# Some Dense Linear Subspaces of Extended Little Lipschitz Algebras

Authors: Davood Alimohammadi; Sirous Moradi
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

Let(X,d) be a compact metric space. In 1987, Bade, Curtis, and Dales obtained a sufficient condition for density of a subspace P of little Lipschitz algebra lip(X,α) in this algebra and in particular showed that Lip(X,1) is dense in lip(X,α), whenever 0<α<1. Let K be a compact subset of X. We define new classes of Lipchitz algebras Lip(X,K,α) for α∈(0,1] and lip(X,K,α) for α∈(0,1), consisting of those continuous complex-valued functions f on X such that f|K∈Lip(K,α) and f|K∈lip(K,α), respectively. In this paper we obtain a sufficient condition for density of a linear subspace P of extended little Lipschitz algebra lip(X,K,α) in this algebra and in particular show that Lip(X,K,1) is dense in lip(X,K,α), whenever 0<α<1.


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## Body

## 1. Introduction

LetΩ be a locally compact Hausdorff space. The linear space of all continuous (bounded continuous) complex-valued functions on Ω is denoted by C(Ω) (Cb(Ω)). It is known that Cb(Ω) under the uniform norm on Ω, that is,(1.1)‖h‖Ω=sup⁡{|h(w)|:w∈Ω}(h∈Cb(Ω)),
is a commutative Banach algebra. The set of all f in C(Ω), which vanish at infinity, is denoted by C0(Ω), which is a closed linear subspace of (Cb(Ω),∥·∥Ω). Clearly, C0(Ω)=Cb(Ω)=C(Ω), whenever Ω is compact. The linear space of all complex regular Borel measures on Ω is denoted by M(Ω). It is known that M(Ω) under the norm ∥μ∥=|μ|(Ω) (μ∈M(Ω)) is a Banach space, where |μ| is the total variation of μ∈M(Ω).The Riesz representation theorem asserts that there exists a linear isometry from(C0(Ω),∥·∥Ω)*, the dual space (C0(Ω),∥·∥Ω) onto (M(Ω),∥·∥). In fact, for each Λ∈(C0(Ω),∥·∥Ω)*, there exists a unique measure μ∈M(Ω) with ∥μ∥=∥Λ∥ such that(1.2)Λ(f)=∫Ωfdμ(f∈C0(Ω)).
Let (X,d) be a compact metric space and α>0. The Lipschitz algebra Lip(X,α) is defined as the set of all complex-valued functions f on X such that(1.3)pα(f)=sup⁡{|f(x)-f(y)|dα(x,y):x,y∈X,x≠y}<∞.
Then lip(X,α) is a subalgebra of C(X). The subalgebra Lip(X,α) of Lip(X,α) is the set of all those complex-valued functions f on X for which lim|f(x)-f(y)|/dα(x,y)=0 as d(x,y)→0 and is called little Lipschitz algebra of order α.We know thatLip(X,1) separates the points of X, 1∈Lip(X,1) and Lip(X,β)⊆lip(X,α)⊆Lip(X,α), where 0<α<β⩽1. Also, if X is infinite and 0<α<1, then lip(X,α)≠Lip(X,α). The algebras Lip(X,α) for α⩽1 and lip(X,α) for α<1 are Banach function algebras on X under the norm ∥f∥α=∥f∥X+pα(f). Since these algebras are self-adjoint, they are uniformly dense in C(X), by the Stone-Weierstrass theorem. We know that if A is a Banach function algebra on a compact Hausdorff space X such that A is self-adjoint and 1/f∈A whenever f∈A and f(x)≠0 for each x∈X, then A is natural, that is, the maximal ideal space of A coincides with X. Hence, if X is infinite, then the Lipschitz algebras Lip(X,α) for α⩽1 and lip(X,α) for α<1, are natural.Extensive study of Lipschitz algebras started with Sherbert [1, 2]. Honary and Moradi introduced new classes of analytic Lipschitz algebras on compact plane sets and determined their maximal ideal spaces [3].Bade et al. have obtained a sufficient condition for density of a linear subspaceP of lip(X,α) in this algebra as follows.Theorem 1.1 (see [4, Theorem  3.6]).
