Certain Subalgebras of Lipschitz Algebras of Infinitely Differentiable Functions and Their Maximal Ideal Spaces

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1 Int. J. Nonlinear Anal. Appl No., 9-22 ISSN: electronic Certain Subalgebras of Lipschitz Algebras of Infinitely Differentiable Functions and Their Maxial Ideal Spaces Davood Aliohaadi a,, Fateeh Nezaabadi a a Departent of Matheatics, Faculty of Science, Arak University, P. O. Box: , Arak, Iran. Dedicated to the Meory of Charalabos J. Papaioannou Counicated by M. Eshaghi Gordji Abstract We study an interesting class of Banach function algebras of infinitely differentiable functions on perfect, copact plane sets. These algebras were introduced by Honary and Mahyar in 999, called Lipschitz algebras of infinitely differentiable functions and denoted by LipX, M, α, where X is a perfect, copact plane set, M = { } is a sequence of positive nubers such that = and n! M n! M n! for, n N {0} and α 0, ]. Let d = li sup n! n and X d = {z C : distz, X d}. Let Lip P,d X, M, α[lip R,d X, M, α] be the subalgebra of all f LipX, M, α that can be approxiated by the restriction to X d of polynoials [rational functions with poles off X d ]. We show that the axial ideal space of Lip P,d X, M, α is X d, the polynoially convex hull of X d, and the axial ideal space of Lip R,d X, M, α is X d, for certain copact plane sets.. Using soe forulae fro cobinatorial analysis, we find the axial ideal space of certain subalgebras of Lipschitz algebras of infinitely differentiable functions. Keywords: Infinitely differentiable functions, Function algebra, Lipschitz algebra, Maxial ideal space, Star-shaped set, Uniforly regular MSC: Priary 46J0; 46J5; Secondary 46J20. Corresponding author Eail addresses: d-aliohaadi@araku.ac.ir Davood Aliohaadi, fthn@yahoo.co Fateeh Nezaabadi Received: April 203 Revised: Deceber 203

2 0 Aliohaadi, Nezaabadi. Introduction and preliinaries Let X be a copact Housdorff space. We denote by CX the coplex algebra of all continuous coplex-valued functions on X. For f CX and a closed subset E of X, we denote the unifor nor f on E by f E ; that is, f E = sup{ fx : x X}. A function algebra on X is a subalgebra A of CX that separates the points of X and contains the constant functions on X. If there is an algebra nor. on A such that A is coplete under the nor. and =, then A is a Banach function algebra on X, and if the given nor is the unifor nor on X, then A is a unifor algebra on X. Let A be a Banach function algebra on X. We denote by MA the axial ideal space of A. We know that MA with the Gelfand topology is a copact Hausdorff space. For each x X, the ap e x : A C, defined by e x f = fx, is an eleent of MA and called the evaluation character on A at x. This fact iplies that f X ˆf MA for all f A, where ˆf is the Gelfand transfor of f. The ap J : X MA, defined by Jx = e x, is injective and continuous, and so X is hoeoorphic to a copact subset of MA. If the ap J is surjective, then A is called a natural Banach function algebra on X and we write MA X. In this case, f X = ˆf MA for all f A. It is known that if A is a Banach function algebra on X, then Ā, the unifor closure A in CX, is a unifor algebra on X and MĀ MA. The following result is proved in [9] and we will use it in sequel. Theore.. Let X be a copact Hausdorff space and let A be a Banach function algebra on X. Then MA = MĀ if and only if ˆf MA = f X, for all f A, where ˆf is the Gelfand transfor of f. For a copact plane set X, we denote the set of all coplex-valued continuous functions on X that are analytic on intx by AX, and the set of all coplex-valued functions on X having an analytic extension to a neighborhood of X by H 0 X. We denote the set of restriction to X of rational functions with poles off X by R 0 X, and the restriction to X of polynoials by P 0 X. The polynoial convex hull of X is denoted by X. By the coordinate functional on X we ean the function Z on X that aps any point to itself. We denote by HX, RX and P X the unifor closure of H 0 X, R 0 X and P 0 X, respectively. It is known that RX = HX, RX is natural and MP X ˆX see[8]. Let X be a perfect, copact plane set. We say that coplex-valued function f on X is coplexdifferentiable at a point a X if the liit f a = li z a z X fz fa z a exists. We call f a the coplex-derivative of f at a. We denoted the nth derivative of f at a X by f n a. We denote the set of n ties continuously coplex-differentiable functions on X by D n X and the set of infinitely coplex-differentiable functions on X by D X. Let M = { } be a sequence of positive nubers. We say that M is an algebra sequence if = and n!! n!, M n M

