Biseparating Maps on Fréchet Function Algebras
Let and be strongly regular normal Fréchet function algebras on compact Hausdorff spaces and , respectively, such that the evaluation homomorphisms are continuous on and . Then, every biseparating map is a weighted composition operator of the form , where is a homeomorphism from onto and is a nonvanishing element of . In particular, is automatically continuous.
1. Introduction and Preliminaries
Assume that and are spaces of complex functions on topological spaces and , respectively. A linear map is called separating or disjointness preserving whenever implies , for all , where the cozero set of an element is defined by . Equivalently, a linear map is separating if for every , the equality implies the equality . Moreover, is called biseparating if it is bijective and both and 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 , Jarosz has studied separating maps between spaces of continuous scalar-valued functions. He showed that if and are compact Hausdorff spaces, , the space of all continuous scalar-valued functions on and , then every bijective separating map is a weighted composition operator of the form , and , where is a homeomorphism from onto and is a nonvanishing continuous complex-valued function on . Later, Font extended this result to the case where and are regular commutative semisimple Banach algebras satisfying Ditkin’s condition . On the other hand, Gau et al. used an algebraic method to study separating maps between spaces of continuous scalar as well as vector-valued 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  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 and are Banach function algebras on compact Hausdorff spaces and , respectively, and is a unital cozero preserving map, then 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 preserves zero set containments, that is, where are realcompact or metric spaces and are locally convex spaces . In fact, preserves zero set containments if and only if and 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 (LMC-algebra) which is also a complete metrizable space. The topology of a Fréchet algebra can be defined by an increasing sequence of submultiplicative seminorms and without loss of generality; we may assume that , for all , if has unit. An F-algebra with a defining sequence of seminorms is denoted by . The set of all characters (nonzero complex homomorphisms) of an F-algebra is denoted by , and the continuous character space, or the spectrum of , denoted by , is the set of all continuous characters on . We always endow and with the Gelfand topology, and is the set of all Gelfand transforms of elements in . The algebra is called functionally continuous whenever .
Note that a sequence in an F-algebra converges to an element if and only if for each as .
Definition 1.1. Let be a nonempty topological space. A subalgebra of is a function algebra on if contains the constants and separates the points of . The algebra is a Fréchet function algebra (Ff-algebra) or a Banach function algebra (Bf-algebra) on if 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 be an Ff-algebra (Bf-algebra) on such that the evaluation homomorphisms are all continuous, where for and . It is clear that the map 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 is a natural Ff-algebra (Bf-algebra) on , and we identify with , 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 ], the class of natural Ff-algebras 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 be a compact metric space and . The algebra of all complex-valued functions on for which
is denoted by , and its subalgebra of those functions with the property is denoted by . It is known that for and for are Bf-algebras on under the norm .
In the case that is a perfect compact plane set which is a finite union of regular sets (see  for the definition), the algebra of all functions with derivatives of all orders (resp., for all ) is denoted by (resp., ) (see, e.g., ). It is interesting to note that and are natural Ff-algebras on which are not Bf-algebras.
For a function algebra on a nonempty topological space and for each nonempty closed subset of , we consider the following subsets of : For , we usually write for and for . Note that is an ideal in and whenever is compact.
Definition 1.3. Let be an Ff-algebra on a topological space .
is said to be regular on if for any nonempty closed subset of and each , there exists such that and , and it is normal if for each nonempty closed subset and nonempty compact subset of with , there exists such that and .
A commutative F-algebra is regular (normal) if and is a regular (normal) Ff-algebra on .
is said to be a strongly regular algebra if for every and with , there exists a sequence in and open neighborhoods of such that for all , and as , or equivalently, for each .
An Ff-algebra on is said to satisfy Ditkin’s condition if for every and with , there exists a sequence in and open neighborhoods of such that for all , and as , or equivalently, for all and .
It is clear that every strongly regular algebra is regular. Moreover, if an Ff-algebra satisfies Ditkin’s condition, then 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 is a Banach algebra such that , where is the linear span of the set consisting of all characteristic functions of the singleton subsets of .
