Biseparating Maps on Fréchet Function Algebras

Let A and B be strongly regular normal Fréchet function algebras on compact Hausdorff spaces X and Y , respectively, such that the evaluation homomorphisms are continuous on A and B. Then, every biseparating map T : A → B is a weighted composition operator of the form Tf h · f ◦φ , where φ is a homeomorphism from Y onto X and h is a nonvanishing element of B. In particular, T is automatically continuous.


Introduction and Preliminaries
Assume that A and B are spaces of complex functions on topological spaces X and Y , respectively.A linear map T : A → B is called separating or disjointness preserving whenever coz f ∩ coz g ∅ implies coz Tf ∩ coz Tg ∅, for all f, g ∈ A, where the cozero set of an element f ∈ A is defined by coz f {x ∈ X : f x / 0}.Equivalently, a linear map T : A → B is separating if for every f, g ∈ A, the equality f • g 0 implies the equality Tf •Tg 0.Moreover, T is called biseparating if it is bijective and both T and T −1 are separating.
The concept of disjointness preserving operators was introduced for the first time in 1940s see 1, 2 .Since then, many authors have extended this concept to various kinds of Banach algebras.For example in 3 , Jarosz has studied separating maps between spaces of continuous scalar-valued functions.He showed that if X and Y are compact Hausdorff spaces, A C X , the space of all continuous scalar-valued functions on X and B C Y , then every bijective separating map T : A → B is a weighted composition operator of the form Tf y h y f ϕ y , y ∈ Y and f ∈ A, where ϕ is a homeomorphism from Y onto X and h is a nonvanishing continuous complex-valued function on Y .Later, Font extended this result to the case where A and B are regular commutative semisimple Banach algebras satisfying Ditkin's condition 4 .On the other hand, Gau et al. used an algebraic method to study separating maps between spaces of continuous scalar as well as vectorvalued functions in 5, 6 .For more information about separating maps one, can refer to 7-15 .
In this paper, we generalize the results of Jarosz in 3 to Fréchet function algebras using a similar method as in 5, 6 .Then, we define the concept of a cozero preserving map and show that if A and B are Banach function algebras on compact Hausdorff spaces X and Y , respectively, and T : A → B is a unital cozero preserving map, then T is automatically continuous.Finally, we will find the relation between cozero preserving, separating and biseparating maps between certain Fréchet function algebras.Recently, Li and Wong have obtained several Banach-Stone type theorems for the vector-valued functions, specially in the case that the bijective linear map where X, Y are realcompact or metric spaces and E, F are locally convex spaces 16 .In fact, T preserves zero set containments if and only if T and T −1 are cozero preserving.In Corollary 2.6, we obtain similar results for Ff-algebras.We now present some definitions and known results which we need in the sequel.A Fréchet algebra F-algebra is a locally multiplicatively convex algebra LMCalgebra A which is also a complete metrizable space.The topology of a Fréchet algebra can be defined by an increasing sequence p n of submultiplicative seminorms and without loss of generality; we may assume that p n 1 1, for all n ∈ N, if A has unit.An F-algebra A with a defining sequence of seminorms p n is denoted by A, p n .The set of all characters nonzero complex homomorphisms of an F-algebra A, p n is denoted by S A , and the continuous character space, or the spectrum of A, p n , denoted by M A , is the set of all continuous characters on A. We always endow S A and M A with the Gelfand topology, and A is the set of all Gelfand transforms f of elements f in A. The algebra A is called functionally continuous whenever S A M A .
Note that a sequence f k k in an F-algebra A, p n converges to an element f ∈ A if and only if for each n ∈ N, Let X be a nonempty topological space.A subalgebra A of C X is a function algebra on X if A contains the constants and separates the points of X.The algebra A is a Fréchet function algebra Ff-algebra or a Banach function algebra Bf-algebra on X if A is a function algebra which is also an F-algebra or a Banach algebra, respectively, with respect to some topology.
Clearly every Bf-algebra is a Ff-algebra.Let A be an Ff-algebra Bf-algebra on X such that the evaluation homomorphisms δ x : A → C are all continuous, where δ x f f x for f ∈ A and x ∈ X.It is clear that the map J : X → M A , x → δ x is continuous and injective.If this map is also surjective and its inverse is continuous, then it is a homeomorphism, and in this case, we say that A is a natural Ff-algebra Bf-algebra on X, and we identify X with M A , through this map.
Note that the evaluation homomorphisms are always continuous in Bf-algebras, but they may not be continuous in Ff-algebras.By 17, Lemma 3.2.5 , the class of natural Ffalgebras and the class of unital commutative semisimple Fréchet algebras are the same.Moreover, all Ff-algebras as well as Bf-algebras are semisimple.
Example 1.2.Let X, d be a compact metric space and α > 0. The algebra of all complexvalued functions f on X for which is denoted by Lip X, α , and its subalgebra of those functions with the property lim d x,y → 0 |f x − f y |/d α x, y 0 is denoted by lip X, α .It is known that Lip X, α for 0 < α ≤ 1 and lip X, α for 0 < α < 1 are Bf-algebras on X under the norm f α f X p α f .In the case that X is a perfect compact plane set which is a finite union of regular sets see 18 for the definition , the algebra of all functions f with derivatives of all orders resp., see, e.g., 19 .It is interesting to note that D ∞ X , Lip ∞ X, α and lip ∞ X, α are natural Ff-algebras on X which are not Bf-algebras.
For a function algebra A on a nonempty topological space X and for each nonempty closed subset S of X, we consider the following subsets of A: For x ∈ X, we usually write I x for I {x} and M x for M {x} .Note that A 00 is an ideal in A and A A 00 whenever X is compact.
Definition 1.3.Let A be an Ff-algebra on a topological space X. i A is said to be regular on X if for any nonempty closed subset S of X and each x ∈ X \ S, there exists f ∈ A such that f x 1 and f S {0}, and it is normal if for each nonempty closed subset E and nonempty compact subset F of X with E ∩ F ∅, there exists ii A is said to be a strongly regular algebra if for every f ∈ A and x ∈ X with f x 0, there exists a sequence {f n } in A 00 and open neighborhoods V n of x such that f n | V n 0 for all n ∈ N, and f n → f as n → ∞, or equivalently, M x I x for each x ∈ X.
iii An Ff-algebra A on X is said to satisfy Ditkin's condition if for every f ∈ A and x ∈ X with f x 0, there exists a sequence {f n } in A 00 and open neighborhoods V n of x such that f n | V n 0 for all n ∈ N, and f n f → f as n → ∞, or equivalently, f ∈ fI x for all x ∈ X and f ∈ M x .
It is clear that every strongly regular algebra is regular.Moreover, if an Ff-algebra satisfies Ditkin's condition, then A is strongly regular.In general, the converse is not true as the following example shows.
Recall that a Banach sequence algebra on a nonempty set S is a Banach algebra A such that c 00 S ⊂ A ⊂ C S , where c 00 S is the linear span of the set {χ s : s ∈ S} consisting of all characteristic functions χ s of the singleton subsets {s} of S. For each f, g ∈ W and θ ∈ −π, π , set

