Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras

and Applied Analysis 3 Corollary 1.3. Let X be a compact Hausdorff space, and let A be a natural Banach function algebra on X. Then every unital endomorphism T of A is induced by a unique continuous self-map φ of X. In particular, if X is a compact plane set and A contains the coordinate function Z, then φ TZ and so φ ∈ A. Definition 1.4. Let X be a compact plane set which is connected by rectifiable arcs, and let δ z,w be the geodesic metric onX, the infimum of the length of the arcs joining z andw.X is called uniformly regular if there exists a constantC such that, for all z,w ∈ X, δ z,w ≤ C|z−w|. The following lemma occurs in 1 but it is important and we will be using it in the sequel. Lemma 1.5 see 1, Lemma 1.5 . Let H and K be two compact plane sets with H ⊆ int K . Then there exists a finite union of uniformly regular sets in int K containing H, namely Y , and then a positive constant C such that for every analytic complex-valued function f on int K and any z,w ∈ H, ∣ ∣f z − f w ∣∣ ≤ C|z −w|(∥∥f∥∥Y ∥ ∥f ′ ∥ ∥ Y ) . 1.3 Let X be a compact plane set. We denote by A X the algebra of all continuous complex-valued functions on X which are analytic on int X , the interior of X, and call it the analytic uniform algebra on X. It is known that A X is a natural uniform algebra on X. Let X and K be compact plane sets such that K ⊆ X. We define A X,K {f ∈ C X : f |K ∈ A K }. Clearly, A X,K A X if K X, and A X,K C X if int K is empty. We know that A X,K is a natural uniform algebra on X see 2 and call it the extended analytic uniform algebra on X with respect to K. Let X, d be a compact metric space. For α ∈ 0, 1 , we denote by Lip X, α the algebra of all complex-valued functions f for which pα,X f sup{|f z − f w |/dα z,w : z,w ∈ X, z/ w} < ∞. For f ∈ Lip X, α , we define the α-Lipschitz norm f by ‖f‖Lip X,α ‖f‖X pα,X f . Then Lip X, α , ‖ · ‖Lip X,α is a unital commutative Banach algebra. For α ∈ 0, 1 , we denote by lip X, α the algebra of all complex-valued functions f on X for which |f z − f w |/dα z,w → 0 as d z,w → 0. Then lip X, α is a unital closed subalgebra of Lip X, α . These algebras are called Lipschitz algebras of order α and were first studied by Sherbert in 3, 4 . We know that the Lipschitz algebras Lip X, α and lip X, α are natural Banach function algebras on X. Let X, d be a compact metric space, and letK be a compact subset ofX. For α ∈ 0, 1 , we denote by Lip X,K, α the algebra of all complex-valued functions f on X for which pα,K f sup{|f z − f w |/dα z,w : z,w ∈ K, z/ w} < ∞. In fact, Lip X,K, α {f ∈ C X : f |K ∈ Lip K,α }. For f ∈ Lip X,K, α , we define ‖f‖Lip X,K,α ‖f‖X pα,K f . Then Lip X,K, α under the algebra norm ‖ · ‖Lip X,K,α is a unital commutative Banach algebra. Moreover, Lip X, α is a subalgebra of Lip X,K, α ; Lip X,K, α Lip X, α if X \ K is finite, and Lip X,K, α C X if K is finite. For α ∈ 0, 1 , we denote by lip X,K, α the algebra of all complex-valued functions f on X for which |f z − f w |/dα z,w → 0 as d z,w → 0 with z,w ∈ K. In fact, lip X,K, α {f ∈ C X : f |K ∈ lip K,α }. Clearly, lip X,K, α is a closed unital subalgebra of Lip X,K, α . Moreover, lip X, α is a subalgebra of lip X,K, α ; lip X,K, α lip X, α if X \K is finite, and lip X,K, α C X if K is finite. The Banach algebras Lip X,K, α and lip X,K, α are Banach function algebras on X and were first introduced by Honary and Moradi in 5 . 