Let(X,d) be a compact metric space, and let P be a linear subspace of lip(X,α). Suppose that there is a constant C such that for each finite subset E of X and each f∈lip(X,α), there exists g∈P with g|E=f|E and with ∥g∥α⩽C∥f∥α. Then P is dense in lip(X,α).They also showed thatLip(X,1) is dense in lip(X,α) [4, Corollary  3.7]. We extend the above results for the more general classes of the Lipschitz algebras by generalizing and using some results that have been given by them.Throughout this work we always assume that(X,d) is a compact metric space, K is nonempty compact subset of X, and α is a positive number.Definition 1.2.
The algebra of all continuous complex-valued functionsf on X for which
(1.4)pα,K=sup⁡{|f(x)-f(y)|dα(x,y):x,y∈K,x≠y}<∞
is denoted by Lip(X,K,α), and the subalgebra of those f∈Lip(X,K,α) for which |f(x)-f(y)|/dα(x,y)→0 as d(x,y)→0, when x,y∈K, is denoted by lip(X,K,α). The algebras Lip(X,K,α) and lip(X,K,α) are called extended Lipschitz algebra and extended little Lipschitz algebra of order α on (X,d) with respect to K, respectively.It is easy to see that these extended Lipschitz algebras are both Banach algebras under the norm∥f∥α,K=∥f∥X+Pα,K(f). In fact, lip(X,K,α) is a Banach function algebra on X for α∈(0,1], and lip(X,K,α) is a Banach function algebra on X for α∈(0,1). Note that if 0<α<β⩽1, then Lip(X,K,β)⊆lip(X,K,α). We always assume that 0<α⩽1 for Lip and 0<α<1 for lip. Note that lip(X,K,α) is a proper subalgebra of Lip(X,K,α) when K is infinite. Because if y∈K, then function f:X→ℂ defined by f(x)=dα(x,y) is an element of Lip(X,K,α) but does not belong to lip(X,K,α).It is clear that wheneverK=X, the new classes of Lipschitz algebras coincide with the standard Lipschitz algebras. Also Lip(X,K,α)=lip(X,K,α)=C(X), whenever K is finite. Hence, we may assume that K is infinite.By the Stone-Weierstrass theorem,Lip(X,K,α) and lip(X,K,α) are both uniformly dense in C(X).LetA be Lip(X,K,α) or lip(X,K,α). For each f∈A and for all n∈ℕ, we have(1.5)‖fn‖α,K⩽‖fn‖X+n(‖f‖X)n-1pα,K(f).
By the spectral radius theorem,(1.6)‖f̂‖M(A)=lim⁡n→∞(‖fn‖α,K)1/n⩽lim⁡n→∞‖f‖X(1+pα,K(f)‖f‖Xn)1/n=‖f‖X,
where ℳ(A) is the maximal ideal space of A and f̂ is the Gelfand transform of f on ℳ(A). Hence, by applying the main theorem in [5], we can show that A is natural, that is, ℳ(A) coincides with X. We can prove this fact with another way. Since every self-adjoint inverse-closed Banach function algebra A on a compact Hausdorff space X is natural, and Banach function algebras Lip(X,K,α) and lip(X,K,α) have the mentioned properties, they are natural.In this paper we obtain a sufficient condition for density of a linear subspaceP of lip(X,K,α) that is dense in this algebra by generalizing some results in [4]. In particular, we show that Lip(X,K,1) is dense in (lip(X,K,α),∥·∥α,K).