1.5
Then, W, , • is a commutative Banach algebra for an equivalent norm .
Identifying W with its algebra of Fourier transforms on Z, W is a strongly regular Banach sequence algebra on Z.Moreover, W # , the unitization of W, is a strongly regular Bfalgebra on Z ∞ , the one point compactification of Z. Now, we show that W # does not satisfy Ditkin's condition.In the following, we write rZ s for the subset {rn s : n ∈ Z} of Z. Set F 1 4Z ∪ {∞}, F 2 4Z 2 ∪ {∞}, and F F 1 ∪ F 2 2Z ∪ {∞}.Define g 0 on T by Then, g 0 ∈ W and with g 0 0 0, and so g 0 ∈ M F .By 20, Example 4.5.33 v , g 0 ∈ M F \ I F .Since g 0 ∈ M F , necessarily g 0 I ∞ ⊂ I F , and so g 0 / ∈ g 0 I ∞ , where I ∞ is the set of all functions in A 00 which are zero on a neighborhood of ∞.

Main Results
We first state the following useful result, which is, in fact, the generalization of 6, Lemma 2.1 and Theorem 2.2 .Lemma 2.1.Let X and Y be compact Hausdorff spaces, A and B normal Ff-algebras on X and Y , respectively, and T : A → B a biseparating map.Then, for each x ∈ X, there exists a unique y ∈ Y such that TI x I y .If we define ϕ : Y → X by ϕ y x, then ϕ is a homeomorphism.
Proof.We omit the proof, since it is similar to the proofs of 6, Lemma 2.1 and Theorem 2.2 .
We now bring the following theorem, which is an extension of the results of Jarosz and Font.
Theorem 2.2.Let A, p n and B, q n be strongly regular normal Ff-algebras on compact Hausdorff spaces X and Y , respectively, such that the evaluation homomorphisms on A and B are continuous.Then, every biseparating map T : A → B is a weighted composition operator of the form where ϕ is a homeomorphism from Y onto X and h is a nonvanishing element of B. In particular, T is automatically continuous.
Proof.By Lemma 2.1, there exists a homeomorphism ϕ from Y onto X defined by ϕ y x, where TI x I y .We first show that TM x ⊆ M y .Suppose on the contrary that there exists f ∈ M x such that Tf y / 0. If x belongs to the interior of f −1 0 , then f ∈ I x , and thus Tf y 0, since TI x I y .Therefore, we may assume that there exists a net {x λ } λ of distinct elements of X converging to x such that f x λ is never zero.Consider the net {y λ } λ in Y such that ϕ y λ x λ .Clearly, y λ converges to y, and by passing through a subnet if necessary, we may assume that there exists a constant ε such that for all λ.Since f x λ → 0, we can find a subsequence {f x n } such that f x n → 0. Since x λ → x, it follows that x n → x.By the normality of X, there exists a neighborhood W n of Consider the sequence {y n } in Y such that ϕ y n x n , for each n ∈ N. By 2.2 , without loss of generality, we may assume that |T f − f x n 1 y n | ≥ δ for some positive δ and for all n ∈ N. Since A is normal, for each n ∈ N, there exists s n ∈ A such that s n 1 on V n and s n 0 on X \ W n .If we take f n n f − f x n 1 , then f n x n 0, and since A is strongly regular, we can find h n in A and a neighborhood Therefore, ∞ n 1 ψ n converges to an element ψ ∈ A. On the other hand, for each n ∈ N, ψ n f n on U n , which implies that ψ n −f n ∈ I x n and T ψ n −f n ∈ I y n .Consequently, Tψ n y n Tf n y n .Since the evaluation homomorphisms are continuous on A, the series ∞ n 1 ψ n x converges to ψ x for each x ∈ X.Hence, ψ ψ n on W n , since the elements of the sequence {W n } are pairwise disjoint and cozψ n ⊂ W n .Therefore,