4 Abstract and Applied Analysis Let X be a compact plane set. We define LipA X, α Lip X, α ∩ A X for α ∈ 0, 1 and lipA X, α Lip X, α ∩ A X for α ∈ 0, 1 . These algebras are called analytic Lipschitz algebras. We know that analytic Lipschitz algebras LipA X, α and lipA X, α under the norm ‖ · ‖Lip X,α are natural Banach function algebras on X see 6 . Let X and K be compact plane sets with K ⊆ X. We define LipA X,K, α Lip X,K, α ∩ A X,K for α ∈ 0, 1 and lipA X,K, α lip X,K, α ∩ A X,K for α ∈ 0, 1 . Then LipA X,K, α and lip X,K, α are closed unital subalgebras of Lip X,K, α and lip X,K, α under the norm ‖ · ‖Lip X,K,α , respectively. Moreover, LipA X,K, α LipA X, α lipA X,K, α LipA X, α if K X, and LipA X,K, α Lip X,K, α lipA X,K, α lip X,K, α if int K is empty. The algebras LipA X,K, α and lip X,K, α are called extended analytic Lipschitz algebras and were first studied by Honary and Moradi in 5 . They showed that the extended analytic Lipschitz algebras LipA X,K, α and lipA X,K, α under the norm ‖ · ‖Lip X,K,α are natural Banach function algebras on X 5, Theorem 2.4 . Behrouzi and Mahyar in 1 studied endomorphisms of some uniform subalgebras of A X and some Banach function subalgebras of LipA X, α and investigated some necessary and sufficient conditions for these endomorphisms to be compact, whereX is a compact plane set and α ∈ 0, 1 . In Section 2, we study unital homomorphisms from natural Banach function subalgebras of LipA X1, K1, α1 to natural Banach function subalgebras of LipA X2, K2, α2 and investigate necessary and sufficient conditions for which these homomorphisms are compact. In Section 3, we determine the spectrum of unital compact endomorphisms of LipA X,K, α . 2. Unital Compact Homomorphisms We first give a sufficient condition for which a continuous map φ : X2 → X1 induces a unital homomorphism T from a subalgebra B1 of A X1, K1 into a subalgebra B2 of A X2, K2 . Proposition 2.1. Let Xj and Kj be compact plane sets with int Kj / ∅ and Kj ⊆ Xj , and let Bj be a subalgebra of A Xj,Kj which is a natural Banach function algebra on Xj under an algebra norm ‖ · ‖j , where j ∈ {1, 2}. If φ ∈ B2 with φ X2 ⊆ int K1 , then φ induces a unital homomorphism T : B1 → B2. Moreover, if Z ∈ B1, then φ TZ. Proof. The naturality of Banach function algebra B2 on X2 implies that σB2 h h X2 , where σA h is the spectrum of h ∈ A in the Banach algebra A. Let f ∈ B1. Since φ ∈ B2, φ X2 ⊆ int K1 , and f is analytic on int K1 , we conclude that f is analytic on an open neighborhood of σB2 φ . By using the Functional Calculus Theorem 2, Theorem 5.1 in Chapter I , there exists g ∈ B2 such that ĝ f ◦ φ̂ on M B2 . It follows that g z ez ( g ) ĝ ez f ( φ̂ ez ) f ( ez ( φ )) f ( φ z ) ( f ◦ φ) z , 2.1 for all z ∈ X2 and so g f ◦ φ. Therefore, f ◦ φ ∈ B2. This implies that the map T : B1 → B2 defined by Tf f ◦φ is a unital homomorphism from B1 into B2, which is induced by φ. Now let Z ∈ B1. Then φ TZ by Proposition 1.2. Abstract and Applied Analysis 5 Corollary 2.2. Let X andK be compact plane sets with int K / ∅ andK ⊆ X. Let B be a subalgebra of A X,K which is a natural Banach function algebra on X under an algebra norm ‖ · ‖B. If φ ∈ B with φ X ⊆ int K , then φ induces a unital endomorphism T of B. Moreover, ifZ ∈ B, then φ TZ. Proposition 2.3. Suppose that αj ∈ 0, 1 , zj ∈ C, 0 < rj < Rj , Gj D zj , Rj , Ωj D zj , rj , Xj Gj , and Kj Ωj , where j ∈ {1, 2}. Then for each ρ ∈ r1, R1 there exists a continuous map φρ : X2 → X1 with φρ X2 D z1, ρ such that φρ ∈ LipA X2, K2, α2 and φρ does not induce any homomorphism from LipA X1,K1, α1 to LipA X2, K2, α2 . Proof. Let ρ ∈ r1, R1 . We define the map φρ : X2 → X1 byand Applied Analysis 5 Corollary 2.2. Let X andK be compact plane sets with int K / ∅ andK ⊆ X. Let B be a subalgebra of A X,K which is a natural Banach function algebra on X under an algebra norm ‖ · ‖B. If φ ∈ B with φ X ⊆ int K , then φ induces a unital endomorphism T of B. Moreover, ifZ ∈ B, then φ TZ. Proposition 2.3. Suppose that αj ∈ 0, 1 , zj ∈ C, 0 < rj < Rj , Gj D zj , Rj , Ωj D zj , rj , Xj Gj , and Kj Ωj , where j ∈ {1, 2}. Then for each ρ ∈ r1, R1 there exists a continuous map φρ : X2 → X1 with φρ X2 D z1, ρ such that φρ ∈ LipA X2, K2, α2 and φρ does not induce any homomorphism from LipA X1,K1, α1 to LipA X2, K2, α2 . Proof. Let ρ ∈ r1, R1 . We define the map φρ : X2 → X1 by


