Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 519893, 9 pages http://dx.doi.org/10.1155/2013/519893

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

Davood Alimohammadi1 and Taher Ghasemi Honary2

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

Correspondence should be addressed to Davood Alimohammadi; [email protected]

Received 5 December 2012; Accepted 1 March 2013

Academic Editor: Dachun Yang

Copyright © 2013 D. Alimohammadi and T. G. 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.

Let 𝑋 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).In1971,Dalesobtainedsome 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).

1. Introduction and Preliminaries Let 𝐴 be a subspace of 𝐶(𝑋) over F containing 1. We ∗ denote by 𝐾F (𝐴) the set of all 𝜑∈𝐴 for which ‖𝜑‖ = F R C 𝑋 ∗ Let denote either or . We always assume that is a 𝜑(1) =,where 1 𝐴 is the dual space of the normed space 𝐶 (𝑋) ∗ compact Hausdorff space. We denote by F the Banach (𝐴, ‖ ⋅ ‖𝑋) over F. In fact, the elements of 𝐴 are F-valued 𝑋 F algebra of all continuous functions from into ,withthe linear functionals on 𝐴 over F.Foreach𝑥∈𝑋the map uniform norm 𝑒𝑥 :𝐴 → C, defined by 𝑒𝑥(𝑓) = 𝑓(𝑥), is a linear mapping over F,whichiscalledtheevaluation map on 𝐴 at 𝑥.Clearly 󵄩 󵄩 󵄨 󵄨 󵄩𝑓󵄩𝑋 = sup {󵄨𝑓 (𝑥)󵄨 :𝑥∈𝑋}(𝑓∈𝐶F (𝑋)) . (1) 𝑒𝑥 ∈𝐾C(𝐴) whenever F = C and Re 𝑒𝑥 ∈𝐾R(𝐴) whenever F = R. 𝐵 𝐶(𝑋) However, we always write 𝐶(𝑋) instead of 𝐶C(𝑋) and we Let be a complex subspace of containing 1. A 𝜑∈𝐵∗ denote the uniform closure of 𝐴 by 𝐴,whenever𝐴 is a subset representing measure for is a complex regular Borel 𝜇 𝑋 𝜑(𝑓) =∫ 𝑓𝑑𝜇 𝑓∈𝐵 of 𝐶(𝑋). measure on such that 𝑋 for all .It 𝜑∈𝐾(𝐵) Let 𝐴 be a real or complex subspace of 𝐶(𝑋). A nonempty is known that every C has a representing measure 𝜑 subset 𝑆 of 𝑋 is called a boundary (Choquet set,resp.)for𝐴 and every representing measure for such is a probability 𝑥∈𝑋 𝛿 (with respect to 𝑋), if for each 𝑓∈𝐴the function |𝑓| (Re 𝑓, measure [2,Section2.1].If ,then 𝑥,thepointmass 𝑥 𝑒 resp.) assumes its maximum on 𝑋 at some 𝑥∈𝑆.Notethat measure at , is a representing measure for 𝑥. We denote 𝐶ℎ(𝐵, 𝑋) 𝑥∈𝑋 𝛿 every closed boundary for 𝐴 is a closed Choquet set for 𝐴 by the set of all for which 𝑥 is the only 𝑒 𝐶ℎ(𝐵, 𝑋) 𝐵 [1, Lemma 1.1] and if 𝐴 is closed under the complex scalar representing measure for 𝑥.If is a boundary for , 𝐵 multiplication, then every Choquet set for 𝐴 is a boundary it is called the Choquet boundary for . 𝐴 𝐶(𝑋) for 𝐴 [1,Section1].WedenotebyΓ(𝐴, 𝑋) the intersection of Let be a real subspace of containing 1. A real part 𝜓∈𝐴∗ all closed boundaries for 𝐴.IfΓ(𝐴, 𝑋) is a boundary for 𝐴,it representing measure for is a regular Borel measure 𝜇 𝑋 𝜓(𝑓) =∫ 𝑓𝑑𝜇 𝑓∈𝐴 is called the Shilov boundary for 𝐴. on such that 𝑋 Re for all .Itis 2 Journal of Function Spaces and Applications

