Applied Mathematical Sciences, Vol. 1, 2007, no. 52, 2565 - 2571

On the Real Banach Function Algebras

Sh. Najafzadeh

Department of Mathematics Maragheh University, Maragheh, Iran [email protected]

A. Ebadian

Department of Mathematics Urmia university, Urmia, Iran [email protected]

Abstract

In this paper we define the real Banach function algebras and extend some results in the real uniform algebras for the real Banach function.

Mathematics Subject Classification: Primary 46J10

Keywords: Real Banach Algebras

1 Introduction

Let X be a compact Hausdorff space, a τ : X −→ X is called a topological involution on X if τ 2(x)=x for all x ∈ X. Let C(X) denote the algebra of continuous complex-valued function on X and define C(X, τ)={f ∈ C(X):f ◦ τ = f}. Then C(X, τ) is a uniformly closed subalgebra of C(X), which contain 1 and separates the points of X. Moreover,

the C(X, τ) is a more general object than C(X) and CÊ(X).

Definition 1.1. Let X be a compact Hausdorff space, τ be a topological involution on X. A real uniform (function) algebra on (X, τ), is a uniformly closed subalgebra of C(X, τ) which contains 1 and separates the points of X. 2566 Sh. Najafzadeh and A. Ebadian

Theorem 1.2. Let X be a compact Hausdorff space, τ be a topological involution on X. Let σ(f)=f ◦ τ (f ∈ C(X)) be the algebra involution induced by τ on C(X). Let A be a real subspace of C(X, τ) and define

B := {f + ig : f,g ∈ A}( complexification of A).

Then: (i): σ(B)=B, A = {h ∈ B : σ(h)=h} = B ∩ C(X, τ) and B = A ⊕ iA. (ii): For f,g in A,

max{fX, gX}≤|f + ig|X ≤fX + gX

(iii): B is uniformly closed (Self adjoint, separates the points of X, contains 1) if and only if A has the same property, respectively. Moreover, B is a complex algebra if and only if A is a real algebra. (iv): If A is a real on (X, τ) then the map α : Carr(A) −→ Carr(B), define by α(φ)(f + ig)=φ(f)+iφ(g), is a homeomorphism of Carr(A)ontoCarr(B) (see [5]).

Theorem 1.3. Let X be a compact Hausdorff space, τ be a topological involution on X. Let σ(f)=f ◦ τ (f ∈ C(X)) be the algebra involution induced by τ on C(X), and B a complex uniform algebra on X.Ifσ(B)=B and suppose A = {h ∈ B : σ(h)=h} then A is a real uniform algebra on (X, τ) and B can be regarded as the complexification of A(see [5]). The theory of real Banach algebras was first introduced by Ingelstam [4]. Later on the theory of real uniform algebras was developed by S. H. Kulkarni and B. V. Limaye [5]. It is interesting to note that, every complex uniform algebra can be regarded as a real uniform algebra on a compact Hausdorff space whit a suitable topological involution τ. Hence the class of real uniform algebras is larger than that of complex uniform algebras. At last Alimohammadi and ebadian studied on others real algebras [1], [2].

2 Real Banach function algebras

Definition 2.1. Let X be a compact Hausdorff space, τ be a topological involution on X. A real Banach function algebra on (X, τ), is a real subalgebra Real Banach function algebras 2567

A of C(X, τ) such that separates the points of X, contains 1 and complete under an algebra .. { ∈ C 1 ≤| |≤ } Example 2.2. Let R>1 and X = z : R z R . Now define −→ − 1 the map τ : X X by τ(z)= z . Then τ is a topological involution on compact metric space X. Let B be the set of all continuous complex- valued function on X which are analytic in interior of X.Ifσ is the algebra involution on C(X) induced by τ then we can show that σ(B)=B, by the Cauchy-Riman equations. Now, set A := {h ∈ B : σ(h)=h}. Then A is a real Banach function algebras on (X, τ) and B = A ⊕ iA. Also, the map α : Carr(A) −→ Carr(B) which defines α(φ)(f + ig)=φ(f)=iφ(g), is a bijection. It is known that Carr(B)={ex : x ∈ X}, where ex is evaluation homomorphism on B at x. Therefore we have Carr(A)={ex : x ∈ X}, where ex is the evaluation homomorphism on A at x.

