii)
T0(A,X,τ)∩Fix(τ)⊆S0(B,X) and T(A,X,τ)∩Fix(τ)⊆S0(B,X).(iv)
IfA is a real uniform function algebra on (X,τ), then T0(A,X,τ)=S0(B,X) and T(A,X,τ)=S(B,X).We now extend Theorem23 for natural real Banach function algebras.Theorem 26.
Let(A,∥·∥) be a natural real Banach function algebra on (X,τ). Then a finite union of pairwise disjoint peak sets (p-sets, resp.) for A is a peak set (p-set, resp.) for A.Proof.
It is sufficient to show that ifS1 and S2 are two disjoint peak sets (p-sets, resp.) for A, then S1∪S2 is a peak set (p-set, resp.) for A. Let B={f+ig:f,g∈A}. Then B is a complex subalgebra of C(X) and there exists an algebra norm |||·||| on B such that (B,|||·|||) is a natural complex Banach function algebra on X by [15, Theorem 1.2]. On the other hand, S1 and S2 are peak sets (p-sets, resp.) for B by Theorem 24(i). Thus S1∪S2 is a peak set (p-set, resp.) for B by Theorem 23. Since S1 and S2 are τ-invariant, S1∪S2 is also τ-invariant. Therefore, S1∪S2 is a peak set (p-set, resp.) for A 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 A in the above theorem are necessary.Now, we show that part (i) of Theorem25 holds for real Banach function algebras with small modification.Theorem 28.
Let(A,∥·∥) be a real Banach function algebra on (X,τ) and B={f+ig:f,g∈A} be its complexification. If S is a peak set (p-set, resp.) for B then S∪τ(S) is a peak set (p-set, resp.) for A-.Proof.
SinceS is a peak set (p-set, resp.) for B, we conclude that τ(S) is also a peak set (p-set, resp.) for B by Theorem 24(ii). Let D={f+ig:f,g∈A-} be the complexification of A-. Clearly, S and τ(S) are peak sets (p-sets, resp.) for D and hence by Theorem 25(i), S∪τ(S) is a peak set (p-set, resp.) for A-.Theorem 29.
Let(B,∥·∥) be a natural complex Banach function algebra on X and σ be the algebra involution on C(X), induced by the topological involution τ. Let σ(B)=B and A={h∈B:σ(h)=h}. If S1 and S2 are τ-invariant peak sets (p-sets, resp.) for B such that S1∩S2=ϕ, then S1∪S2 is a peak set (p-set, resp.) for A.Proof.
By [15, Theorem  1.1], (A,∥·∥) is a natural real Banach function algebra on (X,τ) and B={f+ig:f,g∈A}. On the other hand, S1 and S2 are peak sets (p-sets, resp.) for A by Theorem 24(ii). Therefore, S1∪S2 is a peak set(p-set, resp.) for A by Theorem 26.The following example shows that the uniformly closed condition ofA in Theorem 25(i) and the condition S1∩S2=∅ in Theorem 29 are essential.Example 30.
LetX={(z,w)∈ℂ2:|z|≤1,|w|≤1} and let B be the complex algebra of all continuous complex-valued functions on X which have continuous first-order partial derivatives with respect to z and w. We define the algebra norm ∥·∥ on B by
(18)∥f∥=∥f∥X+∥∂f∂z∥X+∥∂f∂w∥X(f∈B).
Then (B,∥·∥) is a natural complex Banach function algebra on X and although the sets S1={(1,w):|w|≤1} and S2={(z,1):|z|≤1} are peak sets for B, the set S1∪S2 is not a peak set for B, as shown by Dales in [12, Example 1]. Note that S1∩S2 is nonempty.