wellknown that every 𝜓∈𝐾R(𝐴) has a real part representing by 𝑆0(𝐴, 𝑋),or𝑆(𝐴,,respectively.Notethat 𝑋) 𝑆0(𝐴, 𝑋) is the measure [1,Theorem1.5].If𝑥∈𝑋,then𝛿𝑥 is a real part intersection of all boundaries for 𝐴. representing measure for Re 𝑒𝑥. We denote by 𝐶ℎ(𝐴, 𝑋) the If 𝐴 isarealsubalgebraof𝐶(𝑋,,thenevery 𝜏) 𝑝-set and set of all 𝑥∈𝑋for which 𝛿𝑥 is the only real part representing hence every peak set for 𝐴 are 𝜏-invariant. Hence 𝑆0(𝐴, 𝑋) ⊆ measure for Re 𝑒𝑥.If𝐶ℎ(𝐴, 𝑋) is a boundary for 𝐴, it is called 𝑆(𝐴, 𝑋) ⊆ Fix(𝜏).Moreover,𝑆0(𝐴, 𝑋) is contained in every the Choquet boundary for 𝐴. 𝜏-invariant boundary for 𝐴. Let 𝐸 be a nonempty set. A self-map 𝜏:𝐸is →𝐸 called an involution on 𝐸 if 𝜏(𝜏(𝑥)) =𝑥 for all 𝑥∈𝐸.Wedenoteby Definition 1. Let 𝐴 be a real subspace of 𝐶(𝑋,.Thepoint 𝜏) 𝑥∈ Fix(𝜏) the set of all 𝑥∈𝑋for which 𝜏(𝑥) =𝑥.Asubset𝑆 of 𝐸 𝑋 is a 𝜏-peak point (𝜏-p-point,resp.)for𝐴 if the set {𝑥, 𝜏(𝑥)} is called 𝜏-invariant if 𝜏(𝑆).A =𝑆 𝜏-invariant measure on 𝐸 is a peak set (a 𝑝-set, resp.) for 𝐴. We define is a measure 𝜇 on 𝐸 such that 𝜇∘𝜏=𝜇. 𝑇 (𝐴, 𝑋, 𝜏) ={𝑥∈𝑋:𝑥 𝜏 𝐴} , An involution 𝜏 on 𝑋 is called a topological involution on 0 is a -peak point for 𝑋 if 𝜏 is continuous. The map 𝜎: 𝐶(𝑋) → 𝐶(𝑋) defined by (4) 𝑇 (𝐴, 𝑋, 𝜏) ={𝑥∈𝑋:𝑥is a 𝜏-𝑝-point for 𝐴} . 𝜎(𝑓) = 𝑓∘𝜏is an algebra involution on 𝐶(𝑋),whichiscalled 𝜏 𝐶(𝑋) the algebra involution induced by on . We now define It is easy to see that 𝑇0(𝐴, 𝑋, 𝜏) is the intersection of all 𝜏-invariant boundaries for 𝐴. 𝐶 (𝑋, 𝜏) ={𝑓∈𝐶(𝑋) :𝜎(𝑓)=𝑓}. (2) Definition 2. A complex Banach function algebra on 𝑋 is a Then 𝐶(𝑋, 𝜏) is a unital self-adjoint uniformly closed real complex subalgebra of 𝐶(𝑋) which contains 1, separates the subalgebra of 𝐶(𝑋), which separates the points of 𝑋 and points of 𝑋, and it is a unital Banach algebra under an algebra does not contain the constant function 𝑖.Moreover,𝐶(𝑋) = norm ‖⋅‖. If the norm of a complex Banach function algebra 𝐶(𝑋, 𝜏) ⊕ 𝑖𝐶(𝑋, 𝜏) and on 𝑋 is the uniform norm on 𝑋, then it is called a complex 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑋 max {󵄩𝑓󵄩𝑋, 󵄩𝑔󵄩𝑋} ≤ 󵄩𝑓+𝑖𝑔󵄩𝑋 ≤2max {󵄩𝑓󵄩𝑋, 󵄩𝑔󵄩𝑋} (3) uniform function algebra on . Clearly, 𝐶(𝑋) is a complex uniform function algebra on for all 𝑓, 𝑔 ∈ 𝐶(𝑋, 𝜏). In fact, the complex Banach algebra 𝑋. (𝐶(𝑋), ‖𝑋 ⋅‖ ) can be regarded as the complexification of the Let 𝐵 be a unital commutative Banach algebra over F.A real Banach algebra (𝐶(𝑋, 𝜏),𝑋 ‖⋅ ).Notethat𝐶(𝑋, 𝜏) = character of 𝐵 is a nonzero homomorphism 𝜑:𝐵 → C, 𝐶R(𝑋) if and only if 𝜏 is the identity map on 𝑋.Hence where C is regarded as an algebra over F. We denote by the class real Banach algebras of continuous complex-valued 𝑀𝐵 the set of all characters of 𝐵.Itisknownthat𝑀𝐵,with functions 𝐶(𝑋, 𝜏) are, in fact, larger than the class of real the Gelfand topology, is a compact Hausdorff space (see5 [ , Banach algebras of continuous real-valued functions 𝐶R(𝑋). Chapter 11] and [4,Chapter1]). This class of continuous functions was defined explicitly by Kulkarni and Limaye in [3]. Definition 3. A real Banach function algebra on (𝑋, 𝜏) is a real Let 𝐴 be a real subspace of 𝐶(𝑋, 𝜏) containing 1. We subalgebra of 𝐶(𝑋, 𝜏) which contains 1, separates the points denote by Γ(𝐴, 𝑋, 𝜏) the intersection of all 𝜏-invariant closed of 𝑋, and it is a real unital Banach algebra under an algebra boundaries for 𝐴.IfΓ(𝐴, 𝑋, 𝜏) is a boundary for 𝐴,wecall norm ‖⋅‖. If the norm of a real Banach function algebra it the Shilov boundary for 𝐴 with respect to (𝑋, 𝜏).Foreach on (𝑋, 𝜏) is the uniform norm on 𝑋, then it is called a real 𝑥∈𝑋,themeasure(1/2)(𝛿𝑥 +𝛿𝜏(𝑥)) is a positive regular Borel uniform function algebra on (𝑋, 𝜏). measure on 𝑋, which is denoted by 𝑚𝑥.If𝑥∈𝑋,then𝑚𝑥 Clearly, 𝐶(𝑋, 𝜏) is a real uniform function algebra on is a 𝜏-invariant real part representing measure for Re 𝑒𝑥.We (𝑋, 𝜏). denote by 𝐶ℎ(𝐴,𝑋,𝜏) the set of all 𝑥∈𝑋for which 𝑚𝑥 is Sincetheclassofrealuniform(orBanach)function the only 𝜏-invariant real part representing measure for Re 𝑒𝑥. algebras on (𝑋, 𝜏) is larger than the class of complex uniform If 𝐶ℎ(𝐴,𝑋,𝜏) is a boundary for 𝐴,wecallittheChoquet (or Banach) function algebras, it is quite natural to ask which boundary for 𝐴 with respect to (𝑋, 𝜏). properties of a complex algebra can be extended to the For a topological involution 𝜏 on 𝑋 let 𝑋/𝜏 = {{𝑥, 𝜏(𝑥)} : corresponding real algebra. 𝑥∈𝑋}.Forasubset𝐴 of 𝐶(𝑋, 𝜏) we say that Re 𝐴 separates Let 𝐴 be a real (complex, resp.) Banach function algebra the points of 𝑋/𝜏 if for each 𝑥∈𝑋and 𝑦 ∈ 𝑋 \ {𝑥, ,𝜏(𝑥)} on (𝑋, 𝜏) (on 𝑋, resp.). The evaluation character 𝑒𝑥 is an there exists a function 𝑓 in 𝐴 such that Re 𝑓(𝑥) ≠ Re 𝑓(𝑦). element of 𝑀𝐴 for all 𝑥∈𝑋.Wecall𝐴 to be natural if every It is interesting to note that if 𝐴 isarealsubalgebraof𝐶(𝑋, 𝜏) 𝜑∈𝑀𝐴 isgivenbyanevaluationcharacter𝑒𝑥 at some 𝑥∈𝑋. which separates the points of 𝑋 andcontains1,thenRe𝐴 The concept of the Shilov boundary for a real Banach separates the points of 𝑋/𝜏 [4, Lemma 1.3.9]. algebra was studied by Ingelstam, Limaye, and Simha in [6– Let 𝐴 be a real or complex subalgebra of 𝐶(𝑋) and let 𝑆 8]. The concept of Choquet and Shilov boundaries for real be a nonempty subset of 𝑋.Then𝑆 is called a peak set for 𝐴 uniform function algebras was first studied by Kulkarni and if there exists 𝑓∈𝐴such that 𝑆={𝑥∈𝑋:𝑓(𝑥)=1}and Arundhathi in [9]. Later Kulkarni and Limaye introduced the |𝑓(𝑦)| <1 for all 𝑦∈𝑋\𝑆.Wesaythat𝑆 is a 𝑝-set (a peak set in notions of Choquet sets and boundaries for real subspaces the weak sense or a weak peak set)for𝐴 if 𝑆 is the intersection of 𝐶(𝑋) and 𝐶(𝑋, 𝜏) in [1] and obtained interesting results of some collection of peak sets for 𝐴.Ifthepeaksetor𝑝-set 𝑆 on Choquet and Shilov boundaries for uniformly closed real for 𝐴 is a singleton {𝑥},thenwecall𝑥 a peak point or p-point subalgebras of 𝐶(𝑋) or 𝐶(𝑋,.Theyalsoobtainedrelations 𝜏) for 𝐴. The set of all peak points, or 𝑝-points, for 𝐴 is denoted between 𝐶ℎ(𝐴,𝑋,𝜏) and 𝐶ℎ(𝐵, 𝑋) (Γ(𝐴, 𝑋, 𝜏) and Γ(𝐵, 𝑋), Journal of Function Spaces and Applications 3

