The Near-Ring of Lipschitz Functions on a Metric Space

Volume 2010, Article ID 284875,14pages doi:10.1155/2010/284875

Research Article

The Near-Ring of Lipschitz Functions on a Metric Space

Mark Farag

and Brink van der Merwe

1Department of Mathematics, Fairleigh Dickinson University, 1000 River Rd., Teaneck, NJ 07666, USA

2Department of Computer Science, University of Stellenbosch, Stellenbosch 7602, South Africa

Correspondence should be addressed to Mark Farag,[email protected] Received 6 August 2009; Revised 3 April 2010; Accepted 25 April 2010 Academic Editor: Francois Goichot

Copyrightq2010 M. Farag and B. 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.

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 Eells1, de Leeuw2, Sherbert3,4, 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 spaceX, ρto a Banach spaceEoftenRorC, so that additionand multiplication in caseE RorE Cof functions is defined. The Lipschitz number of a functionf, denotedf_L, is used in combination with the infinity norm · _∞ to produce a norm · :max · _L, · _∞. If one identifies a distinguished basepoint inX, then · _Lis used as a norm on the set of basepoint preserving Lipschitz functions mapping X, ρtoE.

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 spaceX, ρto itself and may be regarded as a generalization of the bounded linear operators that are so important in analysis. Rather than takingXto be a Banach space, we only require X,to be an abelian group. By restricting the possible metrics ρ on X, we ensure that the set of zero-preserving Lipschitz functions onXis 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, Clay8, 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 Irish11, 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 2.1see6,12. A functionffrom a metric spaceX1, ρ1to a metric spaceX2, ρ2 is Lipschitz if there exists a constant K ≥ 0 such that for all x, y ∈ X1, ρ2fx, fy ≤ Kρ₁x, y. Iff:X1, ρ₁ → X2, ρ₂is Lipschitz, the Lipschitz number offis defined as

f

L:sup ρ2

fx, f y ρ1

x, y |x, y∈X₁, x /y

. 2.1

Remark 2.2. If f is Lipschitz, then f_L ≥ 0. Also, f_L 0 if and only if f is constant.

For Lipschitz functionsf : X1, ρ1 → X2, ρ2and g : X2, ρ2 → X3, ρ3, we have that g◦f_L≤ g_Lf_L.

Remark 2.3. It is well-known that a Lipschitz functionf : R → Ris absolutely continuous and therefore differentiable almost everywherea.e.. Also, the derivative offis bounded a.e.

in magnitude by the Lipschitz constant, and fora≤b, the differencefb−fais equal to the integral of the derivative offon the intervala, b. Conversely, iff:R → Ris absolutely continuous and thus differentiable a.e. and if|fx| ≤ K a.e., then f is Lipschitz with Lipschitz constant at mostK. We will only use these observations in the case wheref is a continuous function fromRtoRthat is everywhere, except possibly for finitely many points, differentiable; also, the derivative offwill be continuous at all points where it exists. In this casefis Lipschitz if and only if the set{|fx|:fis differentiable atx}is bounded and

f

Lsupfx:f is differentiable atx . 2.2 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 consult12or6 for more on Lipschitz functions.

In what follows, all metric spacesX, ρare also abelian groups under the operation , with identity element 0. We exclude the trivial caseX {0}. If the metricρsatisfies the condition in the next definition, the pointwise addition of two Lipschitz functions is again a Lipschitz function.

Definition 2.4. LetK >0 be a real number. A metricρon a metric spaceXisK-subadditive on Xif

ρab, cd

ρa, c ρb, d, 2.3

for alla, b, c, d∈Xwithρa, c ρb, d/0, bounded, and

sup

ρa,cρb,d/0

ρab, cd

ρa, c ρb, d K. 2.4

Remark 2.5. Ifρis a metric onX and we define the metricρonX×Xviaρa, b,c, d ρa, c ρb, d, thenK-subadditivity ofρis equivalent to having the Lipschitz number of :X×X → Xequal toK.

Example 2.6. LetX, · be a normed vector space and let the metricρbe defined onXby ρx, y x−yfor allx, y∈X. Thenρis 1-subadditive.