3 Certain Subalgebras of Lipschitz Algebras No.,9-22 for all, n N {0}. Let M = { } be an algebra sequence and let dm = li sup n! n. We say that M = { } is an analytic algebra sequence if dm > 0 and a non-analytic algebra sequence if dm = 0. We need the following result about an algebra sequence M = { }. Lea.2. Let M = { } be an algebra sequence. Then i the sequence { n! n } is convergent and ii the series n! ρ n dm = li n! n! n = inf{ n : n N {0}}, is convergent, if ρ > dm. To see the proof of Lea.2i, we refer to [6, Proposition A..26]. For a perfect copact plane set X and an algebra sequence M = { }, a Dales-Davie algebra associated with X and M is defined by f n DX, M = {f D X : X < }, where the nor on DX, M is given by f DX,M = f n X. Since Z DX, M and M is an algebra sequence, DX, M is a nored function algebra on X. A copact subset X of the coplex plane is connected by rectifable arcs if any two points of X can be joined by a rectifiable arc lying within X. For such a set, let δz, w denote the geodesic distance between z and w; that is, the infiu of the lengths of the arcs joining z and w. Clearly δ defines a etric, the geodesic etric, on X. Definition.3. Let X be a copact plane set which is connected by rectifiable arcs, and let δz, w be the geodesic distance between z and w in X. i X is regular if for every z X there exists a constant C z such that δz, w C z z w, for all w X. ii X is uniforly regular if there exists a constant C such that δz, w C z w, for all z, w X. Dales and Davie in [6] proved that if X is finite of union of uniforly regular sets, then DX, M is coplete. In fact, if X is a finite union of regular sets, then D X with the nor f D X = f X f X is coplete and so DX, M with the nor DX,M is coplete. For soe further results see [4]. We soeties require the following condition on X which is called the -condition. There exists a constant C such that for every z, w X and f D X, fz fw C z w f X f X. For exaple, every uniforly regular set satisfies the -condition see [6]. The copleteness of D X is also concluded fro the -condition.

4 2 Aliohaadi, Nezaabadi Let X be a copact plane set and let α 0, ]. We denote by LipX, α the coplex algebra of coplex-valued functions f on X for which p α f = sup{ fz fw : z, w X, z w} is finite. For z w α each f LipX, α, set f α = f X p α f. Then. α is an algebra nor on LipX, α and LipX, α with the nor. α is a natural Banach function algebra on X. These algebras are called Lipschitz algebras of order α and were first studied by Sherbert in [3, 4]. We denote the coplex algebra of coplex-valued functions f on a perfect copact plane set X whose derivatives of all orders exist and f n LipX, α for all n N {0}, by Lip X, α. For a perfect copact plane set X, an algebra sequence M = { } and α 0, ], a Lipschitz algebra of infinitely differentiable functions associated with X, M and α is defined by f n LipX, M, α = {f Lip X, α : X p α f n < }, where the nor on LipX, M, α is given by f LipX,M,α = f n X p α f n. Since Z LipX, M, α and M is an algebra sequence, LipX, M, α is a nored function algebra on X. If D X under the nor f D X = f X f X is coplete, then LipX, M, α with the nor LipX,M,α is coplete and so a Banach function algebra on X [0]. Soe properties of these algebras have studied in [3, 0,, 2]. NOTATION. Let X be a copact plane set. For a point ζ C, the distance between ζ and X is defined by distζ, X = inf{ ζ z : z X}. For a non-negative real nuber d, we set X d = {ζ C : distζ, X d}. For z C and r > 0, Dz, r = {ζ C : ζ z < r} and z, r = {ζ C : ζ z r} are the open and closed disc with center at z and radius r. Abtahi and Honary studied soe properties of certain subalgebras of Dales-Davie algebras in [2]. In this paper we introduce certain subalgebras of Lipschitz algebras of infinitely differentiable functions and deterine their axial ideal spaces. 2. Certain subalgebras Throughout this section, we assue that X is a perfect copact plane set, M = { } is an algebra sequence and α 0, ]. Clearly, LipX, M, α contains the polynoials on X; that is, P 0 X LipX, M, α. Proposition 2.. Suppose that d = dm and X satisfies the -condition. Then H 0 X d is contained in LipX, M, α and, oreover, the ebedding of H 0 X d in LipX, M, α is continuous in the sense that if f H 0 X d and {f n } n= is a sequence in H 0 X and f n f uniforly on a neighborhood U of X d, then f n f in LipX, M, α. Proof. Let f H 0 X d. Then, there exists a neighborhood U of X d such that f and f are analytic on U. Choose ρ > d so that X ρ U. Suppose that z X. Then Cz, ρ U, where Cz, ρ is the

