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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 284875, 14 pages
http://dx.doi.org/10.1155/2010/284875
Research Article

The Near-Ring of Lipschitz Functions on a Metric Space

1Department of Mathematics, Fairleigh Dickinson University, 1000 River Rd., Teaneck, NJ 07666, USA
2Department of Computer Science, University of Stellenbosch, Stellenbosch 7602, South Africa

Received 6 August 2009; Revised 3 April 2010; Accepted 25 April 2010

Academic Editor: Francois Goichot

Copyright © 2010 Mark Farag and Brink van der Merwe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper treats near-rings of zero-preserving Lipschitz functions on metric spaces that are also abelian groups, using pointwise addition of functions as addition and composition of functions as multiplication. We identify a condition on the metric ensuring that the set of all such Lipschitz functions is a near-ring, and we investigate the complications that arise from the lack of left distributivity in the resulting right near-ring. We study the behavior of the set of invertible Lipschitz functions, and we initiate an investigation into the ideal structure of normed near-rings of Lipschitz functions. Examples are given to illustrate the results and to demonstrate the limits of the theory.

1. Introduction and Background

Banach spaces and Banach algebras of scalar-valued Lipschitz functions on a metric space have been studied in some depth by functional analysts for the past half of a century. The papers of Arens and Eells [1], de Leeuw [2], Sherbert [3, 4], and Johnson [5] contain some of the important early work on these topics. The book of Weaver [6] provides a systematic treatment of both the analytic and algebraic results concerning spaces of scalar-valued Lipschitz functions on a metric space. The Lipschitz functions considered therein are usually bounded and map a metric space to a Banach space (often or ), so that addition (and multiplication in case or ) of functions is defined. The Lipschitz number of a function , denoted , is used in combination with the infinity norm to produce a norm . If one identifies a distinguished basepoint in , then is used as a norm on the set of basepoint preserving Lipschitz functions mapping to .

In this paper, we initiate an analogous study of zero-preserving Lipschitz functions on a metric space that is also an abelian group, using pointwise addition of functions as addition and function composition as multiplication. Our Lipschitz functions, therefore, map a metric space to itself and may be regarded as a generalization of the bounded linear operators that are so important in analysis. Rather than taking to be a Banach space, we only require to be an abelian group. By restricting the possible metrics on , we ensure that the set of zero-preserving Lipschitz functions on is a near-ring under pointwise addition of functions and function composition. Near-rings and near-algebras, the nonlinear counterparts of rings and algebras, respectively, have a rich theory of their own. Basic near-ring definitions and results can be found in the books of Pilz [7], Clay [8], and Meldrum [9]; the dissertation of Brown [10] is the seminal work in near-algebras. Closely related to our present work is the dissertation of Irish [11], which considers near-algebras of Lipschitz functions on a Banach space.

In the next section, we give the required definitions and elementary results. Next, we study the behavior of the set of units and also of ideals in normed near-rings of Lipschitz functions under topological closure. We conclude the paper by investigating the ideal structure of near-rings of Lipschitz functions.

2. Definitions, Notation, and Elementary Results

We begin this section by recalling the definition of a Lipschitz function.

Definition 2.1 (see [6, 12]). A function from a metric space to a metric space is Lipschitz if there exists a constant such that for all , . If is Lipschitz, the Lipschitz number of is defined as

Remark 2.2. If is Lipschitz, then . Also, if and only if is constant. For Lipschitz functions and , we have that .

Remark 2.3. It is well-known that a Lipschitz function is absolutely continuous and therefore differentiable almost everywhere (a.e.). Also, the derivative of is bounded a.e. in magnitude by the Lipschitz constant, and for , the difference is equal to the integral of the derivative of on the interval . Conversely, if is absolutely continuous (and thus differentiable a.e.) and if a.e., then is Lipschitz with Lipschitz constant at most . We will only use these observations in the case where is a continuous function from to that is everywhere, except possibly for finitely many points, differentiable; also, the derivative of will be continuous at all points where it exists. In this case is Lipschitz if and only if the set is differentiable at is bounded and In this restricted case these facts about real-valued Lipschitz functions are elementary consequences of the definition of Lipschitz functions, the definition of derivatives of real-valued functions, and the mean value theorem. The interested reader can consult [12] or [6] for more on Lipschitz functions.

In what follows, all metric spaces are also abelian groups under the operation , with identity element . We exclude the trivial case . If the metric satisfies the condition in the next definition, the pointwise addition of two Lipschitz functions is again a Lipschitz function.