Assume thatρis a 1-subaddive metric on the abelian groupX, and define · :X → R∪ {0}byx:ρx,0for allx∈X. We will show that · satisfies the properties given in the next definition.

Definition 2.7see, e.g.,13. A function · :X → R∪ {0}is a norm on the abelian group Xif · satisfies the following criteria:

1x0 if and only ifx0;

2x −xfor allx∈X;

3xy ≤ xyfor allx, y∈X.

Remark 2.8. Let · be a norm on an abelian group. Then the functionρ:X×X → R∪ {0}, defined byρx, y:x−yfor allx, y∈X, is a 1-subadditive metric onX.

Example 2.9. LetX be a multiplicative subgroup of the unit circle in the complex plane, and take “addition” inXto be complex multiplicationso that 1∈Xis the neutral element. If we denote byzthe Euclidean distance between the complex numberszand 1, · is a norm on the abelian groupX.

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

Proposition 2.10. Assume thatρisK-subadditive on the metric spaceX. Then 1ρx−y,0≤Kρx, yandρx, y≤Kρx−y,0;

2ρx, y≤K²ρ−x,−yandρ−x,−y≤K²ρx, y;

3K≥1;

4ifK1, thenρx, y ρx−y,0andρx, y ρ−x,−y;

5ifK 1, then · :X → R∪ {0},defined byx:ρx,0for allx∈X, is a norm on Xandx−yρx, yfor allx, y∈X.

Proof. We prove1and2. The other parts follow immediately from the following:

1

ρ

x−y,0 ρ

x−y, y−y

≤K ρ

x, y ρ

−y,−y Kρ

x, y ,

2.5

and

ρ x, y

ρ x−y

y,0y

≤K ρ

x−y,0 ρ

y, y Kρ

x−y,0 .

2.6

2

ρ x, y

ρ y, x ρ

x−xy, x−yy

≤K ρ

x−x, x−y ρ

y, y

≤K²

ρx, x ρ

−x,−y K²ρ

−x,−y .

2.7

Thusρx, y≤K²ρ−x,−yand therefore alsoρ−x,−y≤K²ρx, y.

Remark 2.11. Note that from part5 ofProposition 2.10andRemark 2.8 we have that any 1-subadditive metric on an abelian groupXis induced by a norm onXand, conversely, any norm onXinduces a 1-subadditive metric.

The following is an example of a metric space with a metric that is notK-subadditive for anyK.

Example 2.12. Consider the metric onRgiven byρa, b:|2^a−2^b|fora, b ∈R. Then ifρis K-subadditive, we would have that 2ⁿ⁻¹ ρn, n−1≤Kfor alln∈N, which is clearly not possible.

Notation 2.13. For a metric spaceX, ρ, we denote byLX,ρ, or simplyLX when there is no ambiguity about the metric, the set of zero-preserving Lipschitz functions onX.

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

Proposition 2.14. Letρbe aK-subadditive metric onX. Then for allf, g∈ LXone has 1f_L≥0 andf_L0 if and only iff0X;

2 −f_L≤K²f_L; 3f◦g_L≤ f_Lg_L; 4fg_L≤Kf_Lg_L.

Proof. The result follows from Definitions2.1and2.4andProposition 2.10.

We now recall the definition of a near-ring.

Definition 2.15. A tripleN,,∗is called arightnear-ring if 1 N,is anot necessarily abeliangroup,

2 N,∗is a semigroup,

3for alla, b, c∈N,ab∗ca∗cb∗c.

A near-ringN,,∗is called zero-symmetric if, for alln ∈ N,n∗0N 0N, where 0N is the neutral element ofN,.

IfG,is any group, thenMG, the set of all self-maps ofG, is a near-ring under pointwise addition and function composition. The set of all zero-preserving self-maps ofG, M0G, is a zero-symmetric sub-near-ring of MG. 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, Meldrum9, and Pilz7.

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

Definition 2.16. A normed near-ringN,·is a near-ringNwith a function·:N → R∪{0}, such that

1x0 if and only ifx0;

2x −xfor allx∈N;

3xy ≤ xyfor allx, y∈N;

4xy ≤ xyfor allx, y∈N.