## 2. Representing Measure

We denote(2.1)Δ(K)={(x,y)∈K×K:x=y},V(K)=(K×K)∖Δ(K),W(X,K)=X∪V(K),W(K)=W(K,K).
Obviously, W(X,K) is a locally compact Hausdorff space. We define the norm |∥·∥| on Cb(W(X,K)) by(2.2)|‖h‖|=‖h|X‖X+‖h|V(K)‖V(K).
Then Cb(W(X,K) is a Banach space under the norm |∥·∥|, since(2.3)‖h‖W(X,K)⩽‖|h|‖⩽2‖h‖W(X,K),
for all h∈Cb(W(X,K)). Moreover, C0(W(X,K)) is a closed linear subspace of (Cb(W(X,K)),|∥·∥|).We define the norm|∥·∥| on M(W(X,K)) by(2.4)‖|μ|‖=max⁡{|μ|(X),|μ|(V(K))}.
Then M(W(X,K)) is a Banach space under the norm ∥|·|∥, since(2.5)‖μ‖⩽‖|μ|‖⩽2‖μ‖,
for all μ∈M(W(X,K)). By applying the Riesz representation theorem, we obtain the following result which is a generalization of Theorem A in [6], and one can prove it by the same method.Theorem 2.1.
For eachΨ∈(C0(W(X,K)),∥|·|∥)*, there exists a unique measure μ∈M(W(X,K)) with ∥|μ∥|=∥|Ψ∥| such that
(2.6)Ψ(h)=∫W(X,K)hdμ(h∈C0(W(X,K))),
where ∥|Ψ|∥=sup{|Ψ(h)|:h∈C0(W(X,K)),|∥h∥|⩽1}.Definition 2.2.
Forf∈C(X), the function TX,K(f):W(X,K)→ℂ defined by
(2.7)TX,K(f)(x)=f(x)(x∈X),TX,K(f)(x,y)=f(x)-f(y)dα(x,y)((x,y)∈V(K))
is called Leeuw’s extension of f on W(X,K).It is obvious thatTX,K(f)∈Cb(W(X,K)) for each f∈Lip(X,K,α).Theorem 2.3.
Take0<α<1.(i)
TX,K is a linear isometry from the extended Lipschitz algebra (Lip(X,K,α),∥·∥α,K) into (Cb(W(X,K)),∥|·|∥).(ii)
TX,K(Lip(X,K,α)) is a closed linear subspace of (C0(W),∥|·|∥).(iii)
For eachΦ∈(lip(X,K,α),∥·∥α,K)*, there exists μ∈M(W(X,K)) such that(2.8)Φ(f)=∫W(X,K)TX,K(f)dμ(f∈lip(X,K,α)),‖|μ|‖=‖Φ‖.Proof.
(i) It is immediate.
(ii) Sincelip(X,K,α) is a linear subspace Lip(X,K,α), TX,K(lip(X,K,α)) is a linear subspace of Cb(W(X,K)) by (i). Let f∈lip(X,K,α), and let ɛ be an arbitrary positive number. There exists δ>0 such that(2.9)|f(x)-f(y)|dα(x,y)<ɛ,
for all x,y∈K with 0<d(x,y)<δ. Set Eɛ=X∪{(x,y)∈K:d(x,y)⩾δ}. Clearly, Eɛ is a compact subset of W(X,K) and
(2.10)|TX,K(f)(x,y)|<ɛ,
for all (x,y)∈W(X,K)∖Eɛ. It follows that TX,K(f)∈C0(W(X,K)). Therefore, TX,K(lip(X,K,α)) is a subset C0(W(X,K)). Since TX,K is a linear isometry and C0(W(X,K)) is a closed linear subspace of (Cb(W(X,K)),∥|·|∥), we conclude that TX,K(lip(X,K,α)) is a closed linear subspace of (C0(W(X,K)),∥|·|∥).
(iii) LetΦ∈(Lip(X,K,α),∥·∥α,K)* and define ηX,K:lip(X,K,α)→TX,K(lip(X,K,α)) by
(2.11)ηX,K(f)=TX,K(f)(f∈lip(X,K,α)).
Then Φo(ηX,K)-1∈(TX,K(lip(X,K,α)),∥·∥)*. By the Hahn-Banach extension theorem, there exists Ψ∈C0(W(X,K),∥|·|∥)* with ∥|Ψ∥|=∥|Φo(ηX,K)-1∥| such that
(2.12)Ψ(h)=Φo(ηX,K)-1(h)(h∈TX,K(lip(X,K,α))).
By Theorem 2.1, there exists μ∈M(W(X,K)) with ∥|μ∥|=∥|Ψ∥| such that(2.13)Ψ(h)=∫W(X,K)hdμ(h∈C0(W(X,K))).
Therefore,
(2.14)Φ(f)=(Φo(ηX,K)-1)(TX,K(f))=∫W(X,K)TX,K(f)dμ(f∈lip(X,K,α)),‖|μ|‖=‖|Φo(ηX,K)-1|‖.
On the other hand,
(2.15)‖|Φo(ηX,K)-1|‖=sup⁡{|(Φ(f))|:f∈lip(X,K,α),‖|TX,K(f)|‖⩽1}=‖Φ‖.
It follows that ∥|μ∥|=∥Φ∥. This completes the proof.Note that the mapTX,K is not an algebra homomorphism and that its image is not a subalgebra of Cb(W(X,K)).Definition 2.4.
ForΦ∈(lip(X,K,α),∥·∥α,K)*, a measure μ∈M(W(X,K))with∥|μ∥|=∥Φ∥ and with
(2.16)Φ(f)=∫W(X,K)TX,K(f)dμ(f∈lip(X,K,α))
is called a representing measure for Φ on W(X,K).Note that a representing measure forΦ on W(X,K) is not unique.