5 Certain Subalgebras of Lipschitz Algebras No., circle with center at z and radius ρ. Let n N {0}. By the Cauchy integral forula, we have f n z = n! 2πi f ζ n dζ, ζ z Cz,p f n z = n! 2πi f ζ n dζ. ζ z Hence, Therefore, f n z n! f Xρ ρ n, f n X n! f X ρ ρ n, Cz,p f n z n! f Xρ ρ n. f n X n! f Xρ ρ n. 2. Since X satisfies the -condition, there exists a positive constant C such that for each g D X and for all z, w X, Applying 2.2 for f n and then 2., we obtain g z g w C z w g X g X. 2.2 f n z f n w C z w f n X f n X = C z w α z w α f n X f n X C z w α diax α n! f X ρ ρ n for all z, w X with z w. This iplies that n! f Xρ ρ n = z w α n!cdiax α ρ n f Xρ f Xρ, p α f n n!cdiax α ρ n f Xρ f Xρ. 2.3 Fro 2. and 2.3, we give f n X p α f n n! ρ n [ f Xρ CdiaX α f Xρ f Xρ ]. 2.4 Since the series n! ρ n conclude that the series where is convergent by Lea.2ii and 2.4 holds for all n N {0}, we f n X p αf n is convergent and f n X p α f n n! λ, ρ n λ = f Xρ CdiaX α f Xρ f Xρ.

6 4 Aliohaadi, Nezaabadi This iplies that f LipX, M, α and f LipX,M,α λ n! ρ n. 2.5 Therefore, H 0 X d is contained in LipX, M, α. Now, suppose that f H 0 X d and {f n } n= is a sequence on H 0 X d such that f n f uniforly in a neighborhood U of X d. By [5, Theore VII.2.], f n f uniforly on every copact subset of U. We can choose ρ > d such that X d X ρ U, so that li f n f Xρ = 0 and li f n f Xρ = 0. Let n N. Then f n f LipX, M, α and, by above arguent, we have where Clearly, li γ n = 0. Therefore, and so the proof is coplete. 0 f n f LipX,M,α γ n =0! ρ M, γ n = f n f Xρ CdiaX α f n f Xρ f n f Xρ. li f n f LipX,M,α = 0, Proposition 2.2. Suppose that X satisfies the -condition. If d is a real nuber with dm d, then R 0 X d is contained in LipX, M, α. Proof. Since M = { } is an algebra sequence, X satisfies the -condition and α 0, ], we conclude that H 0 X dm LipX, M, α by Theore 2.. Now, let d be a real nuber with dm d. Then R 0 X d R 0 X dm. On the other hand, R 0 X dm H 0 X dm. Therefore, R 0 X d is contained in LipX, M, α. Corollary 2.3. If dm = 0, then R 0 X ic contained in LipX, M, α. Abtahi and Honary in [2] proved that if X is a perfect copact plane set such that DX, M, DX,M is coplete, then i H 0 X d DX, M, when d = dm, ii R 0 X d DX, M, if and only if dm d, iii R 0 X dm DX, M if and only if dm = 0. They also defined the closed subalgebras D P,d X, M, D R,d X, M, and D H,d X, M of DX, M to be the closure of P 0 X d, R 0 X d and H 0 X d in DX, M, respectively see [2, Definition 2.4]. Siilarly, we introduce certain subalgebras of LipX, M, α as the following. Definition 2.4. Let d = dm and let X satisfying the -condition. We define Lip P,d X, M,Lip R,dX, M, and Lip H,d X, M to be the closure of P 0 X d, R 0 X d, and H 0 X d in LipX, M, α, respectively. It is clear that Lip P,d X, M, α, Lip R,d X, M, α, and Lip H,d X, M, α are Banach function algebras on X with the nor LipX,M,α. Definition 2.5. Let Y be a plane set and let z 0 Y. We say that Y is star-shaped with respect to z 0, if for each z X the closed segent [z 0, z] is contained in X.