Introduction and Preliminaries
Let A and B be unital commutative semisimple Banach algebras with maximal ideal spaces M A and M B .A homomorphism T : If T is a unital homomorphism from A into B, then T is continuous and there exists a normcontinuous map ϕ : M B → M A such that Tf f • ϕ for all f ∈ A, where g is the Gelfand transform g.In fact, ϕ is equal the adjoint of T * : B * → A * restricted to M B .Note that T * is a weak * -weak * continuous map from B * into A * .Thus ϕ is a continuous map from M B with the Gelfand topology into M A with the Gelfand topology.
Let A be a unital commutative semisimple Banach algebra, and let T be an endomorphism of A, a homomorphism from A into A. We denote the spectrum of T by σ T and define σ T {λ ∈ C : λI − T is not invertible}. 1.1 For a compact Hausdorff space X, we denote by C X the Banach algebra of all continuous complex-valued functions on X. Definition 1.1.Let X be a compact Hausdorff space.A Banach function algebra on X is a subalgebra A of C X which contains 1 X , the constant function 1 on X, separates the points of X, and is a unital Banach algebra with an algebra norm • .If the norm of a Banach function algebra on X is • X , the uniform norm on X, it is called a uniform algebra on X.
Let A and B be Banach function algebras on X and Y , respectively.If ϕ : Y → X is a continuous mapping such that f • ϕ ∈ B for all f ∈ A and if T : A → B is defined by Tf f • ϕ, then T is a unital homomorphism, which is called the induced homomorphism from A into B by ϕ.In particular, if Y X and B A, then T is called the induced endomorphism of A by the self-map ϕ of X.
Let A be a Banach function algebra on a compact Hausdorff space X.For x ∈ X, the map e x : A → C, defined by e x f f x , is an element of M A and is called the evaluation homomorphism on A at x.This fact implies that A is semisimple and f X ≤ f M A for all f ∈ A. Note that the map x → e x : X → M A is a continuous one-to-one mapping.If this map is onto, we say that A is natural.Proposition 1.2.Let X and Y be compact Hausdorff spaces, and let A and B be natural Banach function algebras on X and Y , respectively.Then every unital homomorphism T : A → B is induced by a unique continuous map ϕ : Y → X.In particular, if X is a compact plane set and the coordinate function Z belongs to A, then ϕ TZ and so ϕ ∈ B.
Proof.Let T : A → B be a unital homomorphism.Since A and B are unital commutative semisimple Banach algebras, there exists a continuous map ψ : M B → M A such that Tf f • ψ for all f ∈ A. The naturality of the Banach function algebra A on X implies that the map J A : X → M A , defined by J A x e x , is a homeomorphism and so for all y ∈ Y , we have Tf f • ϕ.Therefore, T is induced by ϕ.Now, let X be a compact plane set, and let Z ∈ A. Then ϕ Z • ϕ TZ, and so ϕ ∈ B. Corollary 1.3.Let X be a compact Hausdorff space, and let A be a natural Banach function algebra on X.Then every unital endomorphism T of A is induced by a unique continuous self-map ϕ of X.In particular, if X is a compact plane set and A contains the coordinate function Z, then ϕ TZ and so ϕ ∈ A. Definition 1.4.Let X be a compact plane set which is connected by rectifiable arcs, and let δ z, w be the geodesic metric on X, the infimum of the length of the arcs joining z and w.X is called uniformly regular if there exists a constant C such that, for all z, w ∈ X, δ z, w ≤ C|z−w|.
The following lemma occurs in 1 but it is important and we will be using it in the sequel.
