Homomorphisms of Certain Banach Function Algebras

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 112, No. 2, May 2002, pp. 331–336. © Printed in India

Homomorphisms of certain Banach function algebras

F BEHROUZI

Institute of Mathematics, University for Teacher Education, 599, Taleghani Avenue, Tehran 15614, Iran E-mail: [email protected]

MS received 28 May 2001; revised 19 December 2001.

n Abstract. In this note, we study homomorphisms with domain D (X) or Lipα(X, d) of which ranges are certain Banach function algebras and determine in which cases these homomorphisms are compact.

Keywords. Compact homomorphism; semisimple Banach algebra.

0. Introduction Let X be a compact Hausdorff space and C(X) be the Banach algebra of all continuous complex-valued functions f on X under the uniform norm kf kX = supx∈X |f(x)|.A subalgebra of C(X) which separates the points of X, contains the constants, and which is a Banach algebra with respect to some algebra norm k·k, is a Banach function algebra on X. If the norm is equivalent to the uniform norm, then it is called uniform algebra. A Banach function algebra A on X is natural if MA, the maximal ideal space of A,isX; equivalently, if every character on A is given by evaluation at a point of X [3]. Let (X, d) be a compact metric space and let α ∈ (0, 1]. The algebra of all complex- valued functions f on X for which |f(x)−f(y)| p (f ) = x,y ∈ X, x 6= y < ∞ α sup dα(x, y) : is denoted by Lipα(X, d). These algebras are called Lipschitz algebras of order α and were ﬁrst studied by Sherbert. The Lipschitz algebras Lipα(X, d) for α ≤ 1 are natural Banach function algebras on X under the norm kf kα =kfkX+pα(f ) [6,7]. For α = 1 we write Lip(X, d) instead of Lip1(X, d). Let X be a perfect compact plane set. A complex-valued function f is called differentiable on X if at each point z0 ∈ X 0 f(z)−f(z0) f (z0) = lim :z∈X, z −→ z 0 z−z0 exists. n The algebra of functions onPX with continuous nth derivatives is denoted by D (X). n n (r) n f ∈ D (X) kf k n = kf kX/r k·k n,D(X) For , let D r=0 !. Clearly under the norm D is a normed function algebra. The algebra Dn(X) is not necessarily complete. For this we consider a type of compact plane set. A compact plane set X is connected by rectiﬁable arcs if any two points of X

331 332 F Behrouzi

can be joined by a rectifiable arc lying within X. For such a set, let δ(z1,z2)denote the geodesic distance between z1 and z2, infimum of the lengths of the arcs joining z1 and z2. Clearly, δ defines a metric, the geodesic metric, on X. A compact plane set X is uniformly regular if it is perfect, connected by rectifiable arcs and if the geodesic metric is uniformly equivalent to the Euclidean metric on X. Thus, if X is uniformly regular, there is a constant C such that

δ(z1,z2)≤C|z1 −z2| (z1,z2)∈X. (∗)

n The algebras (D (X);k·kDn), n = 1, 2,... were introduced by Dales and Davie [2] in 1973. They proved that when X is uniformly regular, these algebras are natural Banach 1 0 function algebras, also for z1,z2 ∈X, f ∈ D (X), |f(z1)−f(z2)|≤Ckf kX|z1−z2|, where C is the same constant in (∗) [1]. Let A, B be commutative semi-simple Banach algebras and T : A −→ B be a homo- ∗ ∗ morphism. As we know, T , the adjoint of T , maps MB into MA ∪{0}.IfT maps MB into ∗ c ˆ ˆ MA and ϕ = T |MB , then Tf = f ◦ϕ for every f in A, where f is the Gelfand transform of f . In this case we say T is induced by ϕ. For example, if T is onto or A, B are unital ∗ with identities 1A, 1B respectively and T(1A)=1B, then T maps MB into MA.

1. Homomorphisms with domain Lipα (X, d) Theorem 1. Let B a Banach function algebra on a compact Hausdorff space X such that ˆ kf kMB =kfkX. Then the following statements are equivalent:

(i) Inclusion map i : B −→ C(X) is compact. (ii) Every homomorphism T from B into a uniform algebra A, induced by ϕ, is compact.

Proof. (i)H⇒ (ii): Let T : B −→ A be a homomorphism induced by ϕ, where A is Y {f }∞ a uniform algebra on compact Hausdorff space . Assume that n n=1 be a bounded B {f }∞ C(X) sequence in . Then there exists subsequence nk k=1 such that it is convergent in . {Tf }∞ A It is sufﬁcient to show that nk k=1 is a Cauchy sequence in .Wehave kTf − Tf k =kTfc − Tfc k nk nj Y nk nj MA ˆ ˆ =kfnk ◦ϕ−fnj ◦ϕkMA ˆ ˆ ≤kfnk −fnjkMB =kfnk −fnjkX.