7 Certain Subalgebras of Lipschitz Algebras No., Lea 2.6. Let z 0 ˆX and let ˆX be star-shaped with respect to z0. Suppose that ρ > 0. If f H 0 X ρ, then there exists a sequence {p n } n= of polynoials such that li p n f Xρ = 0, and li p n f Xρ = 0. Proof. Let f H 0 X ρ. Then f H 0 X ρ. By Runge s theore, there exists a sequence {q n } n= of polynoials such that li q n f Xρ = 0. Let {p n } n= be the sequence of polynoials with p n = q n on X ρ and p n z 0 = fz 0 for all n N. Let z X ρ. Then there exists w z X such that w z z = ρ. We assue that C z = [z 0, w z ] [w z, z]. Then p n z f z = [ C z q n ζ dζ p n z 0 ] [ C z f ζ dζ f z 0 ] = C z [q n ζ f ζ] dζ w z z 0 z w z q n f Xρ ρ dia ˆX q n f Xρ, for all n N. This iplies that p n f Xρ ρ dia ˆX q n f Xρ Hence, the proof is coplete. and so li p n f Xρ = 0. Theore 2.7. Suppose that X satisfies the -condition, z 0 ˆX and ˆX is star-shaped with respect to z 0. If d = dm, then H 0 X d is contained in Lip P,d X, M, α. Proof. For f H 0 X d, there exists ρ > d such that f H 0 X ρ. By Lea 2.6, there exists a sequence {p n } n= of polynoials such that li p n f Xρ = 0, and li p n f Xρ = 0. Let n N. Since H 0 X ρ H 0 X ρ and X satisfies the -condition, we conclude that p n f LipX, M, α by Proposition 2. and p n f LipX,M,α η n by given arguent in the proof of Proposition 2., where Since li η n = 0, we conclude that =0! ρ M, η n = p n f Xρ CdiaX α p n f Xρ p n f Xρ. and so f Lip P,d X, M, α. Therefore, and the proof is coplete. li p n f LipX,M,α = 0, H 0 X d Lip P,d X, M, α,