Lemma 1.5 see 1, Lemma 1.5 .Let H and K be two compact plane sets with H ⊆ int K .Then there exists a finite union of uniformly regular sets in int K containing H, namely Y , and then a positive constant C such that for every analytic complex-valued function f on int K and any z, w ∈ H, Let X be a compact plane set.We denote by A X the algebra of all continuous complex-valued functions on X which are analytic on int X , the interior of X, and call it the analytic uniform algebra on X.It is known that A X is a natural uniform algebra on X.
Let X and K be compact plane sets such that K ⊆ X.We define A X, K {f ∈ C X : We know that A X, K is a natural uniform algebra on X see 2 and call it the extended analytic uniform algebra on X with respect to K.
Let X, d be a compact metric space.For α ∈ 0, 1 , we denote by Lip X, α the algebra of all complex-valued functions f for which p α,X f sup{|f z − f w |/d α z, w : z, w ∈ X, z / w} < ∞.For f ∈ Lip X, α , we define the α-Lipschitz norm f by f Lip X,α f X p α,X f .Then Lip X, α , • Lip X,α is a unital commutative Banach algebra.For α ∈ 0, 1 , we denote by lip X, α the algebra of all complex-valued functions f on X for which |f z − f w |/d α z, w → 0 as d z, w → 0. Then lip X, α is a unital closed subalgebra of Lip X, α .These algebras are called Lipschitz algebras of order α and were first studied by Sherbert in 3, 4 .We know that the Lipschitz algebras Lip X, α and lip X, α are natural Banach function algebras on X.Let X, d be a compact metric space, and let K be a compact subset of X.For α ∈ 0, 1 , we denote by Lip X, K, α the algebra of all complex-valued functions f on X for which For α ∈ 0, 1 , we denote by lip X, K, α the algebra of all complex-valued functions f on X for which |f z − f w |/d α z, w → 0 as d z, w → 0 with z, w ∈ K.In fact, lip X, K, α {f ∈ C X : f| K ∈ lip K, α }.Clearly, lip X, K, α is a closed unital subalgebra of Lip X, K, α .Moreover, lip X, α is a subalgebra of lip X, K, α ; lip X, K, α lip X, α if X \ K is finite, and lip X, K, α C X if K is finite.The Banach algebras Lip X, K, α and lip X, K, α are Banach function algebras on X and were first introduced by Honary and Moradi in 5 .
Let X be a compact plane set.We define Lip A X, α Lip X, α ∩ A X for α ∈ 0, 1 and lip A X, α Lip X, α ∩ A X for α ∈ 0, 1 .These algebras are called analytic Lipschitz algebras.We know that analytic Lipschitz algebras Lip A X, α and lip A X, α under the norm • Lip X,α are natural Banach function algebras on X see 6 .
Let X and K be compact plane sets with K ⊆ X.We define Lip A X, K, α Lip X, K, α ∩ A X, K for α ∈ 0, 1 and lip A X, K, α lip X, K, α ∩ A X, K for α ∈ 0, 1 .Then Lip A X, K, α and lip X, K, α are closed unital subalgebras of Lip X, K, α and lip X, K, α under the norm The algebras Lip A X, K, α and lip X, K, α are called extended analytic Lipschitz algebras and were first studied by Honary and Moradi in 5 .They showed that the extended analytic Lipschitz algebras Lip A X, K, α and lip A X, K, α under the norm • Lip X,K,α are natural Banach function algebras on X 5, Theorem 2.4 .
Behrouzi and Mahyar in 1 studied endomorphisms of some uniform subalgebras of A X and some Banach function subalgebras of Lip A X, α and investigated some necessary and sufficient conditions for these endomorphisms to be compact, where X is a compact plane set and α ∈ 0, 1 .
In Section 2, we study unital homomorphisms from natural Banach function subalgebras of Lip A X 1 , K 1 , α 1 to natural Banach function subalgebras of Lip A X 2 , K 2 , α 2 and investigate necessary and sufficient conditions for which these homomorphisms are compact.In Section 3, we determine the spectrum of unital compact endomorphisms of Lip A X, K, α .