So kTfnk − Tfnj kY −→ 0ask,j −→ ∞ , that is {Tfnk } is a Cauchy sequence in A. (ii)H⇒ (i): is trivial.

Theorem 2. Let (X, d) be a compact metric space. Then every homomorphism from Lipα(X, d) into uniform algebra A is compact.

Proof. As we know, Lipα(X, d) is natural Banach function algebra. So by Theorem 1, it is sufﬁcient to show that every bounded sequence in Lipα(X, d) has a subsequence which C(X) {f }∞ Lip (X, d) is convergent in . For this, let n n=0 be a bounded sequence in α . Then {f }∞ X n n=1 is uniformly bounded and it also is equicontinuous on . So by the Arzela–Ascoli Theorem, {fn} contains a uniformly convergent subsequence. Homomorphisms of certain Banach function algebras 333

Sherbert in [6] proved that if (X1,d1) and (X2,d2) are compact metric spaces then T : Lip(X1,d1) −→ Lip(X2,d2) is a homomorphism if and only if Tf = f ◦ ϕ(f ∈ Lip(X1,d1)) where ϕ : X2 −→ X 1 satisﬁes d1(ϕ(x), ϕ(y)) ≤ Md2(x, y) for some constant M>0, all x,y ∈ X2. Kamowitz and Scheinberg in [4] have also proved

Theorem 3. Let (X1,d1) and (X2,d2) be compact metric spaces and let T : Lip (X1,d1) −→ Lip(X2,d2)be a homomorphism induced by map ϕ : X2 −→ X 1 , that is Tf = f ◦ ϕ. Then T is compact if and only if ϕ is supercontraction, i.e.

d1(ϕ(x), ϕ(y)) lim = 0 as d2(x, y) −→ 0 . d2(x, y)

Using this theorem we have

Theorem 4. Let X be a uniformly regular set and let (Y, d) be a compact metric space. If T : Lip(Y, d) −→ D 1 (X) is a compact homomorphism, induced by ϕ : X −→ Y , then ϕ is a constant map.

d(ϕ(z1), ϕ(z2)) lim = 0as|z1−z2|−→0. |z1 − z2|

Let z0 be a ﬁxed point in X. We deﬁne the map F : Y −→ R by F(y) = d(y,ϕ(z0)). Since F ∈ Lip(Y, d), F ◦ ϕ ∈ D1(X). Moreover, for every z, w ∈ X we have

F ◦ ϕ(w) − F ◦ ϕ(z) d(ϕ(w), ϕ(z)) ≤ . w − z |w − z|

So (F ◦ ϕ)0(z) = 0. Since X is connected, F ◦ ϕ is constant. Hence for every z ∈ X, F ◦ ϕ(z) = F ◦ ϕ(z0) = 0. By the deﬁnition of F,ϕ(z) = ϕ(z0) for all z ∈ X, and proof is complete.

2. Homomorphisms with domain Dn(X) Kamowitz and Scheinberg in [5] have proved that

Theorem 5. Let A be a uniform algebra. Then every homomorphism from C1([0, 1]) into A is compact.

The next result is the generalization of this theorem.

Theorem 6. Let X be a uniformly regular set and let T be a homomorphism from Dn(X) into a uniform algebra A. Then T is compact. 334 F Behrouzi

Proof. Since X is uniformly regular there exists a constant C such that for every f ∈ 1 0 ∞ D (X) and z1,z2 ∈ X, |f(z1)−f(z2)|≤CkfkX|z1−z2|. Let {fk}k= be a sequence n ∞ 1 D (X) kf k n ≤ k = , ,... {f } in such that k D 1 for 1 2 . Thus k k=1 is uniformly bounded {f }∞ and equicontinuous. By Arzela–Ascoli Theorem there exists subsequence kj j=1 which is convergent in C(X). By Theorem 1 every homomorphism from Dn(X) into uniform algebra A is compact.

Remark 7. Let X be a uniformly regular set and let T be a homomorphism from Dn(X) into a uniform algebra A induced by ϕ. Clearly, ϕ ∈ Aˆ and if A is a natural uniform n algebra, ϕ ∈ A. Conversely, assume ϕ ∈ A and ϕ(Mˆ A) ⊆ int(X), since every f ∈ D (X) is analytic on int(X), by Functional Calculus Theorem, f ◦ ϕ ∈ A. Therefore the map S : Dn(X) −→ A deﬁned by Sf = f ◦ ϕ is a homomorphism. We shall require the following lemma which is contained in ([1] Chapter I of Part Six).