8 6 Aliohaadi, Nezaabadi 3. Extensions of infinitely differentiable Lipschitz functions Throughout this section, we assue that X is a perfect copact plane set, M = { } is an analytic sequence with d = dm > 0. Let f DX, M and z X. By Lea.2,we have f k z k! ζ z k f k X d k f k X, k! M k for all ζ z, d and for all k N {0}. This iplies that the power series uniforly convergent on z, d. Therefore, the function F z : z, d C defined by F z ζ = k=0 is continuous on z, d and analytic on Dz, d. Abtahi and Honary obtained the following result in [2]. k=0 f k z k! ζ z k is f k z ζ z k, 3. k! Theore 3. see [2, Theore 3.2]. Let X be a perfect, copact plane set such that DX, M, DX,M is coplete. Then every f D H,d X, M has a unique extension F in AX d and F Xd f DX,M. Theore 3.2 see [2, Corollary 3.3]. Let X be a perfect, copact plane set such that DX, M, DX,M is coplete. Then every f D P,d X, M has a unique extension F in A X d. Theore 3.3 see [2, Corollary 3.5]. Let X be a regular set such that DX, M, DX,M is coplete. Then DX, M is contained in H 0 X. Since Lip X, α D X and f n X f n X p α f n for each f Lip X, α and for all n N {0}, we conclude that LipX, M, α DX, M. Hence, Lip H,d X, M, α D H,d X, M and Lip P,d X, M, α D P,d X, M, when X satisfies the -condition. Therefore, we give the following results. Theore 3.4. Suppose that X satisfies the -condition. Then every f Lip H,d X, M, α has a unique extension F in AX d and F Xd f DX,M f LipX,M,α. Theore 3.5. Suppose that X satisfies the -condition. Then, every f Lip P,d X, M, α has a unique extension F in A X d and F Xd f DX,M f LipX,M,α. Theore 3.6. Let X be a regular set satisfying the -condition. Then LipX, M, α is contained in H 0 X.

9 Certain Subalgebras of Lipschitz Algebras No., Maxial ideal space Throughout this section, we assue that X is a perfect copact plane set, M = { } is an algebra sequence and α 0, ]. Dalse and Davie in [6] proved that if dm = 0, then DX, M is coplete and MD R X, M X and MD P X, M X. Honary and Mahyar proved [8] that if dm = 0, then LipX, M, α is coplete and Lip R,d X, M, α is natural and MLip p,d X, M, α X. Abtahi and Honary obtained the following result in [2]. Theore 4. see [2, Theore 4.]. Let X be a perfect, copact plane set such that DX, M is coplete. Then MD R,d X, M X d and MD P,d X, M X d, where d = dm. We now deterine the axial ideal space of the Banach function subalgebras Lip R,d X, M, α and Lip P,d X, M, α of LipX, M, α. Theore 4.2. Suppose that X satisfies the -condition. If d = dm, then MLip R,d X, M, α X d. Proof. Let ζ X d. We define h ζ : Lip R,d X, M, α C by h ζ f = F ζ, where F is the unique extension of f in AX d given by Theore 3.4. We deduce that h ζ MLip R,d X, M, α by Theore 3.4. Let h MLip R,d X, M, α and let ζ = hz, where Z is the coordinate functional on X. We clai that ζ X d. Since ζ Z Lip R,d X, M, α and hζ Z = ζh hz = ζ ζ = 0, we conclude that the function ζ Z is not invertible in Lip R,d X, M, α. Now, we assue that ζ C \ X d. Then R ζ Z 0X d Lip R,d X, M, α. Hence ζ Z is invertible in Lip R,d X, M, α. This contradiction iplies that our clai is justified. It is easy to see that hf X = fζ for all f R 0 X d. Now, let f Lip R,d X, M, α. Then there is a sequence {f n } n= in R 0 X d such that li f n X f LipX,M,α = This iplies that f n X f uniforly on X. Since f Lip H,d X, M, α, we conclude that f has a unique extension F in AX d and F Xd f LipX,M,α, by Theore 3.4. Then f n F is the unique extension f n X f in AX d and so f n F Xd f n X f LipX,M,α, for all n N. Hence, li f nζ = F ζ, 4.2 by 4.. Fro 4., continuity of h on Lip R,d X, M, α and 4.2, we deduce that hf = li hf n X = li f n ζ = F ζ = h ζ f. Since f assued that an arbitrary eleent of Lip R,d X, M, α, we conclude that h = h ζ. Therefore, MLip R,d X, M, α {h ζ : ζ X d }. Hence, MLip R,d X, M, α X d. Theore 4.3. Suppose that X satisfies the -condition, z 0 ˆX, and ˆX is star-shaped with respect to z 0. If d = dm, then MLip P,d X, M, α X d. Proof. Let ζ X d. We define h ζ : Lip P,d X, M, α C by h ζ f = F ζ, where F is the unique extension of f in A X d given by Theore 3.5. We deduce that h ζ MLip P,d X, M, α by Theore 3.5. Let h MLip P,d X, M, α and let ζ = hz, where Z is the coordinate functional on X. We