Unital Compact Homomorphisms
We first give a sufficient condition for which a continuous map ϕ : Proposition 2.1.Let X j and K j be compact plane sets with int K j / ∅ and K j ⊆ X j , and let B j be a subalgebra of A X j , K j which is a natural Banach function algebra on X j under an algebra norm and f is analytic on int K 1 , we conclude that f is analytic on an open neighborhood of σ B 2 ϕ .By using the Functional Calculus Theorem 2, Theorem 5.1 in Chapter I , there This implies that the map T : Corollary 2.2.Let X and K be compact plane sets with int K / ∅ and K ⊆ X.Let B be a subalgebra of A X, K which is a natural Banach function algebra on X under an algebra norm Proposition 2.3.Suppose that α j ∈ 0, 1 , z j ∈ C, 0 < r j < R j , G j D z j , R j , Ω j D z j , r j , X j G j , and K j Ω j , where j ∈ {1, 2}.Then for each ρ ∈ r 1 , R 1 there exists a continuous map

K, α and ϕ ρ does not induce any endomorphism of Lip
We now give a sufficient condition for a unital homomorphism from a subalgebra B 1 of Lip A X 1 , K 1 , α 1 into a subalgebra B 2 of Lip A X 2 , K 2 , α 2 to be compact.Theorem 2.5.Suppose that α j ∈ 0, 1 , X j and K j are compact plane sets with int K j / ∅ and K j ⊆ X j , and B j is a subalgebra of Lip A X j , K j , α j which is a natural Banach function algebra on X j under the norm • Lip X j ,K j ,α j , where j ∈ {1, 2}.Let ϕ : and so it is compact.
Let ϕ : X 2 → X 1 be a nonconstant mapping with ϕ ∈ B 2 and ϕ X 2 ⊆ Ω 1 .Then the map T : By Montel's theorem, the sequences {f n j } ∞ j 1 and {f n j } ∞ j 1 are uniformly convergent on the compact subsets of int K 1 .Since ϕ X 2 and K 1 are compact sets in the complex plane and ϕ X 2 ⊆ int K 1 , by using Lemma 1.5, we deduce that there exists a finite union of uniformly regular sets in int K 1 containing ϕ X 2 , namely Y , and then a positive constant C such that for every analytic complex-valued function f on int K 1 and any z, w ∈ ϕ X 2 Therefore, there exists a positive constant C such that for all j ∈ N and any z, w ∈ X 2 .Let j, k ∈ N.Then, for all z, w ∈ K 2 with ϕ z / ϕ w , we have

2.7
The above inequality is certainly true for all z, w ∈ K 2 with z / w and ϕ z ϕ w .Therefore, and so Since Y is a compact subset of int K 1 , we deduce that the sequences {f n j } ∞ j 1 and {f n j } ∞ j 1 are convergent uniformly on Y .Therefore, Hence, T is compact.
Let B be a subalgebra of Lip A X, K, α which is a natural Banach function algebra on X with the norm • Lip X 2 ,K 2 ,α 2 , and let ϕ be a self-map of X.If ϕ is constant or ϕ ∈ B with ϕ X ⊆ int K , then ϕ induces a unital compact endomorphism of B.