Tψ y n
Tψ n y n Tf n y n n T f − f x n 1 y n ≥ nδ, 2.4 for all n ∈ N, which is a contradiction, since y n ϕ −1 x n → ϕ −1 x and Tψ y n → Tψ ϕ −1 x .Therefore, TM x ⊆ M y .
By a similar argument, we can show that T −1 M y ⊆ M x and hence TM x M y .Thus ker δ x ker δ y • T , and so there exists a scalar h y such that δ y • T h y δ x .Equivalently, Tf y h y f ϕ y for all f in A and y in Y .In particular, when f 1, we have h T 1, which is a nonvanishing element of B, since T is surjective.Definition 2.3.Let A and B be Ff-algebras on compact Hausdorff spaces X and Y respectively.A linear map T : A → B is called cozero preserving, whenever coz f ⊆ coz g implies coz Tf ⊆ coz Tg .
In 21 , Font has studied the automatic continuity of cozero preserving maps between Fourier algebras.In the following theorems, we generalize the results of Font to Bf-algebras as well as Ff-algebras.Theorem 2.4.Let A and B be Bf-algebras on compact Hausdorff spaces X and Y , respectively, such that B is inverse closed.If T : A → B is a unital cozero preserving surjective map, then T is automatically continuous.

Proof. Let λ /
∈ sp A f .Then, λ1 − f ∈ Inv A , and so coz λ1 − f X.Therefore, we have X coz 1 ⊆ coz λ1 − f .Since T 1 1 and T is cozero preserving, we conclude that Y coz 1 ⊆ coz λ1 − Tf .Since B is inverse closed and λ1 − Tf y / 0 for all y ∈ Y , it follows that λ / ∈ sp B Tf , which implies that sp B Tf ⊆ sp A f , for every f ∈ A. Thus by 20, Theorem 5.1.9iii , S T ⊆ rad B {0}, and hence T is automatically continuous.
We now adopt a similar method as in the proof of 6, Lemma 3.3 to obtain the following results.Proof.Suppose on the contrary that there exist f and g in A such that TfTg 0 but f x g x / 0 for some x ∈ X.So, we can find an open neighborhood V of x such that V ⊆ coz f ∩ coz g .Since A is regular, there exists h ∈ A such that h x 1 and h| X\V 0. It is clear that coz h ⊆ coz f ∩ coz g .So by hypothesis, coz Th ⊆ coz Tf ∩ coz Tg .On the other hand, TfTg 0 implies that coz Tf ∩ coz Tg ∅.It follows that coz Th ∅, that is, Th 0. Now injectivity of T shows that h 0, which is a contradiction.Therefore, T −1 is separating.Let for f, g ∈ A and y ∈ Y , we have coz f ⊆ coz g and y / ∈ coz Tg .By Theorem 2.2, TM x M y .Since Tg y 0, it follows that g x 0, and hence f x 0, that is, f ∈ M x .Thus, Tf y 0 and consequently y / ∈ coz Tf .

Theorem 2 . 5 .
Let A and B be function algebras on compact Hausdorff spaces X and Y , respectively, and T : A → B a cozero preserving injection.If A is regular, then T −1 is separating.

Corollary 2 . 6 .
Let A and B be strongly regular normal Ff-algebras on compact Hausdorff spaces X and Y , respectively, such that evaluation homomorphisms are continuous on A and B. If T : A → B is a linear bijection, then the following statements are equivalent: i T is separating and cozero preserving;ii T is biseparating;iii T and T −1 are both cozero preserving; iv T and T −1 are weighted composition operators.Proof.It suffices to prove ii ⇒ i .The other implications are direct consequences of Theorems 2.2 and 2.5.If ii is satisfied, then all hypotheses of Theorem 2.2 are satisfied.