Proposition 2.17. Assume thatρis aK-subadditive metric onX. Then

1 LX,ρ,,◦(“+” is pointwise addition and “◦” is function composition) is a zero-symmetric near-ring with identity;

2 LX,ρ, · _Lis a normed near-ring ifK1.

Proof. 1We show that iff, g∈ LX, thenf◦g, fg,−f∈ LX. FromRemark 2.2,f◦g∈ LX, and fromProposition 2.10, we have forx, y∈X,

ρ

−fx,−f y

≤K²ρ

fx, f y

, 2.8

and thusf∈ LXimplies that−f∈ LX. Also, ρ

fx gx, f y

g y

≤Kρ

fx, f y

Kρ

gx, g y

, 2.9

sinceρisK-subadditive. Thusf, g∈ LXimplies thatfg ∈ LX. Finally, note that sinceLX

contains only zero-preserving functions, it is a zero-symmetric near-ring.

2This result follows from1andProposition 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 fieldFis a function

| · |:F → R∪ {0}, such that 1|x|0 if and only ifx0;

2|xy||x||y|for allx, y∈F;

3|xy| ≤ |x||y|for allx, y∈F.

IfX is a normed vector space over a normed field, LX 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 A over a field F together with another binary operation “·” is arightnear-algebra overFifA,,·is arightnear-ring and for alla, b∈A and allk∈F,ka·bka·b.

As with near-rings, near-algebras need not be zero-symmetric in general, as seen in the following example15.

Example 2.19. LetX be a vector space over a fieldF. ThenMX, the set of all self-maps of X, is a near-algebra overF which is not zero-symmetric. Also, ifRis any subalgebra of the F-algebra EndFX, then the set of all affine transformations arising from elements ofRand X, that is,AFRX:{Av |A∈R, v∈X}, whereA_vx:Axv, is a sub-near-algebra of MXwhich is also not zero-symmetric.

Definition 2.20see11. A normed near-algebraA, · over a fieldF,| · |with an absolute value| · |is a near-algebra such thatA, · is also a normed vector space overF,| · |, with the norm · satisfyingf·g ≤ fgfor allf, g∈A.

Proposition 2.21. LetX, · be a normed vector space over a normed fieldF,| · |. ThenLXis a normed near-algebra overF.

Proof. This result follows fromDefinition 2.1andProposition 2.17.

In the remainder of the paper, we use the induced topology obtained from · _Lwhen making topological statements aboutLX.

3. Units in L

We note for the reader that there is some overlap between this section and11, Chapter 3.

However, our proofs are less involved and our results are more general since in11only the

case whereX is a Banach space overRorCis considered. In this section we show that ifX is a complete and connected normed abelian group, then the set of units is an open subset of LX. Throughout this discussion, the metric onXwill be the 1-subadditive metric induced by the norm onX, as described inRemark 2.11. We start with a technical lemma that is required for the proof ofLemma 3.2. We denote the identity function inLXby 1X.

Lemma 3.1. LetX, · be a normed abelian group. Also, letf ∈ LX, withf−1_XL < 1, and x, y∈Xwithx /y. Then

0<1−f−1_X

L≤ fx−f y

x−y ≤1f−1_X

L. 3.1

Proof. The statement follows from the fact that we have for allx, y∈Xwithx /ythat f−1X

L≥ fx−x

− f

y

−y x−y

≥ fx−f

y−x−y x−y

fx−f y x−y −1

.

3.2

The next lemma shows that iff ∈ LXis surjective and “close enough” to 1X, thenfis a unit inLX.

Lemma 3.2. LetX, · be a normed abelian group. Also, letf ∈ LXbe such thatf−1_XL<1.

Thenfis injective andf⁻¹_L≤1/1− f−1X_L, wheref⁻¹:fX → X.

Proof. FromLemma 3.1it follows that ifx /y, butfx fy, then 0 < 1− f −1_XL ≤ fx−fy/x−y0, which is not possible. Thusfmust be injective.

Note that

supf⁻¹x−f⁻¹ y

x−y :x, y∈fX, x /y

≤sup

x−y fx−f

y :x, y∈X, x /y

3.3

≤ 1

1−f−1X

L

, 3.4

where 3.3follows by replacing x and y by f⁻¹x and f⁻¹y, respectively, in sup{x− y/fx−fy:x, y∈X, x /y}, and3.4follows fromLemma 3.1. Thusf⁻¹is Lipschitz withf⁻¹_L≤1/1− f−1_XL.