Definition 2.7.
a A sector in D z 0 , r at a point ω ∈ ∂D z 0 , r is the region between two straight lines in D z 0 , r that meet at ω and are symmetric about the radius to ω.
b If f is a complex-valued function on D z 0 , r and ω ∈ ∂D z 0 , r , then ∠lim z → ω f z L means that f z → L as z → ω through any sector at ω.When this happens, we say that L is angular or non-tangential limit of f at ω. c An analytic map ϕ : D z 0 , r → D ρ has an angular derivation at a point ω ∈ ∂D r z 0 , r if for some η ∈ ∂D ρ exists finitely .We call the limit the angular derivative of ϕ at ω and denote it by ∠ϕ ω .
The boundary point η in ii and iii is the same, and δ > 0 in i .Also the limit of the difference quotients in ii coincides with the limit of the derivative in iii , and both are equal to ωηδ.
Note that the existence of the angular derivative ϕ at ω ∈ ∂D z 0 , r , according to Theorem 2.9, is equivalent to lim inf z → ω ϕ D z 0 ,r − |ϕ z | / r − |z − z 0 | < ∞.In this case the angular derivative of ϕ at ω is nonzero.Proposition 2.10.Let X be a compact plane set, and let D z 0 , r ⊆ X and K D z 0 , r .Suppose that c ∈ ∂D z 0 , r and ϕ ∈ Lip A X, K, 1 is a nonconstant function such that |ϕ c | ϕ D z 0 ,r .Then the angular derivative of ϕ at c exists and is nonzero.Let X be a compact plane set.The algebra R X consists of all functions in C X which can be approximated by rational functions with poles off X.It is known that R X is a natural uniform algebra on X.
Example 2.13.Let X be a compact plane set such that C \ X is strongly accessible from the interior.If R X ⊆ B ⊆ C X , then X has a peak boundary with respect to B.
Proof.Let z 0 ∈ C \ X.Since C \ X is strongly accessible from the interior, for each c ∈ ∂ C \ X , there exists a δ > 0 such that |c −z 0 | δ and D z 0 , δ ⊆ int C\X .Now, we define the function Theorem 2.14.Let X 1 be a compact plane set such that G 1 int X 1 is connected, and G 1 X 1 .Suppose that X 1 has peak boundary with respect to Lip A X 1 , 1 .Let Ω 1 ⊆ G 1 be a bounded connected open set in the complex plane, and let K 1 Ω 1 .Let Ω 2 be a bounded connected open set in the complex plane, and let K 2 Ω 2 such that K 2 is strongly accessible from the interior.Suppose that X 2 is a compact plane set such that we conclude that T is induced by ϕ TZ and so ϕ ∈ Lip A X 2 , K 2 , 1 by Proposition 1.2.Suppose that ϕ is nonconstant on Ω 2 .Since ϕ is analytic on Ω 2 , we deduce that ϕ Ω 2 is an open subset of X 1 and so ϕ Ω 2 ⊆ G 1 .We now show that ϕ K 2 ⊆ G 1 .Suppose that ϕ K 2 / ⊆G 1 .Then there exists c ∈ ∂K 2 such that ϕ c ∈ ∂X 1 .Since X 1 has peak boundary with respect to Lip A X 1 , 1 , there exists a nonconstant function h ∈ Lip A X 1 , 1 such that h X 1 h ϕ c 1. We now define the sequence {f n } ∞ n 1 of complex-valued functions on X 1 by

2.15
Thus 16 by 2.14 and 2.15 .This implies that {f n } ∞ n 1 is a bounded sequence in Lip A X 1 , K 1 , 1 .The compactness of homomorphism T implies that there exists a subsequence {f On the other hand, we have Tf n j X 2 ≤ 1/n j for all j ∈ N by 2.14 .Hence, Tf n j X 2 0.