resp.), when 𝐴 is a real uniform function algebra on (𝑋, 𝜏) (i) 𝜑∈𝐾R(𝐴) is an extreme point of 𝐾R(𝐴) if and only and 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}is the complexification of 𝐴. if there exists 𝑥 ∈ 𝐶ℎ(𝐴, 𝑋,𝜏) such that 𝜑=Re 𝑒𝑥.In In Section 2,wefirstshowthat𝐶ℎ(𝐴, 𝑋) = 𝐶ℎ(𝐴, 𝑋) and particular, 𝐶ℎ(𝐴,𝑋,𝜏)is nonempty. Γ(𝐴, 𝑋)𝐴, =Γ( 𝑋) and by applying these equalities we extend (ii) 𝐶ℎ(𝐴,𝑋,𝜏)is the set of all 𝑥∈XsuchthatRe 𝑒𝑥 is an some results due to Kulkarni, Limaye, and Arundhathi, by extreme point of 𝐾R(𝐴). omitting the uniformly closed condition on 𝐴. (iii) 𝐶ℎ(𝐴,𝑋,𝜏)is a 𝜏-invariant Choquet set for 𝐴. The concept of peak points and 𝑝-points for real uniform function algebras was studied by Kulkarni and Arundhathi (iv) The closure of 𝐶ℎ(𝐴,𝑋,𝜏) is the smallest 𝜏-invariant in [9, 10]. Also Dales has obtained many results on the Choquet set for 𝐴. boundaries and peak sets for complex Banach function algebras in [11, 12]. In Sections 3 and 4, we extend some of We now show that 𝐶ℎ(𝐴, 𝑋) = 𝐶ℎ(𝐴, 𝑋) and Γ(𝐴, 𝑋) = their results to real Banach function algebras. 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]. 2. Choquet and Shilov Boundaries Theorem 9. (i) Let 𝐴 be a real subspace of 𝐶(𝑋) containing 1. We begin this section by recalling some well-known results. If Re 𝐴 separates the points of 𝑋,then𝐶ℎ(𝐴, 𝑋) = 𝐶ℎ(𝐴, .𝑋) 𝐴 𝐶(𝑋, 𝜏) 𝐴 Theorem 4 (see [4, Theorem 4.3.3 and Corollary 4.3.4]). Let (ii) Let be a real subspace of containing 1. If Re 𝐵 be a complex subspace of 𝐶(𝑋) containing 1 and separating separates the points of 𝑋/𝜏,then𝐶ℎ(𝐴,𝑋,𝜏)=𝐶ℎ(𝐴, 𝑋,. 𝜏) the points of 𝑋.Then (iii) Let 𝐵 be a complex subspace of 𝐶(𝑋) containing 1. If 𝐵 separates the points of 𝑋,then𝐶ℎ(𝐵, 𝑋) 𝐵,= 𝐶ℎ( 𝑋) and (i) 𝜑∈𝐾C(𝐵) is an extreme point of 𝐾C(𝐵) if and only Γ(𝐵, 𝑋)𝐵, =Γ( 𝑋). if there exists 𝑥 ∈ 𝐶ℎ(𝐵, 𝑋) such that 𝜑=𝑒𝑥.In particular, 𝐶ℎ(𝐵, 𝑋) is nonempty. Proof. (i) For every 𝑥∈𝑋,let𝑒𝑥 and 𝜀𝑥 be the evaluation (ii) 𝐶ℎ(𝐵, 𝑋) is the set of all 𝑥∈𝑋such that 𝑒𝑥 is an maps at 𝑥 on 𝐴 and 𝐴,respectively.ByTheorem 6(ii), we have extreme point of 𝐾C(𝐵). 𝐶ℎ (𝐴, 𝑋) (iii) 𝐶ℎ(𝐵, 𝑋) is a Choquet set for 𝐵 and, in particular, a 𝐵 boundary for . ={𝑥∈𝑋:Re 𝑒𝑥 is an extreme point of 𝐾R (𝐴)}, (iv) The closure of 𝐶ℎ(𝐵, 𝑋) is equal to Γ(𝐵, .𝑋) (5) 𝐶ℎ (𝐴, 𝑋) Theorem 5 (see [1,Corollary1.8]). Let 𝐴 be a real subspace of 𝐶(𝑋).Thentheclosureof𝐶ℎ(𝐴, 𝑋) is contained in every ={𝑥∈𝑋:Re 𝜀𝑥 is an extreme point of 𝐾R (𝐴)} . closed Choquet set as well as every closed boundary for 𝐴.In 𝑥 ∈ 𝐶ℎ(𝐴, 𝑋) 𝜀 = (1 − 𝑡)Φ + 𝑡Ψ 0<𝑡<1 particular, 𝐶ℎ(𝐴, 𝑋) ⊆ Γ(𝐴,. 𝑋) Let and Re 𝑥 ,where and Φ, Ψ ∈ 𝐾R(𝐴).Set𝜑=Φ|𝐴 and 𝜓=Ψ|𝐴.Itiseasyto Theorem 6 (see [1, Lemma 2.1 and Theorem 2.2]). Let 𝐴 be show that 𝜑, 𝜓R ∈𝐾 (𝐴) and Re 𝑒𝑥 = (1−𝑡)𝜑+𝑡𝜓 on 𝐴.Hence arealsubspaceof𝐶(𝑋) containing 1 and Re 𝐴 separates the 𝜑=𝜓=Re 𝑒𝑥.Let𝑓 be an arbitrary element of 𝐴. Then there points of 𝑋.Then exists a sequence {𝑓𝑛} in 𝐴 such that lim𝑛→∞ ‖𝑓𝑛 −𝑓‖𝑋 =0. ∗ Since Φ∈(𝐴) ,wehave (i) The element 𝜑∈𝐾R(𝐴) is an extreme point of 𝐾R(𝐴) 𝜑= 𝑒 𝑥 ∈ 𝐶ℎ(𝐴, 𝑋) ifandonlyif Re 𝑥 for some .In Φ(𝑓)= lim 𝜑(𝑓𝑛) 𝐶ℎ(𝐴, 𝑋) 𝑛→∞ particular, is nonempty. (6) 𝐶ℎ(𝐴, 𝑋) 𝑥∈𝑋 𝑒 = ((𝑓 (𝑥))) = 𝑓 (𝑥) =( 𝜀 )(𝑓). (ii) is the set of all such that Re 𝑥 is an 𝑛→∞lim Re 𝑛 Re Re 𝑥 extreme point of 𝐾R(𝐴). Hence Φ=Re 𝜀𝑥.Similarly,Ψ=Re 𝜀𝑥. Therefore, 𝜀𝑥 is (iii) 𝐶ℎ(𝐴, 𝑋) is a Choquet set for 𝐴. an extreme point of 𝐾R(𝐴) and so 𝑥∈𝐶ℎ(𝐴, .𝑋) (iv) The closure of 𝐶ℎ(𝐴, 𝑋) isthesmallestclosedChoquet Conversely, let 𝑥∈𝐶ℎ(𝐴, 𝑋) and Re 𝑒𝑥 =(1−𝑡)𝜑+𝑡𝜓, set for 𝐴. where 0<𝑡<1and 𝜑, 𝜓R ∈𝐾 (𝐴). ∗ Clearly, if 𝜂∈𝐴 and {ℎ𝑛} is a sequence in 𝐴 with Theorem 7 (see [4, Theorem 4.2.2((a), (b))]). Let 𝐴 be a real lim𝑛→∞ ‖ℎ𝑛‖𝑋 =0,thenlim𝑛→∞𝜂(ℎ𝑛)=0. subspace of 𝐶(𝑋, 𝜏) containing 1. Now let 𝑓∈𝐴 and let {𝑓𝑛} be a sequence in 𝐴 with (i) If 𝑆 is a 𝜏-invariant boundary for 𝐴, then the closure of lim𝑛→∞ ‖𝑓𝑛 −𝑓‖𝑋 =0.Wedefine𝜑̃ and 𝜓̃ on 𝐴 by 𝜑(𝑓)̃ = 𝑆 is a 𝜏-invariant Choquet set for 𝐴. lim𝑛→∞𝜑(𝑓𝑛) and 𝜓(𝑓)̃ = lim𝑛→∞𝜓(𝑓𝑛).Bytheprevious argument 𝜑̃ and 𝜓̃ are well defined. It is easy to see that (ii) 𝐶ℎ(𝐴,𝑋,𝜏)⊆Γ(𝐴,𝑋,𝜏). 𝜑,̃ 𝜓∈𝐾̃ R(𝐴) and (1 − 𝑡)𝜑+𝑡̃ 𝜓=̃ Re 𝜀𝑥 on 𝐴. On the other Theorem 8 (see [4, Theorem 4.1.10 and Corollary 4.1.11] and hand, 𝜀𝑥 is an extreme point of 𝐾R(𝐴) since 𝑥∈𝐶ℎ(𝐴, .𝑋) [1, Section 4]). Let 𝐴 be a real subspace of 𝐶(𝑋, 𝜏) containing Therefore, 𝜑=̃ 𝜓=𝜀̃ 𝑥 on 𝐴 and so 𝜑=𝜓=𝑒𝑥 on 𝐴.Itfollows 1andletRe 𝐴 separate the points of 𝑋/𝜏.Then that 𝑒𝑥 is an extreme point 𝐾R(𝐴) and so 𝑥 ∈ 𝐶ℎ(𝐴, 𝑋). 4 Journal of Function Spaces and Applications