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

Notation 3.3. Fora∈Xandr∈R, we denote byBrathe set{x∈X:x−a< r}. Also, for A⊆X, we denote byAthe topological closure ofA.

Lemma 3.4. LetX, · be a normed abelian group. Assumef ∈ LX withf−1X_L < 1. Then there is anα∈Rsuch thatB_αrfa⊆fBrafor allr∈Randa∈X.

Proof. Iff−1X_L0, we have thatf 1X,and the result follows trivially. Thus we assume that 0< f−1_XL <1. Choose anyαwith 0 < α <1− f−1_XL. Suppose thatB_αrfais not a subset offBra, and letc∈Bαrfa\fBra. Also, letdc, fBra:inf{c− x : x ∈ fBra} > 0. Chooseb ∈ Brawithc−fb < dc, fBra/f−1X_L. Let d:bc−fb. We show next that

1 d∈Bra,

and

2 fd−c< dc, fBra.

Once we have1and2, we have a contradiction with the definition ofdc, fBra, and we thus have thatB_αrfa⊆fBra. The fact thatd∈B_rafollows from the following:

d−af−1X

a− f−1X

b

c−fa

≤f−1_X

L a−bfa−c

<f−1X

L rαr

< r.

3.5

The fact thatfd−c< dc, fBrafollows from the following:

fd−cf−1X

bc−fb

− f−1X

b

≤f−1_X

Lbc−fb−b f−1X

Lc−fb

< d

c, fBra .

3.6

Lemma 3.5. LetX, · be a normed abelian group. Assume thatXis complete,Cis a closed subset ofX,f∈ LX, andf−1_XL<1. ThenfCis closed inX.

Proof. Suppose thaty∈Xandfxn−y → 0 asn → ∞, wherexn∈Cfor alln∈N. From Lemma 3.2fis injective andf⁻¹ :fX → Xis Lipschitz. Note thatxn :n∈Nis Cauchy

sincefxn:n∈Nis Cauchy and since we have the following:

xn−xmf⁻¹fxn−f⁻¹fxm

≤f⁻¹

Lfxn−fxm.

3.7

SinceX is complete andCis closed,xn : n∈ Nconverges to, say,x ∈C. Thus sincef is continuous,fx y, and we therefore have thatfCis closed inX.

We introduce the main theorem of this section.

Theorem 3.6. LetXbe a complete and connected normed abelian group. Then the set of units is open inLX.

Proof. In order to obtain the result, we show that if g ∈ LX is a unit, then all f ∈ LX, with f−g_L < 1/g⁻¹_L, are also units. First note thatf−g_L < 1/g⁻¹_L implies that f◦g⁻¹−1X_L f−g◦g⁻¹_L ≤ f−g_Lg⁻¹_L<1. Sincefis a unit inLXif and only if f◦g⁻¹is a unit inLX, it is enough to show thatf−1_XL<1 implies thatf is a unit inLX. So assume thatf−1X_L<1. FromLemma 3.2,fis injective.Lemma 3.5implies thatfXis closed, thusfX fX. Also,Lemma 3.4implies that the setfX, which is equal tofX, is open. Now sinceX is connected, we have thatfX both open and closed implies that fX X and thus thatf is surjective. To complete the argument, we recall thatLemma 3.2 implies thatf⁻¹is Lipschitz.

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

Example 3.7. Definefn:Q → Qas follows:

f_nx

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

x x <1− 1

n, x²

2

1− 1 n

−1−1/n²

2 x∈

1− 1

n,1

,

x−1 2

1− 1

n

− 1−1/n²

2 x >1.

3.8

Letf_n:R → Rbe the continuous function such that the restriction off_ntoQis equal tof_n. It follows fromRemark 2.3, by calculating piecewise derivatives, thatfn :Q → Qis Lipschitz forn >1. Letknbe a rational number such that 1−1/n < kn

√2<1. Thenf_nkn

√2is rational but not in the range off_n. Thereforef_nis not surjective and thus not a unit inL_Qfor anyn.