Theorem 2.3. Let X be a compact Hausdorff space and (B,.) be a complex Banach function algebra on X. Then there exists a compact Hausdorff space Y , a topological involution τ on Y and a real Banach function algebra (A, .) on (Y,τ) such that (B,.) is isometrically isomorphic to (A, .). In particular if B is a complex uniform algebra on X then A will be a real uniform algebra on (Y,τ). proof. Let Y = X × Z2. Clearly, Y is a compact Hausdorff space with the product topology. We define the map τ : Y −→ Y by τ(x, 0) = (x, 1) , τ(x, 1) = (x, 0) for all x ∈ X. It is easy to see to see that τ is a topological involution on Y . Now we define the map ψ : C(X) −→ C(Y,τ)byψ(f)(x, 0) = f(x),ψ(f)(x, 1) = f(x) for all x ∈ X. Then we can easily show that ψ is an isometric isomorphism of (C(X), .X), regarded as a real , onto (C(Y,τ), .Y ). set,

A = {ψ(f):f ∈ B}.

It is easy to see to see that A is a real subalgebra of C(Y,τ) which contains all real-valued constant functions on Y and separates the points of Y . For each g ∈ A, we define |g| = ψ−1(g). This implies that |.| is an algebra norm on the real algebra A and (A, |.|) is a real Banach algebra. Therefore, (A, |.|) is a real Banach function on (Y,τ). Now, we assume φ is restriction map of ψ on B. It is easy to see that 2568 Sh. Najafzadeh and A. Ebadian

φ is an isometric isomorphism of (B,.), as a real Banach algebra, onto the real Banach algebra (A, |.|), and the proof is complete. The following theorems are generalization of Theorems 1.2, 1.3.

Theorem 2.4. Let X be a compact Hausdorff space, τ be a topological involution on X and σ be the algebra involution induced by τ on C(X). If (B,.) is a complex Banach function algebra on X such that σ(B)=B and A := {h ∈ B : σ(h)=h}. Then: (i): A is a real subalgebra of B. (ii): Every h in B can be expressed uniquely as f + ig whitf, g in A. (iii): There exist a constant C ≥ 1 such that σ(h)≤Ch for all h ∈ B and max{f, g} ≤ Cf + ig for all f, g in A. (iv):(A, .) is a real Banach function algebra on (X, τ). (v): For φ in Carr(A), define α(φ)(f + ig):=φ(f)+iφ(g) for all f,g ∈ A. Then α(φ) ∈ Carr(B) and map α is bijection between Carr(A) and Carr(B). proof. (i) Obvious. h+σ(h) h+σ(h) h−σ(h) (ii) Since σ(B)=B, for all h in B, σ( 2 )= 2 and σ( 2i )= h−σ(h) (h+σ(h)) (h−σ(h)) (h+σ(h)) (h−σ(h)) 2i .Thush = 2 + i( 2i ) whit 2 , 2i in A. Further, if − (h+σ(h)) h = f + ig whit f and g in A, then σ(h)=f ig and hence f = 2 and h−σ(h) g = 2i . This proves the uniqueness of f and g. (iii) It is easy to see that every Banach function algebra is semisimple. Since σ(B)=B, we conclude that σ is an algebra involution on B,asa semisimple commutative Banach algebra. Thus σ is continuous by [3,Theorem 36.2]. Hence , there exists a positive real number C such that σ(h)≤Ch for all h ∈ B. So, h≤Cσ(h) for all h ∈ B. So, C ≥ 1. Now, for all f, g ∈ A, we have

1 1 f =  [(f + ig)+σ(f + ig)]≤ ((f + ig) + Cf + ig) ≤ Cf + ig. 2 2 Similarly, g≤Cf + ig. This proves (iii). ∞ ∞ (iv) Let {fn}n=1 be a Cauchy sequence in (A, .). Hence {fn}n=1 is a Cauchy sequence in (B,.). Since B is complete, there exists h ∈ B such that limn→∞fn − h = 0. By continuity of the real linear map σ, we have limn→∞fn − σ(h) =0,σ(h)=h. Hence h ∈ A and (A, .) is complete. Since σ(1) = 1, 1 ∈ A. By (ii), we conclude that A separates the points of X. Hence (A, .) is a real Banach function algebra on (X, τ). (v) Let φ ∈ Carr(A). It is straightforward to check that α(φ) ∈ Carr(B), Real Banach function algebras 2569

α(φ) |A= φ, and for any ψ ∈ Carr(B), ψ |A∈ Carr(A) with α(ψ |A)=ψ. this proves (v).