(ii) This is proved exactly with the same argument as in Table 1 part (i) and applying Theorem 8(ii). 𝐶(𝑋) 𝐶(𝑋, 𝜏) (iii) With the same argument as in part (i), by applying 𝑋 𝑋/𝜏 𝐶ℎ(𝐵, 𝑋) 𝐵,= 𝐶ℎ( 𝑋) points of points of Theorem 4(ii), we can show that . 𝐶ℎ(𝐴, 𝑋) 𝐶ℎ(𝐴, 𝑋, 𝜏) Γ(𝐵, 𝑋) = Therefore, by Theorem 4(iv),weconcludethat 𝜏 Γ(𝐵, 𝑋) (closed) boundary -invariant (closed) boundary . Γ(𝐴, 𝑋) Γ(𝐴, 𝑋, 𝜏) Wenow recall the following known result and then extend Theorem 10 Theorem 12 it by omitting the uniformly closed condition for 𝐴. Ch(𝐴, 𝑋) Ch(𝐴,𝑋,𝜏) Γ(𝐴, 𝑋) Γ(𝐴, 𝑋,𝜏) Theorem 10 𝐴 (see [1, Lemma 3.1 and Theorem 3.2]). Let be Theorem 5 Theorem 7(ii) a uniformly closed real subalgebra of 𝐶(𝑋) containing 1 and let Choquet set 𝜏-invariant Choquet set Re 𝐴 separate the points of 𝑋.Then [1, Lemma 1.1] Theorem 7(i). (i) Asubset𝑆 of 𝑋 is a boundary for 𝐴 ifandonlyifitisa Choquet set for 𝐴. Theorem 13. 𝐴 𝐶(𝑋, 𝜏) (ii) 𝐶ℎ(𝐴, 𝑋) is a boundary for 𝐴. Let be a real subalgebra of containing 1andletRe 𝐴 separate the points of 𝑋/𝜏.Then (iii) The closure of 𝐶ℎ(𝐴, 𝑋) is equal to Γ(𝐴, .𝑋) (i) Γ(𝐴, 𝑋, 𝜏) =Γ(𝐴, 𝑋,. 𝜏) Theorem 11. Let 𝐴 be a real subalgebra of 𝐶(𝑋) containing 1 and let Re 𝐴 separate the points of 𝑋.Then (ii) 𝐶ℎ(𝐴,𝑋,𝜏)is a 𝜏-invariant boundary for 𝐴. (i) Γ(𝐴, 𝑋)𝐴, =Γ( .𝑋) (iii) The closure of 𝐶ℎ(𝐴,𝑋,𝜏) is equal to Γ(𝐴, 𝑋,,the 𝜏) smallest 𝜏-invariant closed boundary for 𝐴. (ii) 𝐶ℎ(𝐴, 𝑋) is a boundary for 𝐴 and hence for 𝐴. 𝜏 𝑆 𝑋 𝐴 (iii) The closure of 𝐶ℎ(𝐴, 𝑋) is equal to Γ(𝐴, .𝑋) (iv) A -invariant closed subset of is a boundary for ifandonlyifitisaChoquetsetfor𝐴. (iv) A 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 Theorem 11 by replacing entries in the first column by the corresponding entries in the 𝐴 Proof. Since satisfies the hypotheses of Theorem 10, second column of Table 1. 𝐶ℎ(𝐴, 𝑋) is a boundary for 𝐴 and 𝐶ℎ(𝐴, 𝑋)𝐴, =Γ( .𝑋) Moreover, we have 𝐶ℎ(𝐴, 𝑋) = 𝐶ℎ(𝐴, 𝑋) by Theorem 9(ii) Corollary 14. 𝐴 (𝑋, 𝜏) and 𝐶ℎ(𝐴, 𝑋) ⊆ Γ(𝐴, 𝑋) by Theorem 5. Therefore, 𝐶ℎ(𝐴, 𝑋) If isarealBanachfunctionalgebraon , then is a boundary for 𝐴 (hence for 𝐴)andΓ(𝐴, 𝑋) ⊆ Γ(𝐴,. 𝑋) Γ(𝐴, 𝑋)𝐴, ⊆Γ( 𝑋) Γ(𝐴, 𝑋)𝐴, =Γ( 𝑋) Since it follows that (i) 𝐶ℎ(𝐴,𝑋,𝜏)=𝐶ℎ(𝐴, 𝑋,. 𝜏) and, moreover, 𝐶ℎ(𝐴, 𝑋) = Γ(𝐴,. 𝑋) 𝐶ℎ(𝐴,𝑋,𝜏) 𝜏 𝐴 To prove (iv) let 𝑆 be a closed boundary for 𝐴.By[1, (ii) is a -invariant boundary for . 𝑆 𝐴 Lemma 1.1], is a closed Choquet set for .Conversely,if (iii) Γ(𝐴, 𝑋, 𝜏) =Γ(𝐴, 𝑋,. 𝜏) 𝑆 is a closed Choquet set for 𝐴,then𝐶ℎ(𝐴, 𝑋) ⊆𝑆 by Γ(𝐴, 𝑋, 𝜏) = 𝐶ℎ(𝐴,𝑋,𝜏) Theorem 6(iv). Therefore, 𝑆 is a boundary for 𝐴 by (ii). (iv) . It is interesting to note that in the previous theorem if Proof. Since 𝐴 separates the points of 𝑋,weconcludethat Re 𝐴 does not separate the points of 𝑋,aclosedChoquetset Re 𝐴 separates the points of 𝑋/𝜏. Therefore, the result follows may not be a boundary for 𝐴 [1, Lemma 1.1]. by Theorem 9(ii) and Theorem 13. The following result is also known and we bring it here for easy reference. The following result is also known.

Theorem 12 (see [4, Theorem 4.2.5] and [9]). Let 𝐴 be a Theorem 15 (see [9, Theorem 3.7, Corollary 3.8]4 or[ , uniformlyclosedrealsubalgebraof𝐶(𝑋, 𝜏) containing 1 and Theorem 4.3.7]). Let 𝐴 be a real uniform function algebra on let Re 𝐴 separate the points of 𝑋/𝜏.Then (𝑋, 𝜏) and suppose that 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}.Then (i) a 𝜏-invariant subset 𝑆 of 𝑋 is a boundary for 𝐴 if and 𝐶ℎ (𝐴, 𝑋, 𝜏) =𝐶ℎ(𝐵, 𝑋) ,Γ(𝐴, 𝑋, 𝜏) =Γ(𝐵, 𝑋) . (7) only if it is a Choquet set for 𝐴. (ii) 𝐶ℎ(𝐴,𝑋,𝜏)is a 𝜏-invariant boundary for 𝐴. We now extend the previous theorem by omitting the (iii) The closure of 𝐶ℎ(𝐴,𝑋,𝜏) is equal to Γ(𝐴, 𝑋,,the 𝜏) uniformly closed condition for 𝐴. smallest 𝜏-invariant closed boundary for 𝐴. Theorem 16. Let 𝐴 be a real subalgebra of 𝐶(𝑋, 𝜏) containing We now extend the previous theorem by omitting the 1 and separating the points of 𝑋 and let 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}. uniformly closed condition for 𝐴. Then 𝐶ℎ(𝐴,𝑋,𝜏)=𝐶ℎ(𝐵,𝑋)and Γ(𝐴, 𝑋, 𝜏) = Γ(𝐵,. 𝑋) Journal of Function Spaces and Applications 5

