Biseparating Maps on Fréchet Function Algebras

Abstract

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.

1. 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 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 [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 𝑇∶𝐶(𝑋,𝐸)→𝐶(𝑌,𝐹) preserves zero set containments, that is, 𝑍𝑓⊆𝑍𝑔⟺𝑍𝑇𝑓⊆𝑍𝑇𝑔𝑓,𝑔∈𝐶(𝑋,𝐸), (1.1) 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 (LMC-algebra) 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,𝑝_𝑛(𝑓_𝑘−𝑓)→0 as 𝑘→∞.

Definition 1.1. 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 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 (𝑋,𝑑) be a compact metric space and 𝛼>0. The algebra of all complex-valued functions 𝑓 on 𝑋 for which 𝑝_𝛼𝑓𝑓(𝑥)𝑦=sup−𝑓𝑑^𝛼𝑥,𝑦∶𝑥,𝑦∈𝑋,𝑥≠𝑦<∞ (1.2) 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., 𝑓^(𝑘)∈Lip(𝑋,𝛼)𝑓^(𝑘)∈lip(𝑋,𝛼) for all 𝑘∈ℕ) is denoted by 𝐷^∞(𝑋) (resp., Lip^∞(𝑋,𝛼)lip^∞(𝑋,𝛼)) (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: 𝐴₀₀=,𝐼(𝑆)=𝑓∈𝐴∶supp𝑓iscompactinX𝑓∈𝐴₀₀∶,𝑀(𝑆)=(𝑆).supp𝑓∩𝑆=∅𝑓∈𝐴∶𝑓=0 (1.3) For 𝑥∈𝑋, we usually write 𝐼_𝑥 for 𝐼({𝑥}) and 𝑀_𝑥 for 𝑀({𝑥}). Note that 𝐴₀₀ is an ideal in 𝐴 and 𝐴=𝐴₀₀ whenever 𝑋 is compact.

Definition 1.3. Let 𝐴 be an Ff-algebra on a topological space 𝑋.
(i) 𝐴is 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 𝑓∈𝐴 such that 𝑓(𝐸)={1} and 𝑓(𝐹)={0}.
A commutative F-algebra is regular (normal) if 𝑆_𝐴≠∅ and 𝐴 is a regular (normal) Ff-algebra on 𝑆_𝐴.
(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 𝐴₀₀ and open neighborhoods 𝑉_𝑛 of 𝑥 such that 𝑓_𝑛∣_𝑉𝑛=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 𝐴₀₀ and open neighborhoods 𝑉_𝑛 of 𝑥 such that 𝑓_𝑛∣_𝑉𝑛=0 for all 𝑛∈ℕ, and 𝑓_𝑛𝑓→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 𝑐₀₀(𝑆)⊂𝐴⊂ℂ^𝑆, where 𝑐₀₀(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.

Example 1.4 (see [20, Example 4.5.33]). Consider the Banach space (𝐿²(𝕋),‖⋅‖₂) where 𝕋=[−𝜋,𝜋]. Let 𝑆=[−𝜋/2,𝜋/2] and set 𝑊=𝑓∈𝐿²(𝕋)∶𝑓∣_𝑆∈𝐶(𝑆). (1.4) For each 𝑓,𝑔∈𝑊 and 𝜃∈[−𝜋,𝜋], set (𝜃)=1𝑓⋆𝑔2𝜋^𝜋_−𝜋𝑓𝑔𝜙𝑓=𝑓𝜃−𝜙𝑑𝜙,₂+𝑓_𝑆=1√2𝜋^𝜋−𝜋𝑓(𝜃)²𝑑𝜃^1/2+𝑓_𝑆. (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 Bf-algebra 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 𝐹₁=4ℤ∪{∞}, 𝐹₂=(4ℤ+2)∪{∞}, and 𝐹=𝐹₁∪𝐹₂=2ℤ∪{∞}. Define 𝑔₀ on 𝕋 by 𝑔₀(𝜃)|𝜃|≤𝜋=12,𝑔₀(𝜃)𝜋=−12<|𝜃|≤𝜋. (1.6) Then, 𝑔₀∈𝑊 and 𝑔₀(𝑘)=1𝜋^𝜋₀𝑔₀(𝜃)2cos𝑘𝜃𝑑𝜃=𝑘𝜋sin𝑘𝜋2(𝑘∈ℤ,𝑘≠0), (1.7) with 𝑔₀(0)=0, and so 𝑔₀∈𝑀(𝐹). By [20, Example 4.5.33(v)], 𝑔₀∈𝑀(𝐹)⧵𝐼(𝐹). Since 𝑔₀∈𝑀(𝐹), necessarily 𝑔₀𝐼_∞⊂𝐼(𝐹), and so 𝑔₀∉𝑔₀𝐼_∞, where 𝐼_∞ is the set of all functions in 𝐴₀₀ 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 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 𝑇𝑓=ℎ⋅𝑓∘𝜑,𝑓∈𝐴, (2.1) 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 𝑇𝑓𝑦_𝜆≥𝜀>0, (2.2) 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 𝑥_𝑛 such that 𝑊_𝑛∩𝑊_𝑚=∅ if 𝑛≠𝑚. Let 𝑉_𝑛 be a neighborhood of 𝑥_𝑛 such that 𝑉_𝑛⊆𝑉_𝑛⊆𝑊_𝑛. 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 𝑈_𝑛 of 𝑥_𝑛 in 𝑋 such that 𝑈_𝑛⊆𝑉_𝑛, ℎ_𝑛=0 on 𝑈_𝑛 and 𝑝_𝑛(𝑓_𝑛−ℎ_𝑛)≤1/𝑛²(𝑝_𝑛(𝑠_𝑛)+1). If we set 𝜓_𝑛=(𝑓_𝑛−ℎ_𝑛)𝑠_𝑛, then 𝑝_𝑛(𝜓_𝑛)≤1/𝑛², and hence for each 𝑘∈ℕ, 𝑘≥2, we have ^∞_𝑛=1𝑝_𝑘𝜓_𝑛≤^𝑘−1_𝑛=1𝑝_𝑘𝜓_𝑛+^∞_𝑛=𝑘𝑝_𝑛𝜓_𝑛≤^𝑘−1_𝑛=1𝑝_𝑘𝜓_𝑛+^∞_𝑛=𝑘1𝑛²<∞. (2.3) Therefore, ∑^∞_𝑛=1𝜓_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_𝑥𝑛 and 𝑇(𝜓_𝑛−𝑓_𝑛)∈𝐼_𝑦𝑛. Consequently, 𝑇𝜓_𝑛(𝑦_𝑛)=𝑇𝑓_𝑛(𝑦_𝑛). Since the evaluation homomorphisms are continuous on 𝐴, the series ∑^∞_𝑛=1𝜓_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, 𝑇𝜓𝑦_𝑛=𝑇𝜓_𝑛𝑦_𝑛=𝑇𝑓_𝑛𝑦_𝑛=𝑛𝑇𝑓−𝑓𝑥_𝑛1𝑦_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⁻¹𝑀_𝑦⊆𝑀_𝑥 and hence 𝑇𝑀_𝑥=𝑀_𝑦. Thus ker𝛿_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=𝑇1, 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 𝑇1=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 5.1.9(iii)], 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.

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.

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.

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)𝑇is 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. 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).