Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 146758, 15 pages

http://dx.doi.org/10.1155/2011/146758

## Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras

Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

Received 22 May 2011; Accepted 10 August 2011

Academic Editor: Malisa R.Β Zizovic

Copyright Β© 2011 Davood Alimohammadi and Maliheh Mayghani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let and be compact plane sets with . We define , where is analytic on . For , we define and . It is known that is a natural Banach function algebra on under the norm , where . These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of to natural Banach function subalgebras of and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of .

#### 1. Introduction and Preliminaries

We let , , , , and denote the field of complex numbers, the open unit disc, the closed unit disc, and the open and closed discs with center at and radius , respectively. We also denote by .

Let and be unital commutative semisimple Banach algebras with maximal ideal spaces and . A homomorphism is called* unital* if . If is a unital homomorphism from into , then is continuous and there exists a norm-continuous map such that for all , where is the Gelfand transform . In fact, is equal the adjoint of restricted to . Note that is a weak*-weak* continuous map from into . Thus is a continuous map from with the Gelfand topology into with the Gelfand topology.

Let be a unital commutative semisimple Banach algebra, and let be an endomorphism of , a homomorphism from into . We denote the spectrum of by and define

For a compact Hausdorff space , we denote by the Banach algebra of all continuous complex-valued functions on .

*Definition 1.1. *Let be a compact Hausdorff space. A *Banach function algebra* on is a subalgebra of which contains , the constant function 1 on , separates the points of , and is a unital Banach algebra with an algebra norm . If the norm of a Banach function algebra on is , the uniform norm on , it is called a *uniform algebra* on .

Let and be Banach function algebras on and , respectively. If is a continuous mapping such that for all and if is defined by , then is a unital homomorphism, which is called the *induced homomorphism* from into by . In particular, if and , then is called the *induced endomorphism* of by the self-map of .

Let be a Banach function algebra on a compact Hausdorff space . For , the map , defined by , is an element of and is called the *evaluation homomorphism* on at . This fact implies that is semisimple and for all . Note that the map is a continuous one-to-one mapping. If this map is onto, we say that is *natural*.

Proposition 1.2. *Let and be compact Hausdorff spaces, and let and be natural Banach function algebras on and , respectively. Then every unital homomorphism is induced by a unique continuous map . In particular, if is a compact plane set and the coordinate function belongs to , then and so .*

*Proof. *Let be a unital homomorphism. Since and are unital commutative semisimple Banach algebras, there exists a continuous map such that for all . The naturality of the Banach function algebra on implies that the map , defined by , is a homeomorphism and so is continuous. Since is a Banach function algebra on , the map , defined by , is continuous. We now define the map by . Clearly, is continuous. Let . Since
for all , we have . Therefore, is induced by .

Now, let be a compact plane set, and let . Then , and so .

Corollary 1.3. *Let be a compact Hausdorff space, and let be a natural Banach function algebra on . Then every unital endomorphism of is induced by a unique continuous self-map of . In particular, if is a compact plane set and contains the coordinate function , then and so .*

*Definition 1.4. *Let be a compact plane set which is connected by rectifiable arcs, and let be the geodesic metric on , the infimum of the length of the arcs joining and . is called *uniformly regular* if there exists a constant such that, for all , .

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 and be two compact plane sets with . Then there exists a finite union of uniformly regular sets in containing , namely , and then a positive constant such that for every analytic complex-valued function on and any ,
*

Let be a compact plane set. We denote by the algebra of all continuous complex-valued functions on which are analytic on , the interior of , and call it the *analytic uniform algebra* on . It is known that is a natural uniform algebra on .

Let and be compact plane sets such that . We define . Clearly, if , and if is empty. We know that is a natural uniform algebra on (see [2]) and call it the *extended analytic uniform algebra* on with respect to .

Let be a compact metric space. For , we denote by the algebra of all complex-valued functions for which . For , we define the *-Lipschitz* norm by . Then is a unital commutative Banach algebra. For , we denote by the algebra of all complex-valued functions on for which as . Then is a unital closed subalgebra of . These algebras are called *Lipschitz algebras* of order and were first studied by Sherbert in [3, 4]. We know that the Lipschitz algebras and are natural Banach function algebras on .