Julia Lemma. Let U¯ be the closed unit disc in C and let ϕ be a continuously differentiable ¯ ¯ 0 function on U.If|ϕ(c)|=kϕkU¯ for some c ∈ U, then either ϕ is constant or ϕ (c) 6= 0. Theorem 8. Let X, Y be uniformly regular sets and let T : Dn(X) −→ D m (Y ) be a homomorphism induced by a map ϕ:

(i) if n>mthen T is compact. (ii) if int(X) =∅and int(Y) 6= ∅ then ϕ is constant. ¯ ¯ (iii) if n ≤ m, Y = U,ϕ(U) ⊆ int(X), then T is compact if and only if kϕkU¯ < 1 or ϕ is constant.

∞ n {f } D (X) kf k n ≤ k = , ,... Proof. (i) Let k k=1 be a sequence in such that k D 1 for 1 2 . Since X is uniformly regular for every z1,z2 ∈X, j = 0, 1, 2,... ,n−1 and k = 1, 2,... (j) (j) (j+1) (j) we have |fk (z1) − fk (z2)|≤Ckf kX|z1 − z2| and kfk kX ≤ n!, where C (j) is a constant. Hence, for j = 0, 1, 2,... ,n − 1, {fk } is a uniformly bounded and equicontinuous sequence in C(X). By Arzela–Ascoli Theorem, there exists a subsequence {f }∞ {f } g ∈ Dn−1(X) f −→ g Dn−1(X) ϕ ∈ Dm(Y ) ki i=1 of k and such that ki in . Since m

Theorem 9. Let (Y, d) be a compact metric space, X be a uniformly regular set and n>1. n Then every homomorphism T : D (X) −→ Lipα(Y, d) induced by map ϕ : Y −→ X is compact. {f }∞ Dn(X) {f } Proof. Let k k=1 be a bounded sequence in . As we have seen before, k contains {f }∞ Dn−1(X) {f ◦ ϕ}∞ a subsequence kj j=1 which is convergent in . We show that kj j=1 is a Cauchy sequence in Lipα(Y, d).Ifx,y ∈ Y, x 6= y we have

|(fkj ◦ ϕ − fki ◦ ϕ)(x) − (fkj ◦ ϕ − fki ◦ ϕ)(y)| dα(x, y)

0 0 |ϕ(x) − ϕ(y)| ≤ Ckf − f kX , kj ki dα(x, y)

C ϕ ∈ Lip (Y, d) kf 0 − f 0 k → j,i →∞ where is constant. Since α and kj ki X 0as , then p (f ◦ ϕ − f ◦ ϕ) → j,i −→ ∞ {f }∞ α kj ki 0as . Since kj j=0 is a Cauchy sequence in C(X) {f ◦ ϕ}∞ C(Y) {Tf }∞ , then kj j=1 is a Cauchy sequence in . Thus kj j=1 is convergent in Lipα(Y, d).

Remark 10. As we know, in the above theorem ϕ is in Lipα(Y, d). On the other hand, if n ψ : Y −→ X is in Lipα(Y, d), then for x,y ∈ Y with x 6= y and for f ∈ D (X) we have |f ◦ ψ(x) − f ◦ ψ(y)| |ψ(x) − ψ(y)| ≤ Ckf 0k dα(x, y) X dα(x, y) 0 ≤ Ckf kXpα(ψ),

n where C is a constant. Thus the map S : D (X) −→ Lipα(Y, d) by Sf = f ◦ ψ is a homomorphism.

PROPOSITION11 Let W,V be two open set such that W,¯ V¯ are uniformly regular sets and W is connected. Assume that T : Dn(W)¯ −→ D m ( V)¯ is a homomorphism induced by map ϕ : V¯ −→ W¯ . If ϕ is a nonconstant function, then T is one to one.

Proof. Let f ∈ Dn(W),Tf¯ = f ◦ϕ ≡ 0. Thus f = 0onϕ(V)¯ . Since ϕ is analytic on V by open mapping theorem, ϕ(V ) is an open set in W¯ . Since zeros of nonconstant analytic function in a domain is isolated, f = 0onW¯ and proof is complete.

Acknowledgement The author is grateful to H. Mahyar for her advice and criticism. The author would also like to thank the referee for his comments.

References [1] Caratheodory C, Theory of functions (Chelsa Publishing Company) (1960) 336 F Behrouzi

[2] Dales H G and Davie A M, Quasianalytic Banach function algebra, J. Funct. Anal. 13(1) (1973) 28–50 [3] Gamelin T W, Uniform Algebras (Chelsea Publishing Company) (1984) [4] Kamowitz H and Scheinberg S, Some properties of endomorphisms of Lipschitz algebras, Studia. Math. 96(3) (1990) 255–261 [5] Kamowitz H and Scheinberg S, Homormorphisms of Banach algebras with range C1 ([0, 1]), Internat. J. Math. 5(2) (1994) 201–212 [6] Sherbert D R, Banach algebra of Lipschitz algebra, Paciﬁc. J. Math. 2 (1963) 1387–1399 [7] Sherbert D R, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964) 240–272