Proof. Since 𝐴 separates the points of 𝑋,weconcludethat Then Lip(𝑋, 𝜏, 𝛼) and lip(𝑋, 𝜏, 𝛼) are real Banach function Re 𝐴 separates the points of 𝑋/𝜏. Therefore, 𝐶ℎ(𝐴,𝑋,𝜏) = algebras on (𝑋, 𝜏) under the norm || ⋅ ||𝛼 and we have 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 Lip (𝑋, 𝛼) = Lip (𝑋, 𝜏, 𝛼) ⊕𝑖Lip (𝑋, 𝜏, 𝛼) , (𝑋, 𝜏).Since (13) lip (𝑋, 𝛼) = lip (𝑋, 𝜏, 𝛼) ⊕𝑖lip (𝑋, 𝜏, 𝛼) . 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴},( 𝐶 𝑋) =𝐶(𝑋, 𝜏) ⊕𝑖𝐶(𝑋, 𝜏) , These algebras are called real Lipschitz algebras of complex 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝛼 max {󵄩𝑓󵄩 , 󵄩𝑔󵄩 }≤󵄩𝑓+𝑖𝑔󵄩 functions of order and were first studied in [15]. 𝑋 𝑋 𝑋 (𝑋, 1) ⊆ (𝑋, 𝛼) ⊆ (𝑋, 𝛼) 𝑥∈ (8) Since Lip lip Lip and for each 𝑋 the function 𝑓𝑥 :𝑋 → C, defined by 𝑓, 𝑔 ∈ 𝐶(𝑋, 𝜏) 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴} for all ,wehave . 𝑑(𝑥,𝑦) Hence, 𝐶ℎ(𝐴, 𝑋, 𝜏) 𝐵,= 𝐶ℎ( 𝑋) and Γ(𝐴, 𝑋, 𝜏)𝐵, =Γ( 𝑋) 𝑓 (𝑦) =1− , (14) 𝑥 𝑋 by Theorem 15. On the other hand, since 𝐵 is a complex diam subspace of 𝐶(𝑋) which contains 1 and separates the points belongs to Lip(𝑋, 1) and peaks at 𝑥,weconcludethat of 𝑋,wehave𝐶ℎ(𝐵, 𝑋) 𝐵,= 𝐶ℎ( 𝑋) and Γ(𝐵, 𝑋)𝐵, =Γ( 𝑋) by Theorem 9(iii). Thus the result follows. 𝑆0 (Lip (𝑋, 1) ,𝑋) Corollary 17. 𝐴 (𝑋, 𝜏) (15) If is a real Banach function algebra on =𝑆0 (lip (𝑋, 𝛼) ,𝑋)=𝑆0 (Lip (𝑋, 𝛼) ,𝑋) =𝑋. and 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴},then𝐶ℎ(𝐴,𝑋,𝜏)=𝐶ℎ(𝐵,𝑋)and Γ(𝐴, 𝑋, 𝜏) = Γ(𝐵, 𝑋) . If 𝐵 is the algebra Lip(𝑋, 𝛼) for 𝛼 ∈ (0, 1] or lip(𝑋, 𝛼) for 𝛼 ∈ (0, 1),thenbyTheorem 16, 𝐶ℎ(𝐵, 𝑋) = 𝐶ℎ(𝐴, 𝑋,𝜏) At the end of this section, we present two examples of and Γ(𝐴, 𝑋, 𝜏) = Γ(𝐵,,whenever 𝑋) 𝐴=Lip(𝑋, 𝜏,,for 𝛼) real subalgebras of 𝐶(𝑋, 𝜏) which are not uniformly closed 𝛼 ∈ (0, 1],or𝐴=lip(𝑋, 𝜏,,for 𝛼) 𝛼 ∈ (0, 1).Since𝐵=𝐶(𝑋) and then determine their Choquet and Shilov boundaries by and Theorem 9, Theorem 13,andTheorem 16. 𝐶ℎ (𝐵, 𝑋) =𝐶ℎ(𝐵, 𝑋) =Γ (𝐵, 𝑋) =Γ(𝐵, 𝑋) =𝑋, Example 18. Let (𝑋, 𝑑) be a compact metric space and let 0< (16) 𝛼≤1.LetLip(𝑋, 𝛼) be the complex algebra of all complex- 𝐶ℎ(𝐴,𝑋,𝜏)=Γ(𝐴,𝑋,𝜏)=𝑋 valued functions 𝑓 on 𝑋 for which we have . 󵄨 󵄨 Example 19. Let 𝑋 be the closed unit interval [0, 1] with the 󵄨𝑓 (𝑥) −𝑓(𝑦)󵄨 ∞ 𝑝 (𝑓) = { :𝑥,𝑦∈𝑋,𝑥=𝑦}̸ usual Euclidean topology. Let 𝐵=𝐷 (𝑋) be the complex 𝛼 sup 𝑑𝛼 (𝑥, 𝑦) (9) algebra of all (continuous) complex-valued functions with derivatives of all orders on 𝑋.Itisknownthat𝐵 is not is finite. For 0<𝛼<1,wetakelip(𝑋, 𝛼) to be the set of all 𝑓∈ (𝑋, 𝛼) complete under any algebra norm. For this property see, for Lip for which example, Carpenter’s theorem [16]orSinger-Wermertheo- 󵄨 󵄨 𝐵 󵄨𝑓 (𝑥) −𝑓(𝑦)󵄨 rem [17,§18,Theorem16].Since is a self-adjoint complex 󵄨 󵄨 󳨀→ 0 𝑑(𝑥,𝑦)󳨀→0. subalgebra of 𝐶(𝑋) which contains 1 and separates the points 𝑑𝛼 (𝑥, 𝑦) as (10) of 𝑋,wehave𝐵=𝐶(𝑋)by the Stone-Weierstrass theorem 𝐶(𝑋) 𝐶ℎ(𝐶(𝑋), 𝑋)= ‖⋅‖ (𝑋, 𝛼) for complex subalgebras of .Since The Lipschitz norm 𝛼 on Lip is defined by Γ(𝐶(𝑋), 𝑋) =𝑋,byTheorem 9(iii) it follows that 𝐶ℎ(𝐵, 𝑋) = 󵄩 󵄩 󵄩 󵄩 Γ(𝐵, 𝑋) =𝑋 󵄩𝑓󵄩 = 󵄩𝑓󵄩 +𝑝 (𝑓) . . 󵄩 󵄩𝛼 󵄩 󵄩𝑋 𝛼 (11) We now define the topological involution 𝜏 on 𝑋 by 𝜏(𝑥) = 1 −𝑥 and let 𝜎 be the algebra involution induced It is known that (Lip(𝑋, 𝛼), ‖ ⋅‖𝛼) and (lip(𝑋, 𝛼), ‖𝛼 ⋅‖ ) are by 𝜏 on 𝐶(𝑋).Sinceforeveryℎ∈𝐵, 𝜎(ℎ) is a differentiable complex Banach function algebras on 𝑋.Thesealgebrasare (𝑛) 𝑛 (𝑛) 𝛼 function of all orders on 𝑋 and (𝜎(ℎ)) =(−1)𝜎(ℎ ) for called Lipschitz algebras of order and were first studied by 𝑛∈N 𝜎(𝐵) =𝐵 𝐴= Sherbert in [13, 14]. each ,itfollowsthat . If we define 𝜏 (𝑋, 𝑑) {ℎ∈𝐵:𝜎(ℎ)=ℎ},then𝐴 isarealsubalgebraof𝐶(𝑋, 𝜏) Let be a Lipschitz involution on ,thatis,aninvolu- 𝑋 tion 𝜏 on 𝑋 satisfying the Lipschitz condition 𝑑(𝜏(𝑥), 𝜏(𝑦)) ≤ containing 1, separating the points of , and moreover, the 𝐶𝑑(𝑥, 𝑦) 𝐶 𝑥, 𝑦 𝑋 algebra 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}isthecomplexificationof for a positive constant and for all in .Let 𝐴 𝐴 𝜎 be the algebra involution induced by 𝜏 on 𝐶(𝑋).Then .Thenby[15,Theorem1.2], is not complete under any 𝜎( (𝑋, 𝛼)) = (𝑋, 𝛼) 𝜎( (𝑋, 𝛼)) = (𝑋, 𝛼) algebra norm. Since 𝐶ℎ(𝐵, 𝑋) = Γ(𝐵,,weconclude 𝑋)=𝑋 Lip Lip and lip lip ;thatis, 𝐶ℎ(𝐴,𝑋,𝜏)=Γ(𝐴,𝑋,𝜏)=𝑋 these (complex) Lipschitz algebras are 𝜎-invariant. We now that by Theorem 16. 𝐴=𝐶(𝑋,𝜏) define On the other hand, by the Stone- Weierstrass theorem for real subalgebras of 𝐶(𝑋, 𝜏) [3]. Since Lip (𝑋, 𝜏, 𝛼) ={ℎ∈Lip (𝑋, 𝛼) :𝜎(ℎ) =ℎ}, 𝐶ℎ(𝐶(𝑋, 𝜏), 𝑋) = Γ(𝐶(𝑋,,weconcludethat 𝜏),𝑋)=𝑋 (12) 𝐶ℎ(𝐴,𝑋,𝜏) = Γ(𝐴,𝑋,𝜏) =𝑋 by Theorem 9(ii) and lip (𝑋, 𝜏, 𝛼) ={ℎ∈lip (𝑋, 𝛼) :𝜎(ℎ) =ℎ}. Theorem 13(i). 6 Journal of Function Spaces and Applications