10 8 Aliohaadi, Nezaabadi clai that ζ X d. Since ζ Z Lip P,d X, M, α and hζ Z = ζh hz = ζ ζ = 0, we conclude that the function ζ Z is not invertible in Lip P,d X, M, α. Now, we assue that ζ C\ X d. Then H ζ Z 0 X d and so Lip ζ Z P,dX, M, α by Theore 2.7. This contradiction iplies that our clai is justified. It is easy to see that h ζ p = pζ for all p P 0 X d. Now let f Lip P,d X, M, α. Then there is a sequence {p n } n= in P 0 X d such that li p n X f LipX,M,α = This shows that p n X f uniforly on X. Since f Lip P,d X, M, α, we conclude that f has a unique extension F in A X d and F Xd f LipX,M,α by Theore 3.5. Therefore, p n Xd F is the unique extension p n X f in A X d and so that p n Xd F Xd p n Xd f LipX,M,α, for all n N. Hence, li p n ζ = F ζ, 4.4 by 4.3. Fro 4.3, continuity of f on Lip P,d X, M, α and 4.4, we deduce that hf = li hp n X = li p n ζ = F ζ = h ζ f. Since f assued that an arbitrary eleent of Lip P,d X, M, α, we conclude that h = hζ. Therefore, MLip P,d X, M, α {h ζ : ζ X d }. Hence, MLip P,d X, M, α X d. To continue we need soe forulae fro cobinatorial analysis. For, n N with n, we take S, n as the set of all a = a,..., a n N {0} n such that n k= a k = and n k= ka k = n. For any N and any sequence {A k } k= of positive nubers, by [, Forula B3, P. 823], A k =! k= n= a S,n n Π a k! k= n Π A k a k. 4.5 k= The following equality for higher derivatives of coposite functions is known as the Faa di Bruno s Forula See[, P. 823]: F of n = n F of = The following lea is useful and found in [2]. a S,n n! n n Π f k k= Π a k! k! a k n N. 4.6 k= Lea 4.4 see [2, Lea 4.2]. Let K > 0 and let {ε } =0 be a sequence of positive nubers with li ε = 0. Then p li sup p p p ε K p K. =0 Lea 4.5. Let X be a perfect, copact plane set such that LipX, M, α is coplete. Suppose that A L = {f Lip X, α : f LipX, M, α}. i The set A L is a subalgebra of LipX, M, α containing X and separating the points of X. ii If dm = 0, then f MLipX,M,α = f X for all f A L.

11 Certain Subalgebras of Lipschitz Algebras No., Proof. i Clearly, C A L. Let f A L and g = f. Since n have f n α = f α f n α = f α = f α f α n= = f α M f α M <. f n α M n n g n α M n n g n α M n g n α n g n α M for all n N {0}, we Hence, f LipX, M, α and so that A L LipX, M, α. It is clear that Z A L. Thus, A L separates the points of X. Moreover, A L contains the constant functions on X. We now show that A L is a subalgebra of LipX, M, α. Let f, g A L. Clearly, A L is closed under the addition and scalar ultiplication. Let f, g A L. Since f, g LipX, M, α and A L LipX, M, α, we deduce that f, g LipX, M, α. Thus, f g fg LipX, M, α and so fg LipX, M, α. Therefore, fg A L and so A L is a subalgebra of LipX, M, α. ii Let dm = 0. We deduce that LipX, M, α is a Banach function algebra on X. Let f A and j N. Suppose that F z = z j z C. Then f j = F f. By the Faa di Bruno s forula 4.6, we have f j LipX,M,α = F f LipX,M,α = F f α F f n α M n= n = f j α F of f j α f j α n= n= n= in{j,n} = in{j,n} = = j! j! a S, n k= f j f j α n! a k! f k ak α k! k= a S, n k= n! a k! a S, n k= n! a k! f k ak α k! k= f k ak α. k! k=