We first define the topological involutionτ on X by τ(z,w)=(w-,z-). Let σ be the algebra involution on C(X), induced by τ. By the definition of B, for every f∈B and for each fixed w in the closed unit disk 𝔻-, f(w-,z) has continuous derivative on 𝔻- and hence f-(w-,z-) has also continuous derivative on 𝔻- as a function of z. Similarly, for each fixed z∈𝔻-, f(w,z-) has continuous derivative on 𝔻- and hence, as a function of w, f-(w-,z-) has also continuous derivative on 𝔻-. Therefore, f-(w-,z-) has continuous partial derivatives on X with respect to z and w. Since f-(w-,z-)=f-∘τ(z,w), for all (z,w)∈X, it follows that σ(f)=f-∘τ∈B and hence σ(B)=B. If we define A={h∈B:σ(h)=h}, then (A,∥·∥) is a natural real Banach function algebra on (X,τ) and B={f+ig:f,g∈A} is the complexification of A by [15, Theorem  1.1]. Moreover, A is not uniformly closed. Since τ(S1)=S2, the set S1∪τ(S1) is not a peak set for B. Therefore, S1∪τ(S1) is not a peak set for A by Theorem 24(i).We now define the topological involutionτ on X by τ(z,w)=(z-,w-). Clearly, S1 and S2 are τ-invariant. Let σ be the algebra involution on C(X), induced by τ. By a similar argument as in the above case, it is easy to see that σ(B)=B. If we define A={h∈B:σ(h)=h}, then A is a natural real Banach function algebra on (X,τ) and B={f+ig:f,g∈A} by  [15, Theorem  1.1]. Since S1∪S2 is not a peak set for B, it is not a peak set for A by Theorem 24(i).Theorem 31.
LetA be a natural real Banach function algebra on (X,τ) and let B={f+ig:f,g∈A}. Then S0(B,X)⊆T0(A,X,τ) and S(B,X)⊆T(A,X,τ).Proof.
Letx∈S0(B,X). If x=τ(x) then x0∈S0(A,X) by Theorem 24(ii), and hence x∈T0(A,X,τ). If x≠τ(x), then {x,τ(x)} is a peak set for B by Theorem 24(ii) and Theorem 23. Since {x,τ(x)} is τ-invariant, it follows that it is a peak set for A by Theorem 24(ii) and hence x∈T0(A,X,τ). Therefore, the first inclusion holds.
By applying parts (i) and (iii) in Theorem24  and Theorem 29 for p-sets, the second inclusion holds with the same argument.Concerning the existence of ap-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]).
LetB a complex uniform function algebra on X. Then every peak set for B contains a p-point for B. Moreover, if X is first countable, then every peak set for B contains a peak point for B.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 by𝔻, 𝔻¯, and 𝕋 the open unit disk {z∈ℂ:|z|<1}, the closed unit disc {z∈ℂ:|z|≤1}, and the unit circle {z∈ℂ:|z|=1}, respectively. Let B=A(𝔻¯), the complex disc algebra. It is known that (B,∥·∥𝔻¯) is a complex uniform algebra on 𝔻¯ and S0(B;𝔻¯)=𝕋. We define the topological involution τ on 𝔻¯ by τ(z)=z¯. Let A={h∈B:σ(h)=h}, where σ is the algebra involution induced by τ on C(𝔻¯). Since σ(f)(z)=f¯(z¯) is continuous on 𝔻¯ and it is analytic on 𝔻 for each f∈B, it follows that σ(B)=B. In fact, A is the real disc algebra and B={f+ig:f,g∈A} is the complexification of A. Moreover,
(19)S0(A,𝔻¯)=S0(B,𝔻¯)∩Fix(τ)=𝕋∩{-1,1}={-1,1},
by Theorem 24(iii). Hence for each z∈𝕋 with z¯≠z, the set {z,z¯} is a peak set for A which does not contain any peak point for A.
The following is a modification of Theorem32 for real uniform function algebras.Theorem 34.
LetA be a real uniform function algebra on (X,τ). Then every peak set for A contains a τ-p-point for A. Moreover, if X is first countable, every peak set for A contains a τ-peak point for A.Proof.