Definition 2.4. Let be a real number. A metric on a metric space is -subadditive on if for all with , bounded, and

Remark 2.5. If is a metric on and we define the metric on via , then -subadditivity of is equivalent to having the Lipschitz number of equal to .

Example 2.6. Let be a normed vector space and let the metric be defined on by for all . Then is 1-subadditive.

Assume that is a 1-subaddive metric on the abelian group , and define by for all . We will show that satisfies the properties given in the next definition.

Definition 2.7 (see, e.g., [13]). A function is a norm on the abelian group if satisfies the following criteria: (1) if and only if ; (2) for all ; (3) for all .

Remark 2.8. Let be a norm on an abelian group. Then the function , defined by for all , is a 1-subadditive metric on .

Example 2.9. Let be a multiplicative subgroup of the unit circle in the complex plane, and take “addition” in to be complex multiplication (so that is the neutral element). If we denote by the Euclidean distance between the complex numbers and , is a norm on the abelian group .

Next we give some of the elementary properties of -subadditive metrics.

Proposition 2.10. Assume that is -subadditive on the metric space . Then(1) and ; (2) and ; (3); (4)if , then and ; (5)if , then defined by for all , is a norm on and for all .

Proof. We prove () and (). The other parts follow immediately from the following:
() and
() Thus and therefore also .

Remark 2.11. Note that from part () of Proposition 2.10 and Remark 2.8 we have that any 1-subadditive metric on an abelian group is induced by a norm on and, conversely, any norm on induces a 1-subadditive metric.

The following is an example of a metric space with a metric that is not -subadditive for any .

Example 2.12. Consider the metric on given by for . Then if is -subadditive, we would have that for all , which is clearly not possible.

Notation 2.13. For a metric space , we denote by , or simply when there is no ambiguity about the metric, the set of zero-preserving Lipschitz functions on .

Using some of the elementary properties of -subadditive metrics, we obtain the following properties for .

Proposition 2.14. Let be a -subadditive metric on . Then for all one has (1) and if and only if ; (2); (3); (4).

Proof. The result follows from Definitions 2.1 and 2.4 and Proposition 2.10.

We now recall the definition of a near-ring.

Definition 2.15. A triple () is called a (right) near-ring if (1) is a (not necessarily abelian) group, (2) is a semigroup, (3)for all , . A near-ring is called zero-symmetric if, for all , , where is the neutral element of .
If is any group, then , the set of all self-maps of , is a near-ring under pointwise addition and function composition. The set of all zero-preserving self-maps of , , is a zero-symmetric sub-near-ring of . Further examples of near-rings, along with many of the basic results of the theory of near-rings, may be found in the books of Clay [8], Meldrum [9], and Pilz [7].

Following the definition of a normed ring as given in [14], we make the following analogous definition.

Definition 2.16. A normed near-ring is a near-ring with a function , such that(1) if and only if ; (2) for all ; (3) for all ; (4) for all .

Proposition 2.17. Assume that is a -subadditive metric on . Then(1) (“+” is pointwise addition and “” is function composition) is a zero-symmetric near-ring with identity; (2) is a normed near-ring if .

Proof. () We show that if , then . From Remark 2.2, , and from Proposition 2.10, we have for , and thus implies that . Also, since is -subadditive. Thus implies that . Finally, note that since contains only zero-preserving functions, it is a zero-symmetric near-ring.
(2) This result follows from () and Proposition 2.14.

In some of our results, we assume that our metric space is a normed vector space over a field with an absolute value or norm. Recall that an absolute value on a field is a function , such that(1) if and only if ;(2) for all ;(3) for all .

If is a normed vector space over a normed field, is not only a normed near-ring but in fact a normed near-algebra. First we recall the definitions of a near-algebra and a normed near-algebra. We only consider near-algebras with identity.

Definition 2.18 (see [7]). A vector space over a field together with another binary operation “” is a (right) near-algebra over if is a (right) near-ring and for all and all , .

As with near-rings, near-algebras need not be zero-symmetric in general, as seen in the following example [15].

Example 2.19. Let be a vector space over a field . Then , the set of all self-maps of , is a near-algebra over which is not zero-symmetric. Also, if is any subalgebra of the -algebra , then the set of all affine transformations arising from elements of and , that is, , where , is a sub-near-algebra of which is also not zero-symmetric.

Definition 2.20 (see [11]). A normed near-algebra over a field with an absolute value is a near-algebra such that is also a normed vector space over , with the norm satisfying for all .