Theorem 2.5. Let X be a compact Hausdorff space, τ be a topological involution on X and σ be the algebra involution induced by τ on C(X). If (A, .) be a real Banach function algebra on (X, τ) and define B := {f + ig : f,g ∈ A}. Then: (i): σ(B)=B and A = {h ∈ B : σ(h)=h} = B ∩ C(X, τ). (ii): B is a complex subalgebra of C(X) and B = A ⊕ iA. (iii): There is an algebra norm |.| on B such that f = |f| for all f ∈ A and

max{f, g} ≤ |f + ig| ≤ 2max{f, g} for all f, g ∈ A. (iv) (B,|.|) is a complex Banach function algebra on X. proof. (i) Let h = f + ig ∈ B, f, g ∈ A. Since σ is an algebra involution, σ(h)=σ(f + ig)=f − ig ∈ B. Also, if σ(h)=h, then h = f ∈ A. This proves (i). (ii) Obvious. (iii) B can be viewed as the complexification of A, by (ii). Therefore by Theorem , there exists an algebra norm |.| on B such that f = |f + i0| = |f| for all f ∈ A and

max{f, g} ≤ |f + ig| ≤ 2max{f, g} for all f,g ∈ A. (iv) Since A separates the points of X and 1 ∈ A, B also separates the points of X and 1 ∈ B. Since (A, .) is Banach, (B,|.|) is also Banach by [3,theorem13.3]. Therefore (B,|.|) is a complex Banach function algebra on X and the proof is complete. In view of property (ii) of Theorem 2.4, every h can be written uniquely as h = f + ig for some f, g in A. Hence we can regard B as the complexification of A whenever A is a real algebra. { ∈ C 1 ≤| |≤ } 1 Example 2.6. Let R>1 and X = z : R z R .ByD (X), we denote the complex algebra of all complex-valued function on X whit  1 continuous derivation on X. Set h = hX + h X for all h in D (X). 1 Then (D (X), .) is a complex Banach function algebra on X. Let AP be the closure of all polynomials with real coefficients and AR be the closure of 2570 Sh. Najafzadeh and A. Ebadian all rational function with real coefficients which have poles off X in D1(X). If τ is the topological involution on X which define by τ(z)=z, then AP and AR whit the norm . are real Banach function algebras on X, τ). Set BP = {f + ig : f,g ∈ AP } and BR = {f + ig : f,g ∈ AR}. Then σ(BP )=BP and σ(BR)=BR by the above theorem. It is easy to see that BP and BR are the closure of the polynomials whit complex coefficients and the closure of the rational functions whit complex coefficients which have poles off X, respectively. It is known that,

Carr(BP )={ex : x ∈ X} = {x ∈ C : |x|≤1}.

Hence, Carr(AP )={ex : x ∈ X}.

Example 2.7. Let X = {(z, w) ∈ C2 : |z|≤1, |w|≤1} and let B be the algebra of all continuous complex-valued functions on X which have continuous first-order partial derivatives whit respect to z and w. Let we define the norm on B by

f = fX + fzX + fwX

for all f ∈ B, then (B,.) is a natural Banach function algebra on X.Now we define the topological involution on X by τ(z, w)=(z, w) for all (z, w) ∈ X and take σ(f)=f ◦ τ for all f ∈ B. Since σ(B)=B, we define

A = {f ∈ B : σ(f)=f}.

It is easy to see that (A, .) is a natural real Banach function algebra on (X, τ) which is not uniformly closed and B = {f + ig : f,g ∈ A}.

Acknowledgment: We wish to thank the Urmia University and Tabriz Uni- versity Research Council for the financial support.

References

[1] D. Alimohammadi and A. Ebadian, Hedberg’s theorem in real lips- chitz algebras , Indian J. pure appl. Math., 32(10), (2001), 1479-1493. Real Banach function algebras 2571

[2] D. Alimohammadi and A. Ebadian, Second dual of real lipschitz al- gebras of complex functions, to appear in J. Analysis Sci. Tech.

[3] F. F. Bonsall and J. Duncan, Complete normed algebras, Springer- verlag, New York. (1973).

[4] L. Ingelestam, Real Banach algebras, Ark. Math, 5 (1964), 239-342.

[5] S. H. Kulkarni and B. V. limaye, Real function algebras, Marcel Dekker, Inc. (1992).

Received: October 15, 2007