2.19
By 2.18 and 2.19 , g 0. Therefore, by 2.17 we have lim

2.20
This implies that lim Assume that ε > 0. By 2.21 , there exists a natural number N such that for each j ∈ N with In particular,

2.23
This implies that Since c ∈ ∂K 2 and K 2 is strongly accessible from the interior, there exists an open disc D D z 0 , r such that c ∈ ∂D and D \ {c} ⊆ int K 2 .Since ϕ is analytic on int D ⊆ int K 2 and h is analytic on ϕ D ⊆ int X 1 , we deduce that h • ϕ is analytic on int D .On the other hand, we can easily show that

2.28
Abstract and Applied Analysis 11 Hence, by 2.28 we have Since ε is assumed to be a positive number, we conclude that ∠ h • ϕ c 0, contradicting to ∠ h • ϕ c / 0. Hence, our claim is justified.Since ϕ is nonconstant on K 2 , ϕ is a nonconstant analytic function on connected open set D. This implies that ϕ D is a connected open set in the complex plane.This implies that h is constant on connected open set G 1 .The continuity of h on X 1 G 1 follows that h is constant on G 1 X 1 .This contradicting to h is nonconstant on X 1 .Therefore, ϕ K 2 ⊆ G 1 .
Corollary 2.15.Let X be a compact plane set such that G int X is connected and G X. Let Ω ⊆ G be a bounded connected open set in the complex plane, and let K Ω. Suppose that K is strongly accessible from the interior and X has peak boundary with respect to Lip A X,

2.31
We now define the function f c : X → C by

2.32
It is easily seen that f c ∈ Lip A X, K, 1 and f c is not analytic at c. Definition 2.17.Let X and K be compact plane sets such that K ⊆ X.We say that K has peak K-boundary with respect to B ⊆ A X, K if for each c ∈ ∂K there is a function h ∈ B such that h is nonconstant on K and h X h c 1. We now assume that 0 < r < 1.For each c ∈ ∂K, set z 0 1 r c/r.Then z 0 ∈ C \ D.

2.33
It is easily seen that h ∈ Lip A D, K, Let Ω 2 be a bounded connected open set in the complex plane, and let K 2 Ω 2 such that K 2 is strongly accessible from the interior.Suppose that X 2 is a compact plane set such that 1 is a unital compact homomorphism and TZ is one-to-one on Ω 2 , then T is induced by a continuous mapping ϕ : X 2 → X 1 such that ϕ TZ and ϕ K 2 ⊆ Ω 1 int K 1 .
Proof.Since Lip A X 1 , K 1 , 1 and Lip A X 2 , K 2 , 1 are, respectively, natural Banach function algebras on X 1 and X 2 , T : Since ϕ is a one-to-one analytic mapping on Ω 2 , we conclude that ϕ Ω 2 is an open set in the complex plane.This follows that ϕ Then there exists λ ∈ Ω 2 such that ϕ λ ∈ G 1 \ K 1 .By Lemma 2.16, there exists a function f ∈ Lip A X 1 , K 1 , 1 such that f is not analytic at ϕ λ .But f • ϕ Tf ∈ Lip A X 2 , K 2 , 1 , so that f • ϕ is analytic on Ω 2 .Since f is continuous on ϕ Ω 2 and ϕ is a one-to-one analytic function on Ω 2 , we conclude that f is analytic on ϕ Ω 2 by Lemma 2.19.This contradicts to the fact f is not analytic at ϕ λ ∈ ϕ Ω 2 .Therefore, ϕ Ω 2 ⊆ K 1 so ϕ Ω 2 ⊆ int K 1 Ω 1 since ϕ Ω 2 is an open set in the complex plane.Since ϕ is continuous on K 2 , ϕ Ω 2 ⊆ Ω 1 , K 2 Ω 2 , and K 1 Ω 1 , we can easily show that ϕ K 2 ⊆ K 1 .We now show that ϕ K 2 ⊆ Ω 1 .Suppose that ϕ K 2 / ⊆ Ω 1 .Then there exists c ∈ ∂K 2 such that ϕ c ∈ ∂K 1 .Since K 1 has peak K 1 -boundary with respect to Lip A X 1 , K 1 , 1 , there exists a function h ∈ Lip A X 1 , K 1 , 1 such that h is not constant on K 1 and h X 1 h ϕ c 1.