LetS be a peak set for A. If we take B={f+ig:f,g∈A} then B is a complex uniform function algebra on X by [4, Theorem 1.3.20] and S is also a peak set for B. Hence S∩S(B,X)≠∅ and when X is first countable, S∩S0(B,X)≠∅, by Theorem 32. Since S(B,X)=T(A,X,τ) and S0(B,X)=T0(A,X,τ), we have S∩T(A,X,τ)≠∅, and if X is first countable, then S∩T0(A,X,τ)≠∅.


## 4. On the Density ofτ-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]).
LetB be a complex uniform function algebra on X. Then S(B,X)=Ch(B,X). Moreover, if X is first countable, then S0(B,X)=Ch(B,X) and so Γ(B,X) is the closure of S0(B,X).Theorem 36 (see [11, Theorem 2.3] and [21]).
LetB be a Banach function algebra on a compact metrizable space X. Then Γ(B,X) is the closure of S0(B,X).The following example shows that the above theorem does not hold even for real uniform algebras on compact metrizable spaces, in general.Example 37.
LetB be the complex disc algebra and let A be the real disc algebra, considered in Example 33. It is known that S0(B,𝔻¯)=Γ(B;𝔻¯)=𝕋. Therefore, Γ(A,𝔻¯,τ)=𝕋 by Theorem 16. On the other hand, S0(A,𝔻¯)={-1,1} as shown in Example 33. Hence, the closure of S0(A,𝔻¯) is not equal to Γ(A,X,τ).Theorem 38 (see [10] or [4, Theorem 4.2.4]).
LetA be a uniformly closed real subalgebra of C(X,τ) containing 1. Then
(20)Ch(A,X,τ)=T(A,X,τ).Remark 39.
LetA be a real uniform function algebra on (X,τ) and B={f+ig:f,g∈A}. Then we have Ch(A,X,τ)=T(A,X,τ) by Theorem 38, Ch(A,X,τ)=Ch(B,X) by Theorem 15, and Ch(B,X)=S(B,X) by Theorem 35. Therefore, T(A,X,τ)=S(B,X), which has already been obtained in Theorem 25(iv).Theorem 40.
LetA be a uniformly closed real subalgebra of C(X,τ) containing 1 and let ReA separate the points of X/τ. Then
(21)T(A,X,τ)¯=Γ(A,X,τ).
In particular, if X is a first countable space, then
(22)T0(A,X,τ)¯=Γ(A,X,τ).Proof.
By Theorem38 we have T(A,X,τ)=Ch(A,X,τ) and since Ch(A,X,τ)¯=Γ(A,X,τ) by Theorem 13(iii), we conclude that
(23)T(A,X,τ)¯=Γ(A,X,τ).
In the case that X is first countable, every finite subset of X is a Gδ-set. Hence T(A,X,τ)⊆T0(A,X,τ) and so T(A,X,τ)=T0(A,X,τ). Therefore, T0(A,X,τ)¯=Γ(A,X,τ) by the previous argument.Theorem 41.
LetX be a compact metrizable space and let τ be a topological involution on X. If (A,∥·∥) is a natural real Banach function algebra on (X,τ), then
(24)T0(A,X,τ)¯=Γ(A,X,τ).Proof.
LetB={f+ig:f,g∈A}. The naturality of A implies that S0(B,X)⊆T0(A,X,τ) by Theorem 31. On the other hand, there exists an algebra norm |||·||| on B such that (B,|||·|||) is a complex Banach function algebra on X by [15, Theorem 1.2]. Since X is a metrizable compact space, we have S0(B,X)¯=Γ(B,X) by Theorem 36. On the other hand, Γ(A,X,τ)=Γ(B,X) by Theorem 16. Therefore, S0(B,X)¯=T0(A,X,τ)¯=Γ(A,X,τ).



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