Disjointness Preserving and Functional Type Disjointness Preserving Operators

Abstract

A new concept called functional type disjointness preserving operators is introduced and structure of disjointness preserving and functional type disjointness preserving operators on some function spaces are analysed.

1. Introduction

There are many articles for construction of unbounded disjointness preserving operators. The paper [1] of Abramovich and Lipecki develops techniques to construct unbounded disjointness preserving linear functionals on any infinite dimensional -lattice, and this is also mentioned in Remark  2.7 of the paper [2]. This paper [2] is devoted to construct unbounded disjointness preserving linear functionals on function spaces , when is a locally compact Hausdorff space. The paper [3] of Jeang and Wong presents a simplified procedure to construct unbounded disjointness preserving functionals on function spaces . Jarosz [4] presents a construction of disjointness preserving operators on , where is a compact Hausdorff space. Disjointness preserving operators are defined on many structures, namely, lattices, function spaces, algebras, and so forth. The present paper is restricted to study disjointness preserving operators on function spaces. A linear operator between two function spaces is said to be disjointness preserving if whenever . The second section of the paper exercises the standard techniques applicable to derive an essential structure of disjointness preserving operators. This is done on the function spaces for a normal space , and on . Although results hold for both real and complex cases, it is assumed that the results on are over real fields.

For a disjointness preserving linear functional we have or , whenever . Jarosz [4] constructed an unbounded disjointness preserving operator from onto , for any given infinite compact Hausdorff spaces and , and for any given linear subspace of such that (cardinality of continuum). These mappings have the following property: implies or . So, let us give a new name to this type of mappings.

Definition 1. A linear operator from a function space into a function space is said to be a functional type disjointness preserving (FTDP) if implies or .

An essential structure of FTDP-mappings is derived on some function spaces in Section 3.

For a real or complex valued function on , we define the cozero set of by , and zero set of by and if is a topological space, then we define the support of by , closure of cozero of . For a nonempty set , will denote the linear space of all real or complex valued functions defined on . To each , will denote the evaluation functional defined on by , for every . All the topological spaces to be considered are Hausdorff spaces. For a topological space , will denote the linear space of all real or complex valued bounded continuous functions; will denote the linear space of all real or complex valued continuous functions with compact support in ; will denote the linear space is continuous on , and for every there is a compact subset of such that , for every . Let us use the usual notation to denote the linear space of all real valued functions defined on the real line which are infinitely many times differentiable at every point in . will denote the linear space of all test functions on (see [5]). That is, has compact support in . Let us use the notation to denote the linear space: for every , there is a compact subset of such that , for all . We consider the linear spaces , and as normed spaces with the supremum norms on them. If is normal, then is dense in . For if and , then there is a such that , , has value on , has value on , , and . Similarly, is also dense subspace of , which follows from Lemma 1 of Section 1.8 in [6].

2. Disjointness Preserving Mappings

The following theorem is a variation of the results obtained by Jarosz [4] and by Jeang and Wong [3].

Theorem 2 (see [3, 4]). Let be a linear disjointness preserving mapping from (or ) to where is a compact Hausdorff space (or locally compact Hausdorff space ) and is a nonempty set. Let Then there is a function (or is the one point compactification of ) such that for all (or ), for some fixed scalar , and for all such that on (or ) , for all . Moreover, if every function in the range of is a bounded function on , then is a finite subset of (or , resp., and for ).

These results depend on the Urysohn lemma and existence of partitions of unity. Let us also use them in deriving the following theorem.

Theorem 3. Let be a normal space and be a nonempty set. Let be a linear disjointness preserving mapping from into . Let , , and be defined as in Theorem 2. Then there is a function such that on for some fixed scalar , and for all such that on , for all .

Proof. To each , define for every open neighbourhood of , there is an such that and . First, we claim that to each , contains at most one point. On the contrary, suppose for some , with . Let and be two disjoint open neighbourhoods of and respectively. Then there are such that and , . This is a contradiction to the assumption that is disjointness linear preserving. So, contains at most one point, for every . We next claim that is nonempty for every . On the contrary we assume that is empty for some . Note that . Fix . Let . To each , let be an open neighbourhood of such that whenever and . Then we can find a finite subfamily of such that the subfamily covers . Let be a continuous decomposition of the identity subordinate to . Then, by the Tietze extension theorem, there are functions in such that and , for every . Then and and hence , for every . So, . Thus on , which is a contradiction. Hence is a singleton set for each . Define a function by . To each , write , and . If and , then and there is a with such that . Since is disjointness preserving,  . Thus . Since is normal, if , , and , then there is a such that , , has value on , and has value on , and . Thus is dense in . If , then is closed and hence . So and on , for some . This proves the theorem.

Corollary 4. Let be a normal space and a nonempty set. Let be a linear disjointness preserving mapping from to . Suppose is continuous on , for every . Let Then there is a function such that on for some fixed scalar and for all .