Next we show thatf_nconverges to 1_Q, the identity inLQ, and thus the set of units inLQis not open. Let 1_Rbe the identity function onR. We show thatfnconverges to 1_Q, by showing thatf_n−1_RLconverges to 0. FromRemark 2.3we have that it is enough to show that the absolute value of the derivative ofh_nf_n−1_Ris boundedwhere it is definedby a constant M_n, whereM_n → 0 asn → ∞. This is clearly the case on−∞,1−1/nand on1,∞. But also on1−1/n,1the derivative ofhnisx−1; so on1−1/n,1the absolute value of the derivative ofh_nis bounded by 1/n. Thus we conclude that the required constantsM_nexist.

4. Continuity of Multiplication and Closure of Ideals

In the first example in this section, we show that iff, g, g_n∈ LXfor alln∈Nwithg_n → gin LXasn → ∞, then it is not necessarily the case thatf◦g_nconverges tof◦ginLX. Since

g_n◦f−g◦f

Ngn−g◦f

N≤g_n−g

Nf

N 4.1

in any normedrightnear-ring N, · _N, we have that iff, g, gn ∈ Nfor alln ∈ Nwith g_n → g in N as n → ∞, then g_n ◦f converges to g◦ f. 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 in11.

Example 4.1. In this example we show that it is not necessarily the case thatf◦g−f◦h_L≤ f_Lg−h_L for f, g, h ∈ LX, and also it is not the case that if g_n converges to g as n approaches infinity, thenf◦gnconverges tof◦g.

LetX R be endowed with the Euclidean metricd, so that LX is a normed near- algebra, and definef, g, h:R → Ras follows:

fx:dx,−1,1;

gx:x;

hx:kx, withk >1 fixed.

4.2

FromRemark 2.3,f◦g−f◦h_Lk, whereasf_L1 andg−h_Lk−1.

For eachn∈N, letgnx: 11/nx. Then by replacinghwithgnandkby11/n, we obtain thatf◦g−f◦gn_L 11/n, whereasf_L 1 andg−gn_L 1/n. Thusgn

converges tog, but it is not the case thatf◦g_nconverges tof◦g.

Notation 4.2. LetI be a nonempty indexing set, and for i ∈ I, letAi and Bi be nonempty subsets ofX. DefineLXAi → Bi :i∈Iby{f ∈ LX |fAi⊆Bifori∈I}. If I is finite, we will use the notationLXA1 → B₁, . . . , A_n → B_n.

Remark 4.3. Note that we use the notationLXS → {0}, instead of the familiar notation Ann_LXS.

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

Proposition 4.4. Fori∈I, letA_iand B_ibe nonempty subsets of the normed abelian groupX, · , with theB_i’s closed. Then the setLXAi → B_i :i∈Iis a closed subset ofLX.

Proof. Let fn : n ∈ N be a sequence in LXAi → Bi : i ∈ I,and let f ∈ LX be such that f_n converges f. To show that LXAi → B_i : i ∈ I is closed, we need to show that f ∈ LXAi → B_i :i∈I. Leta_i∈A_i. Thenf_nai∈B_i. Sincefai−f_nai ≤ f−f_nLai and f−fn_Lconverges to 0, we conclude that fnaiconverges to fai. Since each Bi is closed, we conclude thatfai∈B_i, and thusf ∈ LXAi → B_i:i∈I.

Theorem 4.5. The closure of a right ideal of a normed near-ringNis again a right ideal ofN.

Proof. Denote the norm onN by · _N. LetI ⊆ N be a right ideal andf, ginI the closure of I. Assume that fn and gn are sequences inI converging tof and g, respectively.

Thenf−g−fn−g_nN ≤ f−f_nN g−g_nN, and the right side of this inequality converges to 0. Thusf−g∈I. Next leth∈N. Thenfn◦h−f◦h_N fn−f◦h_N ≤ fn−f_Nh_N, and again the right side of the inequality converges to 0 asn → ∞. Thus sincef_n◦h∈I for alln∈N, we conclude thatf◦h∈I. It follows thatI is a right ideal of N.