Let be a compact metric space, and let be a compact subset of . For , we denote by the algebra of all complex-valued functions on for which . In fact, . For , we define . Then under the algebra norm is a unital commutative Banach algebra. Moreover, is a subalgebra of ; if is finite, and if is finite. For , we denote by the algebra of all complex-valued functions on for which as with . In fact, . Clearly, is a closed unital subalgebra of . Moreover, is a subalgebra of ; if is finite, and if is finite. The Banach algebras and are Banach function algebras on and were first introduced by Honary and Moradi in [5].

Let be a compact plane set. We define for and for . These algebras are called *analytic Lipschitz algebras*. We know that analytic Lipschitz algebras and under the norm are natural Banach function algebras on (see [6]).

Let and be compact plane sets with . We define for and for . Then and are closed unital subalgebras of and under the norm , respectively. Moreover, if , and if is empty.

The algebras and are called *extended analytic Lipschitz algebras* and were first studied by Honary and Moradi in [5]. They showed that the extended analytic Lipschitz algebras and under the norm are natural Banach function algebras on [5, Theorem 2.4].

Behrouzi and Mahyar in [1] studied endomorphisms of some uniform subalgebras of and some Banach function subalgebras of and investigated some necessary and sufficient conditions for these endomorphisms to be compact, where is a compact plane set and .

In Section 2, we study unital homomorphisms from natural Banach function subalgebras of to natural Banach function subalgebras of and investigate necessary and sufficient conditions for which these homomorphisms are compact. In Section 3, we determine the spectrum of unital compact endomorphisms of .

#### 2. Unital Compact Homomorphisms

We first give a sufficient condition for which a continuous map induces a unital homomorphism from a subalgebra of into a subalgebra of .

Proposition 2.1. *Let and be compact plane sets with and , and let be a subalgebra of which is a natural Banach function algebra on under an algebra norm , where . If with , then induces a unital homomorphism . Moreover, if , then .*

*Proof. *The naturality of Banach function algebra on implies that , where is the spectrum of in the Banach algebra . Let . Since , , and is analytic on , we conclude that is analytic on an open neighborhood of . By using the Functional Calculus Theorem [2, Theoremββ5.1 in Chapter I], there exists such that on . It follows that
for all and so . Therefore, . This implies that the map defined by is a unital homomorphism from into , which is induced by . Now let . Then by Proposition 1.2.

Corollary 2.2. *Let and be compact plane sets with and . Let be a subalgebra of which is a natural Banach function algebra on under an algebra norm . If with , then induces a unital endomorphism of . Moreover, if , then .*

Proposition 2.3. *Suppose that , , , , , , and , where . Then for each there exists a continuous map with such that and does not induce any homomorphism from to .*

*Proof. *Let . We define the map by
Clearly, is a continuous mapping, , and . We now define the function by
Then, . Since and
we conclude that . Therefore, does not induce any homomorphism from to . Hence, the proof is complete.

Corollary 2.4. *Suppose that , , , , , , and . Then for each , there exists a continuous self-map of with such that and does not induce any endomorphism of .*

We now give a sufficient condition for a unital homomorphism from a subalgebra of into a subalgebra of to be compact.

Theorem 2.5. *Suppose that , and are compact plane sets with and , and is a subalgebra of which is a natural Banach function algebra on under the norm , where . Let be a continuous mapping. If is constant or with , then induces a unital compact homomorphism .*

*Proof. *If is constant, then the map defined by is a unital homomorphism from into with , and so it is compact.

Let be a nonconstant mapping with and . Then the map defined by is a unital homomorphism from to by Proposition 2.1. To prove the compactness of , let be a bounded sequence in with for all . This implies that is a bounded sequence in which is equicontinuous on . By Arzela-Ascoliβs theorem, has a subsequence such that is convergent in . Since for all , is convergent in . By Montel's theorem, the sequences and are uniformly convergent on the compact subsets of . Since and are compact sets in the complex plane and , by using Lemma 1.5, we deduce that there exists a finite union of uniformly regular sets in containing , namely , and then a positive constant such that for every analytic complex-valued function on and any
Therefore, there exists a positive constant such that
for all and any . Let . Then, for all with , we have
The above inequality is certainly true for all with and . Therefore,
and so

Since is a compact subset of , we deduce that the sequences and are convergent uniformly on . Therefore, is a Cauchy sequence on , that is is convergent in . Hence, is compact.