−1 3. Peak Sets and Peak Points 𝜂∘Φ =𝑒(𝑥,0).SinceΦ(𝐵) =,itfollowsthat 𝐴 𝜂=𝑒𝑥 and thus 𝐵 is natural. In this section we extend some of the known results concern- Now let 𝑆 be a peak set (𝑝-set, resp.) for 𝐵.Itiseasytosee 𝑝 ing peak sets and -sets for complex Banach (uniform, resp.) that 𝑆×{0,1}is a peak set (𝑝-set, resp.) for 𝐴. function algebras to those for real Banach (uniform, resp.) Conversely, let 𝐸 be a peak set for 𝐴. Then there exists a function algebras. subset 𝑆 of 𝑋 such that 𝐸 = 𝑆 × {0, 1}.Wecaneasilyshowthat We know that every complex uniform function algebra 𝑆 is a peak set for 𝐵. 𝑋 on can be regarded as a real uniform function algebra If 𝑃 is a 𝑝-set for 𝐵,then𝑃=∩𝛼∈𝐼𝑆𝛼,where𝑆𝛼 is a peak on a compact Hausdorff space with a suitable topological set for 𝐵.Since𝑃×{0,1}=𝛼∈𝐼 ∩ (𝑆𝛼 × {0, 1}),itfollowsthat 𝜏 𝑌 involution [3]. We may take as a disjoint union of two 𝑃×{0,1}is a 𝑝-set for 𝐴. 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 Concerning the union of peak sets and 𝑝-sets for real or of 𝑋 [4, page 29]. Now a natural question arises here whether complex subspaces of 𝐶(𝑋), the following results are well- the above fact holds for Banach function algebras. The answer known. is affirmative as the following result shows. However, its proof issimilartothatofKulkarniandLimayeforuniformfunction Theorem 21 (see [4, Remark 2.2.10]). Let 𝐴 be a uniformly algebras. closed real subalgebra of 𝐶(𝑋) containing 1. If 𝑆 and 𝑇 are peak sets (𝑝-sets, resp.) for 𝐴,then𝑆∪𝑇is a peak set (𝑝-set, resp.) Theorem 20. Let (𝐵, ||| ⋅ |||) be a complex Banach function for 𝐴. algebra on a compact Hausdorff space 𝑋. Then there exists a compact Hausdorff space 𝑌,atopologicalinvolution𝜏 on 𝑌, Theorem 22 (see [18, II. Corollary 12.8]). Let 𝐵 be a complex 𝑋 {𝑆 }∞ 𝑝 and a real Banach function algebra (𝐴, ‖ ⋅ ‖) on (𝑌,𝜏) such that uniform function algebra on .If 𝑗 𝑗=1 is a sequence of -sets (𝐵, ||| ⋅ |||) ∞ , regarded as a real Banach algebra, is isometrically for 𝐵 and 𝑆=⋃𝑗=1 𝑆𝑗 is closed, then 𝑆 is a 𝑝-set for 𝐵. isomorphic to (𝐴, ‖ ⋅ ‖).Moreover, (i) the algebra 𝐴 is natural if and only if 𝐵 is natural. Note that the above two theorems do not hold for complex Banach function algebras, in general. In fact, Dales (ii) There exists a one-to-one correspondence between the has proved the following result in [12]andhasshown,by 𝑝 𝐵 set of all peak sets ( -sets, resp.) for and the set of all counterexamples, that the restriction imposed on 𝐵 and the 𝑝 𝐴 (𝑌,𝜏) peak sets ( -sets, resp.) for ,withrespectto . peak sets (𝑝-sets, resp.) are necessary. 𝑌 𝑋×{0,1} Proof. Let be the compact Hausdorff space with Theorem 23. Let 𝐵 be a natural complex Banach function 𝜏:𝑌 →𝑌 theproducttopology.Wedefinethemap by algebra on 𝑋. Then a finite union of pairwise disjoint peak sets 𝜏(𝑥, 0) = (𝑥, 1) 𝜏(𝑥, 1) = (𝑥, 0) 𝑥∈𝑋 and for all .Clearly, (𝑝-sets, resp.) for 𝐵 is a peak set (𝑝-set, resp.) for 𝐵. 𝜏 is a topological involution on 𝑌. Now, we define the map Ψ : 𝐶(𝑋) → 𝐶(𝑌,𝜏) by The following two results state the relation between the 𝑝 𝐶(𝑋, 𝜏) Ψ(𝑓)(𝑥,) 0 =𝑓(𝑥) , peak sets ( -sets, resp.) for a real subspace of and those of its complexification, which are modifications of[4, (17) Ψ(𝑓)(𝑥,) 1 = 𝑓 (𝑥) (𝑓 ∈ 𝐶 (𝑋) ,𝑥∈𝑋). Theorem 2.2.11]. Theorem 24. Let 𝐴 be a real subspace of 𝐶(𝑋, 𝜏) containing 1 Then Ψ is an isometrically isomorphism from (𝐶(𝑋), ‖𝑋 ⋅‖ ), and 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}. regarded as a real Banach algebra, onto (𝐶(𝑌), ‖𝑌 ⋅‖ ) and Ψ(1) = 1 𝐴=Ψ(𝐵) 𝐴 .Set .Itfollowsthat is a real subalgebra (i) If 𝑆 is a peak set (𝑝-set, resp.) for 𝐴,thenitisapeak 𝐶(𝑌,𝜏) 𝑌 of which contains 1 and separates the points of .We set (𝑝-set, resp.) for 𝐵.Moreover,𝜏(𝑆) is also a peak set ‖ 𝑔 ‖= |||Ψ−1(𝑔)||| 𝑔∈𝐴 ‖⋅‖ define for all .Then is a complete (𝑝-set, resp.) for 𝐴 and 𝜏(𝑆). =𝑆 𝐴 ‖ 1 ‖= 1 𝐴 algebra norm on and . Therefore, is a real Banach 𝑆 𝑝 𝐵 𝜏(𝑆) function algebra on (𝑌,𝜏).Now,wedefinethemapΦ:𝐵 → (ii) If is a peak set ( -set, resp.) for ,then is also a 𝐴 Φ=Ψ| Φ peak set (𝑝-set, resp.) for 𝐵.Incase𝜏(𝑆), =𝑆 𝑆 is, in by 𝐵.Clearly, is an isometric isomorphism from 𝑝 𝐴 (𝐵, ||| ⋅ ,|||) regarded as a real Banach algebra, onto (𝐴, ‖ ⋅ ‖). fact, a peak set ( -sets, resp.) for . Let 𝐵𝑟 denote 𝐵 as a real algebra. Then it is easy to show (iii) 𝑆0(𝐴, 𝑋)0 =𝑆 (𝐵, 𝑋) ∩ Fix(𝜏) and 𝑆(A,𝑋)=𝑆(𝐵,𝑋)∩ 𝑀 =𝑀 ∪{𝜂:𝜂∈𝑀 } 𝐴 (𝜏) that 𝐵𝑟 𝐵 𝐵 .Toprovethenaturalityof ,let Fix . 𝜑∈𝑀𝐴.Then𝜑∘Φ∈𝑀𝐵 .If𝐵 is natural, either there exists 𝑟 Theorem 25. 𝐴 𝑥∈𝑋such that 𝜑∘Φ=𝑒𝑥 or there exists 𝑦∈𝑋such that 𝜑∘ Let be a uniformly closed subalgebra of 𝐶(𝑋, 𝜏) 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴} Φ=𝑒𝑦.Wecaneasilyshowthat𝜑 is the evaluation character containing 1 and let . (𝑥, 0) (𝑦, 1) 𝐴 at or at , respectively. Therefore, is natural. 𝑆 𝑝 𝐵 𝑆∪𝜏(𝑆) 𝐴 𝜂∈𝑀 𝜂∘Φ−1 ∈ (i) If is a peak set ( -set, resp.) for ,then is a Conversely, let be natural and 𝐵.Hence peak set (𝑝-set, resp.) for 𝐴. 𝑀 𝑥∈𝑋 𝜂∘Φ−1 𝐴. Therefore, either there exists such that 𝑆 (𝐵, 𝑋) ⊆𝑇 (𝐴, 𝑋, 𝜏) 𝑆(𝐵, 𝑋) ⊆ 𝑇(𝐴, 𝑋,𝜏) is the evaluation character at some (𝑥, 0) on 𝐴 or it is the (ii) 0 0 and . evaluation character at some (𝑦, 1) on 𝐴.Since𝜂∈𝑀𝐵 and 𝐵 (iii) 𝑇0(𝐴, 𝑋, 𝜏) ∩ Fix(𝜏) ⊆0 𝑆 (𝐵, 𝑋) and 𝑇(𝐴, 𝑋, 𝜏)∩ is a complex algebra, the latter case does not occur, and hence Fix(𝜏) ⊆0 𝑆 (𝐵, 𝑋). Journal of Function Spaces and Applications 7