Remark 4.6. Recall from the previous section that the set of units is open in LX if X is a complete, connected, normed abelian group. In such a case, if S is a proper subset ofLX

that is closed under either left or right function composition by an arbitrary function inLX, then the closure ofSwill also be a proper subset ofLX.

5. Ideals in L

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

Example 5.1. LetFbe a set of functions fromR₀ toR₀. Denote byIF_Xthe set

f∈ LX: there existsF ∈ Fsuch thatfx≤Fxforx∈X . 5.1

Assume that we have the following conditions on the functions inF:

iifF, G∈ F, then there is anH∈ FwithFt Gt≤Htfor allt∈R₀; iiifr ∈R₀ andF∈ F, thenrF∈ F;

iiiifr ∈R₀ andF∈ F, then there is aG∈ Fsuch thatFrt≤Gtfor allt∈R₀; ivthe functions inFare nondecreasing.

We assume thatXis a normed abelian group and show nextin partthatIFXis an ideal in LX.

We show that ifg∈ IFXandf, h∈ LX, thenf◦gh−f◦h∈ IFX. The other cases are handled similarly. Sinceg ∈ IFX, there exists someG∈ IFXsuch that for allx∈X,gx ≤ Gx. We need to show that there is anH∈ IF_X, such thatf◦gh−f◦hx ≤Hx.

For anyx∈Xwe have f◦

gx hx

−f◦hx≤f

Lgx hx−hx f

Lgx

≤f

LGx.

5.2

Thusf◦gh−f◦hx ≤HxforHf_LG.

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

Example 5.2. LetXbe a normed abelian group. In this example we considerBLX, the set of bounded Lipschitz functions inLX. First we show thatBLXis a two-sided ideal. IfFconsists of all bounded nondecreasing functions fromR₀ toR₀, thenIFX BLX, and it thus follows from the previous example thatBLX is a two-sided ideal inLX. Next we consider the case whenX R. We show thatBL_Ris not closed. Define for eachn∈ Nthe Lipschitz function f_n:R → Ras follows:

f_nx

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

0 x≤0, x x∈0,1,

√x x∈ 1, n²

, n x > n².

5.3

Letf:R → Rbe defined as follows:

fx

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

0 x≤0, x x∈0,1,

√x x >1.

5.4

Then fromRemark 2.3it follows thatf−fn_L → 0 asn → ∞. Thus we have a sequence of bounded functions that converges to an unbounded function, which implies thatBL_Ris not closed inLR.

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

Lemma 5.3. Assume thatXis a vector space over a field containingR. LetS⊆Xwith 0∈S. Fixx0

inX,and definedS, x, for allx∈X, bydS, x inf_s∈Ss−x. Then the functionf_S,x0:X → X, defined byfS,x0x:dS, xx0for all x ∈X, is Lipschitz. AlsofS,x0 f_S,x

0, whereSis the closure ofS.

Proof. We show thatf_S,x0is a Lipschitz function and leave the proof of the equalityf_S,x0f_S,x to the reader. First note that it is easy to verify that|dS, x−dS, y| ≤ x−y. Thus 0

f_S,x0x−f_S,x0

ydS, xx0−d S, y

x₀ dS, x−d

S, yx0

≤x−yx0,

5.5

andfS,x0is therefore a Lipschitz function.

We will use the next result to conclude that if x, y ∈ X with x /0, then there is an f∈ LXsuch thatfx y.

Corollary 5.4. Assume thatX is a vector space over a field containing R. Letx, y, z ∈ X with x nonzero andx /z. Then there is anf∈ LXsuch thatfx yandfz 0.

Proof. Note thatcf_{0,z},yx yandcf_{0,z},yz 0 for an appropriatec∈F, wheref_{0,z},yis as inLemma 5.3.

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

Proof. Letx∈ X\ {0}and letS {0, x}. LetfS,xbe as inLemma 5.3, and denote by 1X the identity function onX. Now note that we have thatf_S,x◦1X1_Xx/0f_S,x◦1_Xx f_S,x◦ 1_Xx, sincexx/∈S.

Corollary 5.6. Assume thatXis a vector space over a field containingR. ThenXis a type-2 primitive LX-module.

Proof. FromCorollary 5.4,LXxXfor allx /0. Also,fX0 forf ∈ LXimplies thatf 0.