Example 1.4 (see [20, Example ]). Consider the Banach space where . Let and set
For each and , set
Then, is a commutative Banach algebra (for an equivalent norm).
Identifying with its algebra of Fourier transforms on is a strongly regular Banach sequence algebra on . Moreover, , the unitization of , is a strongly regular Bf-algebra on , the one point compactification of . Now, we show that does not satisfy Ditkin’s condition. In the following, we write for the subset of . Set , , and . Define on by Then, and with , and so . By [20, Example (v)], . Since , necessarily , and so , where is the set of all functions in which are zero on a neighborhood of .
2. 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 and be compact Hausdorff spaces, and normal Ff-algebras on and , respectively, and a biseparating map. Then, for each , there exists a unique such that . If we define by , then 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 and be strongly regular normal Ff-algebras on compact Hausdorff spaces and , respectively, such that the evaluation homomorphisms on and are continuous. Then, every biseparating map is a weighted composition operator of the form where is a homeomorphism from onto and is a nonvanishing element of . In particular, is automatically continuous.
Proof. By Lemma 2.1, there exists a homeomorphism from onto defined by , where . We first show that . Suppose on the contrary that there exists such that . If belongs to the interior of , then , and thus , since . Therefore, we may assume that there exists a net of distinct elements of converging to such that is never zero. Consider the net in such that . Clearly, converges to , and by passing through a subnet if necessary, we may assume that there exists a constant such that
for all . Since , we can find a subsequence such that . Since , it follows that . By the normality of , there exists a neighborhood of such that if . Let be a neighborhood of such that . Consider the sequence in such that , for each . By (2.2), without loss of generality, we may assume that for some positive and for all . Since is normal, for each , there exists such that on and on . If we take , then , and since is strongly regular, we can find in and a neighborhood of in such that , on and . If we set , then , and hence for each , , we have
Therefore, converges to an element . On the other hand, for each , on , which implies that and . Consequently, . Since the evaluation homomorphisms are continuous on , the series converges to for each . Hence, on , since the elements of the sequence are pairwise disjoint and . Therefore,
for all , which is a contradiction, since and . Therefore, .
By a similar argument, we can show that and hence . Thus , and so there exists a scalar such that . Equivalently, for all in and in . In particular, when , we have , which is a nonvanishing element of , since is surjective.
Definition 2.3. Let and be Ff-algebras on compact Hausdorff spaces and respectively. A linear map is called cozero preserving, whenever implies .
In , 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 and be Bf-algebras on compact Hausdorff spaces and , respectively, such that is inverse closed. If is a unital cozero preserving surjective map, then is automatically continuous.
Proof. Let . Then, , and so . Therefore, we have . Since and is cozero preserving, we conclude that . Since is inverse closed and for all , it follows that , which implies that , for every . Thus by [20, Theorem (iii)], , and hence is automatically continuous.
We now adopt a similar method as in the proof of [6, Lemma 3.3] to obtain the following results.
Theorem 2.5. Let and be function algebras on compact Hausdorff spaces and , respectively, and a cozero preserving injection. If is regular, then is separating.
Proof. Suppose on the contrary that there exist and in such that but for some . So, we can find an open neighborhood of such that . Since is regular, there exists such that and . It is clear that . So by hypothesis, . On the other hand, implies that . It follows that , that is, . Now injectivity of shows that , which is a contradiction. Therefore, is separating.
Corollary 2.6. Let and be strongly regular normal Ff-algebras on compact Hausdorff spaces and , respectively, such that evaluation homomorphisms are continuous on and . If is a linear bijection, then the following statements are equivalent: (i)is separating and cozero preserving; (ii) is biseparating; (iii) and are both cozero preserving; (iv) and are weighted composition operators.
Proof. It suffices to prove . The other implications are direct consequences of Theorems 2.2 and 2.5. If is satisfied, then all hypotheses of Theorem 2.2 are satisfied. Let for and , we have and . By Theorem 2.2, . Since , it follows that , and hence , that is, . Thus, and consequently .
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