(iv) If 𝐴 is a real uniform function algebra on (𝑋, 𝜏),then derivatives with respect to 𝑧 and 𝑤. We define the algebra 𝑇0(𝐴, 𝑋, 𝜏)0 =𝑆 (𝐵, 𝑋) and 𝑇(𝐴, 𝑋, 𝜏) = 𝑆(𝐵,. 𝑋) norm ‖⋅‖on 𝐵 by 󵄩 󵄩 󵄩 󵄩 We now extend Theorem 23 for natural real Banach 󵄩 󵄩 󵄩 󵄩 󵄩𝜕𝑓 󵄩 󵄩 𝜕𝑓 󵄩 󵄩𝑓󵄩 = 󵄩𝑓󵄩𝑋 + 󵄩 󵄩 + 󵄩 󵄩 (𝑓 ∈ 𝐵) . (18) function algebras. 󵄩 𝜕𝑧 󵄩𝑋 󵄩𝜕𝑤󵄩𝑋 Theorem 26. (𝐴, ‖ ⋅ ‖) Let be a natural real Banach function Then (𝐵, ‖ ⋅ ‖) is a natural complex Banach function algebra (𝑋, 𝜏) algebra on . Then a finite union of pairwise disjoint peak on 𝑋 and although the sets 𝑆1 = {(1, 𝑤) : |𝑤| ≤1} and 𝑆2 = 𝑝 𝐴 𝑝 𝐴 sets ( -sets, resp.) for is a peak set ( -set, resp.) for . {(𝑧, 1) : |𝑧| ≤1} are peak sets for 𝐵,theset𝑆1 ∪𝑆2 is not 𝐵 𝑆 𝑆 apeaksetfor ,asshownbyDalesin[12,Example1].Note Proof. It is sufficient to show that if 1 and 2 are two disjoint 𝑆 ∩𝑆 𝑝 𝐴 𝑆 ∪𝑆 𝑝 that 1 2 is nonempty. peak sets ( -sets, resp.) for ,then 1 2 is a peak set ( -set, We first define the topological involution 𝜏 on 𝑋 by resp.) for 𝐴.Let𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}.Then𝐵 is a complex 𝜏(𝑧, 𝑤) =(𝑤, ) 𝜎 𝐶(𝑋) 𝐶(𝑋) |||⋅||| z .Let be the algebra involution on , subalgebra of andthereexistsanalgebranorm on induced by 𝜏. By the definition of 𝐵,forevery𝑓∈𝐵 𝐵 such that (𝐵, ||| ⋅ |||) is a natural complex Banach function and for each fixed 𝑤 in the closed unit disk D, 𝑓(𝑤, 𝑧) algebra on 𝑋 by [15, Theorem 1.2]. On the other hand, 𝑆1 and D 𝑓(𝑤, 𝑧) 𝑆2 are peak sets (𝑝-sets, resp.) for 𝐵 by Theorem 24(i). Thus has continuous derivative on and hence has also D 𝑧 𝑆1 ∪𝑆2 is a peak set (𝑝-set, resp.) for 𝐵 by Theorem 23.Since continuous derivative on as a function of .Similarly,for 𝑆1 and 𝑆2 are 𝜏-invariant, 𝑆1 ∪𝑆2 is also 𝜏-invariant. Therefore, each fixed 𝑧∈D, 𝑓(𝑤, 𝑧) has continuous derivative on D 𝑆1 ∪𝑆2 isapeakset(𝑝-set, resp.) for 𝐴 by Theorem 24(ii). and hence, as a function of 𝑤, 𝑓(𝑤, 𝑧) has also continuous derivative on D. Therefore, 𝑓(𝑤, 𝑧) has continuous partial Remark 27. By using the counter examples in [7]andconsid- derivatives on 𝑋 with respect to 𝑧 and 𝑤.Since𝑓(𝑤, 𝑧) = ering Theorem 20, we can show that all restrictions imposed 𝑓∘𝜏(𝑧,𝑤) (𝑧, 𝑤) ∈𝑋 𝜎(𝑓) = 𝑓∘𝜏∈𝐵 on 𝐴 in the above theorem are necessary. ,forall ,itfollowsthat and hence 𝜎(𝐵). =𝐵 If we define 𝐴={ℎ∈𝐵:𝜎(ℎ)=ℎ},then (𝐴, ‖ ⋅ ‖) (𝑋, 𝜏) Now, we show that part (i) of Theorem 25 holds for real is a natural real Banach function algebra on 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴} 𝐴 Banach function algebras with small modification. and is the complexification of by [15, Theorem 1.1]. Moreover, 𝐴 is not uniformly closed. Since Theorem 28. Let (𝐴, ‖ ⋅ ‖) be a real Banach function algebra 𝜏(𝑆1)=𝑆2,theset𝑆1 ∪𝜏(𝑆1) is not a peak set for 𝐵. Therefore, on (𝑋, 𝜏) and 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}be its complexification. 𝑆1 ∪𝜏(𝑆1) is not a peak set for 𝐴 by Theorem 24(i). If 𝑆 is a peak set (𝑝-set, resp.) for 𝐵 then 𝑆∪𝜏(𝑆)is a peak set 𝜏 𝑋 (𝑝-set, resp.) for 𝐴. We now define the topological involution on by 𝜏(𝑧, 𝑤) =(𝑧, 𝑤).Clearly,𝑆1 and 𝑆2 are 𝜏-invariant. Let 𝜎 be Proof. Since 𝑆 is a peak set (𝑝-set, resp.) for 𝐵,wecon- the algebra involution on 𝐶(𝑋),inducedby𝜏.Byasimilar clude that 𝜏(𝑆) is also a peak set (𝑝-set, resp.) for 𝐵 by argument as in the above case, it is easy to see that 𝜎(𝐵). =𝐵 𝐴={ℎ∈𝐵:𝜎(ℎ)=ℎ} 𝐴 Theorem 24(ii). Let 𝐷={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴} be the If we define ,then is a natural real 𝐴 𝑆 𝜏(𝑆) 𝑝 Banach function algebra on (𝑋, 𝜏) and 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴} complexification of .Clearly, and are peak sets ( - 𝑆 ∪𝑆 𝐵 sets, resp.) for 𝐷 and hence by Theorem 25(i), 𝑆∪𝜏(𝑆)is a by [15,Theorem1.1].Since 1 2 is not a peak set for ,itis not a peak set for 𝐴 by Theorem 24(i). peak set (𝑝-set, resp.) for 𝐴. Theorem 31. Let 𝐴 be a natural real Banach function algebra Theorem 29. Let (𝐵, ‖ ⋅ ‖) be a natural complex Banach on (𝑋, 𝜏) and let 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}.Then𝑆0(𝐵, 𝑋) ⊆ function algebra on 𝑋 and 𝜎 be the algebra involution on 𝐶(𝑋), 𝑇0(𝐴, 𝑋, 𝜏) and 𝑆(𝐵, 𝑋) ⊆ 𝑇(𝐴, 𝑋,𝜏). inducedbythetopologicalinvolution𝜏.Let𝜎(𝐵) =𝐵 and 𝐴={ℎ∈𝐵:𝜎(ℎ)=ℎ}.If𝑆1 and 𝑆2 are 𝜏-invariant peak Proof. Let 𝑥∈𝑆0(𝐵, 𝑋).If𝑥=𝜏(𝑥)then 𝑥0 ∈𝑆0(𝐴, 𝑋) sets (𝑝-sets, resp.) for 𝐵 such that 𝑆1 ∩𝑆2 =𝜙,then𝑆1 ∪𝑆2 is by Theorem 24(ii), and hence 𝑥∈𝑇0(𝐴,𝑋,𝜏).If𝑥 ≠ 𝜏(𝑥), apeakset(𝑝-set, resp.) for A. then {𝑥, 𝜏(𝑥)} is a peak set for 𝐵 by Theorem 24(ii) and (𝐴, ‖ ⋅ ‖) Theorem 23.Since{𝑥, 𝜏(𝑥)} is 𝜏-invariant, it follows that it is Proof. By [15,Theorem1.1], is a natural real Banach 𝐴 𝑥∈𝑇(𝐴, 𝑋, 𝜏) function algebra on (𝑋, 𝜏) and 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}.On apeaksetfor by Theorem 24(ii) and hence 0 . Therefore, the first inclusion holds. the other hand, 𝑆1 and 𝑆2 are peak sets (𝑝-sets, resp.) for 𝐴 by 𝑆 ∪𝑆 𝑝 By applying parts (i) and (iii) in Theorem 24 and Theorem 24(ii). Therefore, 1 2 is a peak set( -set, resp.) 𝑝 for 𝐴 by Theorem 26. Theorem 29 for -sets, the second inclusion holds with the same argument. The following example shows that the uniformly closed condition of 𝐴 in Theorem 25(i) and the condition 𝑆1 ∩𝑆2 =0 Concerning the existence of a 𝑝-point (a peak point, in Theorem 29 are essential. respectively) in every peak set for a complex uniform func-