ThusXis a 2-primitiveLX-module.

Corollary 5.7. Assume thatXis a vector space over a field containingR. Leta₁, . . . , a_nandb₁, . . . , b_n be elements inXwith distinct and nonzeroa_i’s. Then there exists anf∈ LXsuch thatfai b_ifor i1, . . . , n.

Proof. Since X is a type-2 primitive LX-module, the Betsch-Wielandt density theorem for near-ringssee, e.g.,7 can be applied whenLX is not a ring. By the density theorem, if a1, . . . , an and b1, . . . , bn are inX and the ai’s are distinct and all nonzero, then there exists f∈ LXsuch thatfai b_i.

We conclude by exhibiting some of the maximal left ideals inLX. With the exception of the statement thatL{x0} → {0}is closed, the argument is solely based on the fact that Xis a 2-primitiveLX-module.

Theorem 5.8. Assume thatX is a vector space over a field containingR. Letx₀ ∈ X withx₀/0.

ThenL{x0} → {0}, often denoted by AnnL{x0}, is a maximal closed left ideal that is not an ideal.

Proof. FromProposition 4.4we have thatLX{x0} → {0}is closed. It is easy to verify that LX{x0} → {0} is a left ideal. Next we show that it is maximal. Assume thatf ∈ LX \ LX{x0} → {0}. Then fromCorollary 5.4we can find ag∈ LXsuch thatgfx0 x0. But thenid_X g◦f idX−g◦fandidX−g◦f∈ LX{x0} → {0},implying that the left ideal generated byf andLX{x0} → {0}is all ofLX. It follows thatLX{x0} → {0}is a maximal left ideal.

Finally we show thatLX{x0} → {0}is not an ideal. Let 0, xbe two distinct elements inX, withx /x0. FromCorollary 5.4we have functionsf, g ∈ LX withfx x, fx0 0, andgx0 x. Then it follows thatf ∈ LX{x0} → {0}, butf◦g /∈ LX{x0} → {0}, since we havefgx0 x.

Acknowledgment

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

References

1 R. F. Arens and J. Eells Jr., “On embedding uniform and topological spaces,” Pacific Journal of Mathematics, vol. 6, pp. 397–403, 1956.

2 K. de Leeuw, “Banach spaces of Lipschitz functions,” Studia Mathematica, vol. 21, pp. 55–66, 1961.

3 D. R. Sherbert, “Banach algebras of Lipschitz functions,” Pacific Journal of Mathematics, vol. 13, pp.

1387–1399, 1963.

4 D. R. Sherbert, “The structure of ideals and point derivations in Banach algebras of Lipschitz functions,” Transactions of the American Mathematical Society, vol. 111, pp. 240–272, 1964.

5 J. A. Johnson, “Banach spaces of Lipschitz functions and vector-valued Lipschitz functions,”

Transactions of the American Mathematical Society, vol. 148, pp. 147–169, 1970.

6 N. Weaver, Lipschitz Algebras, World Scientific, River Edge, NJ, USA, 1999.

7 G. Pilz, Near-Rings: The Theory and Its Applications, vol. 23 of North-Holland Mathematics Studies, North- Holland, Amsterdam, The Netherlands, 2nd edition, 1983.

8 J. R. Clay, Nearrings: Geneses and Application, Oxford Science Publications, The Clarendon Press/Oxford University Press, New York, NY, USA, 1992.

9 J. D. P. Meldrum, Near-rings and Their Links with Groups, vol. 134 of Research Notes in Mathematics, Pitman, Boston, MA, USA, 1985.

10 H. Brown, Near algebras, Ph.D. dissertation, Ohio State University, 1966.

11 J. Irish, Normed near algebras and finite dimensional near algebras of continuous functions, Ph.D.

dissertation, University of New Hampshire, 1975.

12 R. M. Dudley, Real Analysis and Probability, vol. 74 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, MA, USA, 2002.

13 J. Aaronson, An Introduction to Infinite Ergodic Theory, vol. 50 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1997.

14 S. Warner, Topological Rings, vol. 178 of North-Holland Mathematics Studies, North-Holland, Amster- dam, The Netherlands, 1993.

15 H. Heatherly, “Near Algebras,” private communication.