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 continuous function 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 uniform algebra 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 norm .. { ∈ 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 Banach algebra, 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).