2 tion algebra, the following result holds. Example 30. Let 𝑋={(𝑧,𝑤)∈C :|𝑧|≤1,|𝑤|≤1}and let 𝐵 be the complex algebra of all continuous complex-valued Theorem 32 (see [2, Corollary 2.4.6]). Let 𝐵 acomplex functions on 𝑋 which have continuous first-order partial uniform function algebra on 𝑋. Then every peak set for 𝐵 8 Journal of Function Spaces and Applications contains a 𝑝-point for 𝐵.Moreover,if𝑋 is first countable, then The following example shows that the above theorem every peak set for 𝐵 contains a peak point for 𝐵. does not hold even for real uniform algebras on compact metrizable spaces, in general. Note that the above theorem is not true for complex Banach function algebras, in general (see [11, 19, 20]). Example 37. Let 𝐵 be the complex disc algebra and let 𝐴 be The following example shows that the above theorem does therealdiscalgebra,consideredinExample 33.Itisknown not hold even for real uniform function algebras, in general. 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 Example 33. We denote by D, D,andT the open unit disk 𝑆 (𝐴, D) {𝑧 ∈ C :|𝑧|<1} {𝑧 ∈ C :|𝑧|≤1} in Example 33.Hence,theclosureof 0 is not equal to ,theclosedunitdisc , Γ(𝐴, 𝑋,. 𝜏) and the unit circle {𝑧 ∈ C :|𝑧|=1},respectively.Let𝐵= 𝐴(D), the complex disc algebra. It is known that (𝐵, ‖ ⋅‖D) is Theorem 38 (see [10]or[4, Theorem 4.2.4]). Let 𝐴 be a a complex uniform algebra on D and 𝑆0(𝐵; D)=T. We define uniformlyclosedrealsubalgebraof𝐶(𝑋, 𝜏) containing 1. Then the topological involution 𝜏 on D by 𝜏(𝑧) = 𝑧.Let𝐴={ℎ∈ 𝐵:𝜎(ℎ)=ℎ},where𝜎 is the algebra involution induced by 𝐶ℎ (𝐴, 𝑋, 𝜏) =𝑇(𝐴, 𝑋, 𝜏) . (20) 𝜏 on 𝐶(D).Since𝜎(𝑓)(𝑧) = 𝑓(𝑧) is continuous on D and it is 𝐴 analytic on D for each 𝑓∈𝐵,itfollowsthat𝜎(𝐵).Infact, =𝐵 Remark 39. Let bearealuniformfunctionalgebraon (𝑋, 𝜏) 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴} 𝐴 is the real disc algebra and 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}is the and .Thenwehave 𝐶ℎ(𝐴,𝑋,𝜏) = 𝑇(𝐴,𝑋,𝜏) 𝐶ℎ(𝐴,𝑋,𝜏) = complexification of 𝐴.Moreover, by Theorem 38, 𝐶ℎ(𝐵, 𝑋) by Theorem 15,and𝐶ℎ(𝐵, 𝑋) = 𝑆(𝐵, 𝑋) by Theorem 35. Therefore, 𝑇(𝐴, 𝑋, 𝜏) = 𝑆(𝐵,,whichhas 𝑋) 𝑆0 (𝐴, D)=𝑆0 (𝐵, D)∩Fix (𝜏) = T ∩ {−1, 1} = {−1, 1} , already been obtained in Theorem 25(iv). (19) Theorem 40. Let 𝐴 be a uniformly closed real subalgebra of 𝐶(𝑋, 𝜏) containing 1 and let Re 𝐴 separate the points of 𝑋/𝜏. by Theorem 24(iii). Hence for each 𝑧∈T with 𝑧 =𝑧̸ ,theset Then {𝑧, 𝑧} is a peak set for 𝐴 which does not contain any peak point for 𝐴. 𝑇 (𝐴, 𝑋, 𝜏) =Γ(𝐴, 𝑋, 𝜏) . (21) The following is a modification of Theorem 32 for real uniform function algebras. In particular, if 𝑋 is a first countable space, then Theorem 34. Let 𝐴 be a real uniform function algebra on 𝑇0 (𝐴, 𝑋, 𝜏) =Γ(𝐴, 𝑋, 𝜏) . (22) (𝑋, 𝜏). Then every peak set for 𝐴 contains a 𝜏-𝑝-point for 𝐴. 𝑋 𝐴 Moreover, if is first countable, every peak set for contains Proof. By Theorem 38 we have 𝑇(𝐴, 𝑋, 𝜏) = 𝐶ℎ(𝐴, 𝑋,𝜏) a 𝜏-peak point for 𝐴. and since 𝐶ℎ(𝐴,𝑋,𝜏) = Γ(𝐴,𝑋,𝜏) by Theorem 13(iii), we Proof. Let 𝑆 be a peak set for 𝐴.Ifwetake𝐵={𝑓+𝑖𝑔: conclude that 𝑓, 𝑔 ∈ 𝐴} then 𝐵 is a complex uniform function algebra 𝑇 (𝐴, 𝑋, 𝜏) =Γ(𝐴, 𝑋, 𝜏) . on 𝑋 by [4, Theorem 1.3.20] and 𝑆 isalsoapeaksetfor𝐵. (23) Hence 𝑆∩𝑆(𝐵,𝑋)=0̸ and when 𝑋 is first countable, 𝑆∩ In the case that 𝑋 is first countable, every finite subset 𝑆0(𝐵, 𝑋) =0̸ ,byTheorem 32.Since𝑆(𝐵, 𝑋) = 𝑇(𝐴, 𝑋,𝜏) and of 𝑋 is a 𝐺𝛿-set. Hence 𝑇(𝐴, 𝑋, 𝜏)0 ⊆𝑇 (𝐴, 𝑋, 𝜏) and so 𝑆0(𝐵, 𝑋)0 =𝑇 (𝐴, 𝑋, 𝜏),wehave𝑆∩𝑇(𝐴,𝑋,𝜏)=0̸ ,andif𝑋 is 𝑇(𝐴, 𝑋, 𝜏) =𝑇 (𝐴, 𝑋, 𝜏) 𝑇 (𝐴, 𝑋, 𝜏) = Γ(𝐴, 𝑋,𝜏) first countable, then 𝑆∩𝑇0(𝐴,𝑋,𝜏)=0̸ . 0 . Therefore, 0 by the previous argument.

4. On the Density of 𝜏-Peak Points Theorem 41. Let 𝑋 be a compact metrizable space and let 𝜏 in the Shilov Boundary be a topological involution on 𝑋.If(𝐴, ‖ ⋅ ‖) is a natural real Banach function algebra on (𝑋, 𝜏),then We first state some known results on the density of the set of peak points in the Shilov boundary for complex Banach 𝑇0 (𝐴, 𝑋, 𝜏) =Γ(𝐴, 𝑋, 𝜏) . (24) function algebras. Proof. Let 𝐵={𝑓+𝑖𝑔:𝑓,𝑔∈𝐴}.Thenaturalityof𝐴 Theorem 35 𝐵 (see [2, Theorem 2.3.4]). Let be a complex implies that 𝑆0(𝐵, 𝑋)0 ⊆𝑇 (𝐴,𝑋,𝜏) by Theorem 31.Onthe 𝑋 𝑆(𝐵, 𝑋) = 𝐶ℎ(𝐵, 𝑋) uniform function algebra on .Then . other hand, there exists an algebra norm ||| ⋅ ||| on 𝐵 such 𝑋 𝑆 (𝐵, 𝑋) = 𝐶ℎ(𝐵, 𝑋) Moreover, if is first countable, then 0 and that (𝐵, ||| ⋅ |||) is a complex Banach function algebra on 𝑋 by Γ(𝐵, 𝑋) 𝑆 (𝐵, 𝑋) so is the closure of 0 . [15,Theorem1.2].Since𝑋 is a metrizable compact space, we 𝑆 (𝐵, 𝑋) = Γ(𝐵, 𝑋) Theorem 36 𝐵 have 0 by Theorem 36. On the other hand, (see [11, Theorem 2.3] and [21]). Let be a Γ(𝐴, 𝑋, 𝜏) = Γ(𝐵, 𝑋) 𝑆 (𝐵, 𝑋) = Banach function algebra on a compact metrizable space 𝑋. by Theorem 16. Therefore, 0 𝑇 (𝐴, 𝑋, 𝜏) = Γ(𝐴, 𝑋,𝜏) Then Γ(𝐵, 𝑋) is the closure of 𝑆0(𝐵, 𝑋). 0 . Journal of Function Spaces and Applications 9

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