2.34
We now claim that ϕ z 0 ∈ σ T .If ϕ z 0 / ∈ σ T , then there exists a nonzero linear operator S : B → B such that T − ϕ z 0 I S I. this is a contradiction.Hence, our claim is justified.We now show that ϕ z 0 n ∈ σ T for all n ∈ N. If ϕ z 0 0 or |ϕ z 0 | 1, the proof is complete.Suppose that ϕ z 0 / 0 and |ϕ z 0 | / 1.If ϕ z 0 j / ∈ σ T for some j ∈ N with j > 1, then there exists a nonzero linear operator S j : B → B such that T − ϕ z 0 j I S j I.

3.5
Since Z − z 0 1 X j ∈ B, h j S j Z − z 0 1 X j ∈ B and so h j • ϕ − ϕ z 0 j h j Z − z 0 1 X j , 3.6 by 3.5 .By j − 1 times differentiation at z 0 , we have and by j times differentiation at z 0 , we have 0 ϕ z 0 j h j j ϕ z 0 − ϕ z 0 j h j j z 0 j!, 3.8 this is a contradiction.Thus, ϕ z 0 n ∈ σ T for all n ∈ N.This completes the proof.Proof.Since F is a finite set and ϕ F F, there exist z 0 ∈ F and m ∈ N such that ϕ m z 0 z 0 .Since ϕ X ⊆ int K , so z 0 ∈ int K .If ϕ is constant, then the proof is complete.When ϕ is We let C, D {z ∈ C : |z| < 1}, D {z ∈ C : |z| ≤ 1}, D λ, r {z ∈ C : |z − λ| < r}, and D λ, r {z ∈ C : |z − λ| ≤ r} denote the field of complex numbers, the open unit disc, the closed unit disc, and the open and closed discs with center at λ and radius r, respectively.We also denote D 0, r by D r .

1 .
and, by Theorem 2.9, the proof is complete.Definition 2.11. a A plane set X at c ∈ ∂X has an internal circular tangent if there exists a disc D in the complex plane such that c ∈ ∂D and D \ {c} ⊆ int X .b A plane set X is called strongly accessible from the interior if it has an internal circular tangent at each point of its boundary.Such sets include the closed unit disc D and D z 0 , r \ n k 1 D z k , r k , where closed discs D z k , r k are mutually disjoint in D z 0 , r .c A compact plane set X has peak boundary with respect to B ⊆ C X if for each c ∈ ∂X there exists a nonconstant function h ∈ B such that h X h c Example 2.12.The closed unit disc D has peak boundary with respect to A D because, if c ∈ ∂D, then the function h : D → C defined by h z 1/2 1 cz belongs to A D and satisfies h D h c 1.

Lemma 2 .
16. Let G and Ω be bounded connected open sets in the complex plane with Ω ⊆ G, and let X G and K Ω.Then for each c ∈ G \ K there exists a function f c ∈ Lip A X, K, 1 such that f c is not analytic at c. Proof.Let c ∈ G \ K. Then there is a positive number r such that {z ∈ C : |z − c| ≤ r} ⊆ G \ K.

Example 2 .
18. Let r ∈ 0, 1 and K D r .Suppose that Lip A D, K, 1 ⊆ B ⊆ A D, K .Then K has peak K-boundary with respect to B. Proof.We first assume that r 1.If for each c ∈ ∂D the function h : D → C defined by h z 1/2 1 cz , then h ∈ B, h is nonconstant on K D and h c 1 h D .

Corollary 3 . 3 .
Let B and T satisfy the conditions of Theorem 3.2, and let B be a natural Banach function algebra with the norm • α,K .If F is a finite set such that ϕ F F, then there exist z 0 ∈ F and m ∈ N such that{λ m : λ ∈ σ T } {0, 1} ∪ ϕ m z 0 n : n ∈ N .3.9 then, by Proposition 2.10, ∠ h • ϕ c exists and is nonzero and since 1 and h D 1 h c .Let Ω be a connected open set in the complex plane, and let ϕ be a one-to-one analytic function on Ω.If f is a continuous complex-valued function on ϕ Ω and f • ϕ is analytic on Ω, then f is an analytic function on ϕ Ω .Proof.By 8, Theorem 7.5 and Corollary 7.6 in Chapter IV , we deduce that ϕ Ω is a connected open set in the complex plane, ϕ z / 0 for all z ∈ Ω, and ϕ −1