12 20 Aliohaadi, Nezaabadi After interchanging the order of suation, we have f j LipX,M,α f j α j j = f j α j j = f j α j j = f j α j j = f j α j j = f j α! f j α! f j α! f j α f j α! f k α k! n! n= a S, n a k! k= k= P k a k k= n= a S, n k! a P n k= k= n= a S, n a k! k= P n P n! P n= a S, n a k! k= k= n= a S, n a k! k= k= f k α k= f k α km k km k ak ak f k α km k ak f k ak α. P M k Choosing A k = f k α P M k for all k N and applying Forula 4.5, we have f j LipX,M,α f j α = f j α = f j α j = j = j = j j j If f = 0, then f j LipX,M,α f j α and so f j α f j α f j α k= k=0 f k α P M k f k α P M k f LipX,M,α. P f MLipX,M,α li j f j α j = ˆf MLipX,α = f X. Suppose that f 0. Take K = f X and define the sequence {ε } =0 by ε 0 =, and ε =! M f LipX,M,α Then li ε = dm f LipX,M,α = 0. By Lea 4.4, Hence, li sup j li sup j f α j j j = =0 On the other hand, for all j N we have j f j LipX,M,α f j α j j = j ε K j N. K. j f LipX,M,α f j α P j ak K. f j α f LipX,M,α. P

13 Certain Subalgebras of Lipschitz Algebras No., Therefore, li sup f j LipX,M,α j j K = f X, and so f MLipX,M,α f X. Since LipX, M, α is a Banach function algebra on X, we have f X f MLipX,M,α. Consequently, f MLipX,M,α = f X. Theore 4.6. Let M = { } be a non-analytic algebra sequence, and let X be a uniforly regular set. Suppose that A L = {f Lip X, α : f LipX, M, α}, and B L = A L L, the closure of A L in LipX, M, α. Then B L is a natural Banach function subalgebra of LipX, M, α on X. Proof. By Lea 4.5, A L is a subalgebra of LipX, M, α containing X and separating the points of X. Since M = { } is a non-analytic algebra sequence, LipX, M, α is a Banach function algebra on X. Thus, B L is a Banach function algebra on X with the nor. Lip X,M,α and so ĝ MLipX,M,α = ĝ MBL, g B L. 4.7 We now show that B L is natural. Using Theore., it is enough to show that B L, the unifor closure of B L in CX, is natural and ˆf MBL = f X for all f B L. Since M = { } is a non-analytic sequence, R 0 X A L. It is clear that B L D X. Since X is a uniforly regular set, D X RX by [4,Theore.5iv]. Thus, R 0 X B L RX so that B L = RX, and B L is natural by naturality of RX. Let f B L. Then there exists a sequence {f n } n= in A such that li f n f LipX,M,α = 0. Since f n f X f n f MBL = f n f MLipX,M,α f n f LipX,M,α, for all n N, we conclude that li f n MLipX,M,α = f MLipX,M,α and li f n X = f X. Since f n A L for all n N, we have f n MLipX,M,α = f n X for all n N by Lea 4.5. Therefore, f MLipX,M,α = f X. Consequently, f n MBL = f X by 4.7. This copletes the proof. Question. Is the subalgebra A L = {f Lip X, α : f LipX, M, α} dense in LipX, M, α? References [] M. Abraowitz and I. Stegun, Handbook of Matheatical Functions with Forulas, Graphs and Matheatical Tables, U. S. Departent of Coerce, Washington, 964. [2] M. Abtahi and T. G. Honary, Properties of certain subalgebras of Dalse-Davie algebras, Glasgow Math. J [3] D. Aliohaadi, Non-weakly aenable subalgebras of AK and D K, For East J. Math. Sci , [4] W. J. Bland and J. F. Feinestein Copletion of nored algebras of differentiable functions, Studia Matheatica , 89-. [5] J. B. Conway, Functions of One Coplex Variable, Second Edition Springer-Verlag, New York, 995. [6] H. G. Dales, Banach Algebras and Autoatic Continuity, London Matheatical Society Monographs. New Series, Volue 24, The Clarendon Press, Oxford, 2000.

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