Corollary 2.6. *Suppose that , and are compact plane sets with , and . Let be a subalgebra of which is a natural Banach function algebra on with the norm , and let be a self-map of . If is constant or with , then induces a unital compact endomorphism of .*

*Definition 2.7. *
β
(a)A *sector* in at a point is the region between two straight lines in that meet at and are symmetric about the radius to . (b)If is a complex-valued function on and , then means that as through any sector at . When this happens, we say that is *angular* (or *non-tangential*) *limit* of at . (c)An analytic map has an *angular derivation* at a point if for some
exists (finitely). We call the limit the *angular derivative* of at and denote it by .

Lemma 2.8. *Let , and let be an analytic function and defined by . Then has angular derivation at if and only if has angular derivation at . Moreover,
*

The following result is a modification of Julia-Caratheodoryβs theorem. For further details and proof of Julia-Caratheodoryβs theorem, see [7, pages 295β300].

Theorem 2.9. *Take . Let be a nonconstant analytic function and . Then the following are equivalent: *(i)*, *(ii)* exists for some , *(iii)* exists and .*

The boundary point in (ii) and (iii) is the same, and 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 .

Note that the existence of the angular derivative at , according to Theorem 2.9, is equivalent to . In this case the angular derivative of at is nonzero.

Proposition 2.10. *Let be a compact plane set, and let and . Suppose that and is a nonconstant function such that . Then the angular derivative of at exists and is nonzero.*

*Proof. *Let . For every we have
Therefore, , and, by Theorem 2.9, the proof is complete.

*Definition 2.11. *
β
(a)A plane set at has an *internal circular tangent* if there exists a disc in the complex plane such that and . (b)A plane set 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 and , where closed discs are mutually disjoint in . (c)A compact plane set has *peak boundary* with respect to if for each there exists a nonconstant function such that .

*Example 2.12. *The closed unit disc has peak boundary with respect to because, if , then the function defined by belongs to and satisfies .

Let be a compact plane set. The algebra consists of all functions in which can be approximated by rational functions with poles off . It is known that is a natural uniform algebra on .

*Example 2.13. *Let be a compact plane set such that is strongly accessible from the interior. If , then has a peak boundary with respect to .

*Proof. *Let . Since is strongly accessible from the interior, for each , there exists a such that and . Now, we define the function by
Then , .

Theorem 2.14. *Let be a compact plane set such that is connected, and . Suppose that has peak boundary with respect to . Let be a bounded connected open set in the complex plane, and let . Let be a bounded connected open set in the complex plane, and let such that is strongly accessible from the interior. Suppose that is a compact plane set such that . If is a unital compact homomorphism, then is induced by a continuous mapping such that is constant on or . Moreover, .*

*Proof. *Since and are, respectively, natural Banach function algebras on and , is a unital homomorphism, is a compact plane set, and , we conclude that is induced by and so by Proposition 1.2. Suppose that is nonconstant on . Since is analytic on , we deduce that is an open subset of and so . We now show that . Suppose that . Then there exists such that . Since has peak boundary with respect to , there exists a nonconstant function such that . We now define the sequence of complex-valued functions on by . Let . Then
Thus
by (2.14) and (2.15). This implies that is a bounded sequence in . The compactness of homomorphism implies that there exists a subsequence of and a function in such that
This implies that
On the other hand, we have for all by (2.14). Hence,
By (2.18) and (2.19), . Therefore, by (2.17) we have
This implies that
Assume that . By (2.21), there exists a natural number such that for each with
In particular,
This implies that
Since and is strongly accessible from the interior, there exists an open disc such that and . Since is analytic on and is analytic on , we deduce that is analytic on . On the other hand, we can easily show that
Therefore, . Since , we conclude that
We claim that is constant on . If is nonconstant on , then, by Proposition 2.10, exists and is nonzero and since , . If is a sector in at , then
by (2.24). Thus
But
Hence, by (2.28) we have
Since is assumed to be a positive number, we conclude that , contradicting to . Hence, our claim is justified. Since is nonconstant on , is a nonconstant analytic function on connected open set . This implies that is a connected open set in the complex plane. This implies that is constant on connected open set . The continuity of on follows that is constant on . This contradicting to is nonconstant on . Therefore, .

Corollary 2.15. *Let be a compact plane set such that is connected and . Let be a bounded connected open set in the complex plane, and let . Suppose that is strongly accessible from the interior and has peak boundary with respect to . If is a unital compact endomorphism of , then is induced by a continuous self-map of such that is constant on or . Moreover, .*

Lemma 2.16. *Let and be bounded connected open sets in the complex plane with , and let and . Then for each there exists a function such that is not analytic at .*

*Proof. *Let . Then there is a positive number such that
We now define the function by
It is easily seen that and is not analytic at .