Proposition 2.21. Let be a normed vector space over a normed field . Then is a normed near-algebra over .

Proof. This result follows from Definition 2.1 and Proposition 2.17.

In the remainder of the paper, we use the induced topology obtained from when making topological statements about .

3. Units in

We note for the reader that there is some overlap between this section and [11, Chapter 3]. However, our proofs are less involved and our results are more general since in [11] only the case where is a Banach space over or is considered. In this section we show that if is a complete and connected normed abelian group, then the set of units is an open subset of . Throughout this discussion, the metric on will be the 1-subadditive metric induced by the norm on , as described in Remark 2.11. We start with a technical lemma that is required for the proof of Lemma 3.2. We denote the identity function in by .

Lemma 3.1. Let be a normed abelian group. Also, let , with , and with . Then

Proof. The statement follows from the fact that we have for all with that

The next lemma shows that if is surjective and “close enough” to , then is a unit in .

Lemma 3.2. Let be a normed abelian group. Also, let be such that . Then is injective and , where .

Proof. From Lemma 3.1 it follows that if , but , then , which is not possible. Thus must be injective.
Note that where (3.3) follows by replacing and by and , respectively, in , and (3.4) follows from Lemma 3.1. Thus is Lipschitz with .

The next two lemmas will be used in the proof of the main result of this section.

Notation 3.3. For and , we denote by the set . Also, for , we denote by the topological closure of .

Lemma 3.4. Let be a normed abelian group. Assume with . Then there is an such that for all and .

Proof. If , we have that and the result follows trivially. Thus we assume that . Choose any with . Suppose that is not a subset of , and let . Also, let . Choose with . Let . We show next that (1),and(2). Once we have () and (), we have a contradiction with the definition of , and we thus have that . The fact that follows from the following: The fact that follows from the following:

Lemma 3.5. Let be a normed abelian group. Assume that is complete, is a closed subset of , , and . Then is closed in .

Proof. Suppose that and as , where for all . From Lemma 3.2   is injective and is Lipschitz. Note that is Cauchy since is Cauchy and since we have the following: Since is complete and is closed, converges to, say, . Thus since is continuous, , and we therefore have that is closed in .

We introduce the main theorem of this section.

Theorem 3.6. Let be a complete and connected normed abelian group. Then the set of units is open in .

Proof. In order to obtain the result, we show that if is a unit, then all , with , are also units. First note that implies that . Since is a unit in if and only if is a unit in , it is enough to show that implies that is a unit in . So assume that . From Lemma 3.2, is injective. Lemma 3.5 implies that is closed, thus . Also, Lemma 3.4 implies that the set , which is equal to , is open. Now since is connected, we have that both open and closed implies that and thus that is surjective. To complete the argument, we recall that Lemma 3.2 implies that is Lipschitz.

We conclude this section by giving an example to show that the completeness of is an essential hypothesis in the preceding theorem.

Example 3.7. Define as follows: Let be the continuous function such that the restriction of to is equal to . It follows from Remark 2.3, by calculating piecewise derivatives, that is Lipschitz for . Let be a rational number such that . Then is rational but not in the range of . Therefore is not surjective and thus not a unit in for any . Next we show that converges to , the identity in , and thus the set of units in is not open. Let be the identity function on . We show that converges to , by showing that converges to . From Remark 2.3 we have that it is enough to show that the absolute value of the derivative of is bounded (where it is defined) by a constant , where as . This is clearly the case on and on . But also on the derivative of is so on the absolute value of the derivative of is bounded by . Thus we conclude that the required constants exist.

4. Continuity of Multiplication and Closure of Ideals

In the first example in this section, we show that if for all with in as , then it is not necessarily the case that converges to in . Since in any normed (right) near-ring , we have that if for all with in as , then converges to . Thus right multiplication is a continuous function in a normed near-ring, but left multiplication is not. An example, similar to the next example, but more involved, is given in [11].

Example 4.1. In this example we show that it is not necessarily the case that for , and also it is not the case that if converges to as approaches infinity, then converges to .
Let be endowed with the Euclidean metric , so that is a normed near-algebra, and define as follows: From Remark 2.3, , whereas and .
For each , let . Then by replacing with and by , we obtain that , whereas and . Thus converges to , but it is not the case that converges to .

Notation 4.2. Let be a nonempty indexing set, and for , let and be nonempty subsets of . Define by for . If is finite, we will use the notation .