Proof. Consider the restriction of to . Then we conclude that on , for some fixed scalar , because on if and only if on . Continuity assumption on also implies that the relation is true on also because is dense in .
The classical Urysohn lemma and the result on existence of partitions of unity have their version in (see: [6, Section  1.8 Lemma  1] and [5, Theorem  6.20]). So we have the following version of the previous theorem to the space .

Theorem 5. Let be a nonempty set. Let be a linear disjointness preserving mapping from to . Let , , and be defined as in Theorem 2. Then there is a function such that on for some fixed scalar and for all and on , for all .

Proof. To each , define for every open neighbourhood of , there is an such that and . It is easy to verify that contains at most one point, for every . We next claim that is nonempty for every . On the contrary we assume that is empty for some . Note that . Fix . Let . To each , let be an open neighbourhood of such that whenever and . Then we can find a finite subfamily of such that the subfamily covers . By [5, Theorem  6.20] there are functions in such that , for every , on and , for all . Then and and hence , for every . So, . Thus on , which is a contradiction. Hence is a singleton set for each . Define a function by . To each , write , and . If and , then and there is a with such that . Since is disjointness preserving,. Thus . If , , and , then by [6, Lemma 1 of Section  1.8] there is a such that , , has value on , and has value on , and . Thus is dense in . If , then is closed and hence . So and on , for some . This proves the theorem.

Observe that is dense in in view of [6, Lemma  1 in Section  1.8]. So, we have the following corollary.

Corollary 6. Let be a nonempty set. Let be a linear disjointness preserving mapping from to . Suppose is continuous on , for every . Let , be defined as in Corollary 4. Then there is a function such that on for some fixed scalar , and for all .

Remark 7. One may change into any open region in in Theorem 5 and Corollary 6. may be replaced in Corollary 6 by any of its linear subspace containing . may be replaced in Corollary 4 by any of its linear subspace containing .

3. FTDP-Mappings

Theorem 8. Let be a compact Hausdorff space and a nonempty set. Let be a nonzero disjointness preserving mappings. Let , and be defined as in Theorem 2. Consider the following statements. (i) is a FTDP-mapping.(ii)There is a unique such that, for every open neighbourhood of , there is a with such that in .(iii) is a singleton subset of .(iv), in is a singleton set. Then the implications (i)(ii), (i)(iii), (i)(iv), and (iii)(ii) are true. If (i) is true, then the singleton sets in (iii) and (iv) are for given in (ii). If (ii) is true, then .

Proof. (i)(ii): Suppose is a FTDP-mapping. Suppose there are two distinct points with the property given in (ii). We can find two disjoint open neighbourhoods , of , , respectively. Then there are with such that and . This is a contradiction. So, uniqueness of in (ii) is established. If there is no in with the property mentioned in (ii), then for each , there is a neighbourhood of such that in whenever . Then we find a subcover of for . Then we find a partition for unity in such that , . Then for every , we have . Thus on , a contradiction. So (ii) is true.
Note that is nonempty, because on . Moreover if and only if on . We claim that . On the contrary, we assume that . Find an open neighbourhood of and a function such that , in , and . Fix any . Let . Then there is an open neighbourhood of such that . Find a function such that and in . So , because is disjointness preserving. Thus , for any . So , where in . Therefore, we conclude that , if (ii) is true.
(i)(iii): Suppose is a FTDP-mapping. Since , is nonempty. Suppose for some . Find two disjoint open sets and such that and . Find such that , , and , . Then in , but and in . This is a contradiction. So, is a singleton set, and .
(i)(iv): Suppose (i) is true. Consider given in (ii). Suppose in , and . Then find such that and in . Thus , and , a contradiction. Thus if . If and , find two disjoint open neighbourhoods and of and , respectively. Find such that and . Then . So, , in , where the member mentioned in (ii). This proves (iv).
(iii)(ii): Suppose (iii) is true, and suppose . Then for every and for every open neighbourhood of there is a function such that and , and hence in . Let be such that . Find two disjoint neighbourhoods of , of , , respectively. Fix  arbitrarily. Find a function such that and . Then for every function with , we have , and hence , since is disjointness preserving. Thus for every function with , we have , for every , and hence we have in . This of course proves (ii), and it is proved that is the member mentioned in (ii).

Corollary 9. Let be a compact Hausdorff space and be a nonempty set. Let a nonzero FTDP- mapping. Then the following are equivalent.(i) is continuous for every .(ii) on for some fixed function , for some fixed , and for all .

Proof. Suppose (i) is true. Suppose . Then for every , we have , and hence . Thus, . This proves (ii). Another implication is obvious.

Remark 10. Theorem 8 is extendable to when is normal and to . Corollary 9 is extendable to , , , and when is normal. Theorem 8 is extendable to when is locally compact with an additional assumption that the singleton set may be in . Corollary 9 is extendable to when is locally compact.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.