*Definition 2.17. *Let and be compact plane sets such that . We say that has *peak **-boundary* with respect to if for each there is a function such that is nonconstant on and .

*Example 2.18. *Let and . Suppose that . Then has peak -boundary with respect to .

*Proof. *We first assume that . If for each the function defined by , then , is nonconstant on and .

We now assume that . For each , set . Then . Define the function by
It is easily seen that and .

Lemma 2.19. *Let be a connected open set in the complex plane, and let be a one-to-one analytic function on . If is a continuous complex-valued function on and is analytic on , then is an analytic function on .*

*Proof. *By [8, Theorem 7.5 and Corollary 7.6 in Chapter IV], we deduce that is a connected open set in the complex plane, for all , and is an analytic function on . Since on , we conclude that is analytic on .

Theorem 2.20. *Let be a compact plane set such that is connected and . Suppose that is a compact subset of such that is connected, , and has peak -boundary with respect to . Let be a bounded connected open set in the complex plane, and let such that is strongly accessible from the interior. Suppose that is a compact plane set such that . If is a unital compact homomorphism and is one-to-one on , then is induced by a continuous mapping such that and .*

*Proof. *Since and are, respectively, natural Banach function algebras on and , is a unital homomorphism, is a compact plane set, and , we conclude that is induced by and so by Proposition 1.2.

To prove , we first show that . Since is a one-to-one analytic mapping on , we conclude that is an open set in the complex plane. This follows that since . Suppose that . Then there exists such that . By Lemma 2.16, there exists a function such that is not analytic at . But , so that is analytic on . Since is continuous on and is a one-to-one analytic function on , we conclude that is analytic on by Lemma 2.19. This contradicts to the fact is not analytic at . Therefore, so since is an open set in the complex plane. Since is continuous on , , , and , we can easily show that . We now show that . Suppose that . Then there exists such that . Since has peak -boundary with respect to , there exists a function such that is not constant on and
Applying the similar argument used in the proof of Theorem 2.14, we can prove that is constant on . This contradiction shows that .

Corollary 2.21. *Let be a compact plane set such that is connected and . Let be a compact subset of such that is connected and . Suppose that has peak -boundary with respect to and is strongly accessible from the interior. If is a unital compact endomorphism of and is a one-to-one mapping on , then is induced by a continuous self-map of such that and .*

#### 3. Spectrum of Unital Compact Endomorphisms

In this section we determine the spectrum of a unital compact endomorphism of a subalgebra of which is a natural Banach function algebra with the norm .

The following result is a modification of [9, Theorem 1.7] for unital compact endomorphisms of natural Banach function algebras.

Theorem 3.1. *Let be a compact Hausdorff space and a natural Banach function algebra on . If is a unital compact endomorphism of induced by a self-map , then is finite, and if is connected, is singleton where is the th iterate of , that is, and . If , then is a fixed point for . In fact, if , then .*

Theorem 3.2. *Suppose that is a compact plane set with , is a connected open set in the complex plane with , and . Let be a subalgebra of containing the coordinate function which is a natural Banach function algebra on with an algebra norm . Let be a unital compact endomorphism of induced by a self-map of . If and is a fixed point of , then
*

*Proof. *Clearly 0 and also since . If is constant then the proof is complete. Let . The compactness of implies that there exists such that . Since , . We claim that for some . If for all , then on an open disc with center and so on . By maximum modules principle, it follows that on since , , and for all . This contradicts to . Hence, our claim is justified. Let . Then for all and . By times differentiation of , we have , and therefore . Then .

Conversely, first we show that, if with , then . Let and . The compactness of implies that there exists such that . It follows that . Since and is analytic on the connected open set , we conclude that is constant on by maximum modules principle. Since , , and , we deduce that is constant on . Applying again implies that since .

We now claim that . If , then there exists a nonzero linear operator such that
Since , and so
by (3.2). By differentiation at , we have
this is a contradiction. Hence, our claim is justified.

We now show that for all . If or , the proof is complete. Suppose that and . If for some with , then there exists a nonzero linear operator such that
Since , and so
by (3.5). By times differentiation at , we have
and by times differentiation at , we have