Remark 4.3. Note that we use the notation , instead of the familiar notation .

The next proposition shows, for example, that the left ideal, obtained by considering the set of functions in that annihilates a certain subset of , is closed.

Proposition 4.4. For , let and be nonempty subsets of the normed abelian group , with the 's closed. Then the set is a closed subset of .

Proof. Let be a sequence in and let be such that converges . To show that is closed, we need to show that . Let . Then . Since and converges to , we conclude that converges to . Since each is closed, we conclude that , and thus .

Theorem 4.5. The closure of a right ideal of a normed near-ring is again a right ideal of .

Proof. Denote the norm on by . Let be a right ideal and in the closure of . Assume that and are sequences in converging to and , respectively. Then , and the right side of this inequality converges to 0. Thus . Next let . Then , and again the right side of the inequality converges to 0 as . Thus since for all , we conclude that . It follows that is a right ideal of .

Remark 4.6. Recall from the previous section that the set of units is open in if is a complete, connected, normed abelian group. In such a case, if is a proper subset of that is closed under either left or right function composition by an arbitrary function in , then the closure of will also be a proper subset of .

5. Ideals in

This section contains some partial results on the ideal structure of . In the first example we show that ideals in are in abundance.

Example 5.1. Let be a set of functions from to . Denote by the set Assume that we have the following conditions on the functions in : (i)if , then there is an with for all ; (ii)if and , then ; (iii) if and , then there is a such that for all ; (iv) the functions in are nondecreasing. We assume that is a normed abelian group and show next (in part) that is an ideal in .
We show that if and , then . The other cases are handled similarly. Since , there exists some such that for all , . We need to show that there is an , such that . For any we have Thus for .

In the next example we consider the set of bounded functions in .

Example 5.2. Let be a normed abelian group. In this example we consider , the set of bounded Lipschitz functions in . First we show that is a two-sided ideal. If consists of all bounded nondecreasing functions from to , then , and it thus follows from the previous example that is a two-sided ideal in . Next we consider the case when . We show that is not closed. Define for each the Lipschitz function as follows: Let be defined as follows: Then from Remark 2.3 it follows that as . Thus we have a sequence of bounded functions that converges to an unbounded function, which implies that is not closed in .

In the next few results we show that the Betsch-Wielandt density theorem for near-rings can be applied to .

Lemma 5.3. Assume that is a vector space over a field containing . Let with . Fix in and define , for all , by . Then the function defined by for all , is Lipschitz. Also , where is the closure of .

Proof. We show that is a Lipschitz function and leave the proof of the equality to the reader. First note that it is easy to verify that . Thus and is therefore a Lipschitz function.

We will use the next result to conclude that if with , then there is an such that .

Corollary 5.4. Assume that is a vector space over a field containing . Let with nonzero and . Then there is an such that and .

Proof. Note that and for an appropriate , where is as in Lemma 5.3.

Corollary 5.5. Assume that is a vector space over a field containing . Then is not a ring.

Proof. Let and let . Let be as in Lemma 5.3, and denote by the identity function on . Now note that we have that , since .

Corollary 5.6. Assume that is a vector space over a field containing . Then is a type-2 primitive -module.

Proof. From Corollary 5.4, for all . Also, for implies that . Thus is a 2-primitive -module.

Corollary 5.7. Assume that is a vector space over a field containing . Let and be elements in with distinct and nonzero 's. Then there exists an such that for .

Proof. Since is a type-2 primitive -module, the Betsch-Wielandt density theorem for near-rings (see, e.g., [7]) can be applied when is not a ring. By the density theorem, if and are in and the 's are distinct and all nonzero, then there exists such that .

We conclude by exhibiting some of the maximal left ideals in . With the exception of the statement that is closed, the argument is solely based on the fact that is a 2-primitive -module.

Theorem 5.8. Assume that is a vector space over a field containing . Let with . Then , often denoted by , is a maximal closed left ideal that is not an ideal.

Proof. From Proposition 4.4 we have that is closed. It is easy to verify that is a left ideal. Next we show that it is maximal. Assume that . Then from Corollary 5.4 we can find a such that . But then and implying that the left ideal generated by and is all of . It follows that is a maximal left ideal.
Finally we show that is not an ideal. Let be two distinct elements in , with . From Corollary 5.4 we have functions with and . Then it follows that , but , since we have .

Acknowledgment

The authors would like to thank the anonymous referee for helping them to make a substantial improvement to the quality of this paper.

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