## 3. Main Results

In this section, by generalizing Theorem1.1, we obtain a sufficient condition for which a linear subspace of lip(X,K,α) is dense in this algebra. In particular, we show that Lip(X,K,1) is dense in (lip(X,K,α),∥·∥α,K).Theorem 3.1.
letP be a linear subspace of lip(X,K,α) which satisfies the following conditions:(a)
ifh∈C(X) with h|K=0, then h∈P¯, where P¯ is the closure of P in (lip(X,K,α),∥·∥α,K).(b)
there is a constantC such that for each finite subset E of K and each f∈lip(X,K,α), there exists g∈P with g|E=f|E and with ∥g∥α,K⩽C∥f∥α,K. Then P is dense in (lip(X,K,α),∥·∥α,K).Proof.
We first show that ifPK={g∈C(K):g=f|Kforsomef∈P}, then PK is dense in the little Lipschitz algebra lip(K,α). Clearly, PK is a linear subspace of lip(K,α). Let E be a Finite subset of K, and let f∈lip(K,α). By Tietze’s extension theorem [7, Theorem  20.4], there exists F∈C(X) such that F|K=fand∥F∥X=∥f∥K. Clearly, F∈lip(X,K,α). by (b), there exists G∈P such that G|E=F|Eand∥G∥α,K⩽C∥F∥α,K. We define g=G|K. Then g∈PK, g|E=f|E and
(3.1)‖g‖α=‖g‖K+pα(g)⩽‖G‖X+pα(g)=‖G‖X+pα,K(G)=‖G‖α,K⩽C‖F‖α,K=C(‖F‖X+pα,K(F))=C(‖f‖K+pα(f))=C‖f‖α.
Thus PK is dense in (lip(K,α),∥·∥α) by Theorem 1.1.
To prove the density ofP in (lip(X,K,α),∥·∥α,K), it is enough to show that if Φ∈(lip(X,K,α),∥·∥α,K)* with Φ(f)=0 for all f∈P, then Φ(f)=0 for all f∈lip(X,K,α).
LetΦ∈(lip(X,K,α),∥·∥α,K)* such that Φ(f)=0 for all f∈P. Continuity of Φ implies that Φ(f)=0 for all f∈P¯. By Theorem 2.3, there exists μ∈M(W(X,K)) such that
(3.2)Φ(F)=∫W(X,K)TX,K(F)dμ(F∈lip(X,K,α)),
where TX,K(F) is Leeuw’s extension of F∈lip(X,K,α) on W(X,K). We claim that
(3.3)Φ(F)=∫W(K)TX,K(F)dμ(F∈lip(X,K,α)).
Let F∈lip(X,K,α). We define the sequence {Yn}n=1∞ of the subsets of X by
(3.4)Yn={x∈X:d(x,K)⩾1n}.
Then Yn is a compact subset of X for each n∈ℕ, Y1⊆Y2⊆⋯⊆X∖K, and ⋃n=1∞Yn=X∖K. Let n∈ℕ. By Urysohn’s lemma, there exists Fn∈C(X) such that ∥Fn∥X=1, Fn|K=0, and Fn|Yn=1. Define Gn=FnF. Then Gn∈C(X) and Gn|K=0. Hence, Gn∈P¯ by (a) and so Φ(Gn)=0. Thus
(3.5)∫W(X,K)TX,K(Gn)dμ=0.
Let χ̃X∖K be the characteristic function of X∖K on W(X,K). It is easy to see that
(3.6)lim⁡n→∞TX,K(Gn)(w)=(TX,K(F)⋅χ̃X∖K)(w),
for all w∈W(X,K). Since ∥TX,K(Gn)∥W(X,K)=1 for all n∈ℕ and 1∈L1(W(X,K),|μ|), we conclude that
(3.7)lim⁡n→∞∫W(X,K)TX,K(Gn)dμ=∫W(X,K)TX,K(F)⋅χ̃X∖Kdμ,
by Lebesgue’s dominated convergence theorem. Thus
(3.8)∫W(X,K)TX,K(F)⋅χ̃X∖Kdμ=0,
by (3.5) and (3.7). It follows that
(3.9)∫X∖KTX,K(F)dμ=0.
Thus (3.3) is justified, by (3.2) and (3.9).
We now define the functionΨ:lip(K,α)→ℂ, by(3.10)Ψ(g)=∫W(K)TK,K(g)dμ|W(K).
Clearly, Ψ is a linear functional on lip(K,α). Since
(3.11)|Ψ(g)|⩽‖TK,K(g)‖W(K)|μ|(W(K))⩽‖g‖α|μ|(W(K)),
for all g∈lip(K,α), we deduce that Ψ∈(lip(K,α),∥·∥α)*. We claim that Ψ(g)=0 for all g∈PK. If g∈PK, there exists f∈P such that f|K=g, and so
(3.12)Ψ(g)=∫W(K)TK,K(g)dμ|W(K)=∫W(K)TK,K(f)dμ=Φ(f)=0.
Therefore, our claim is justified. It follows that Ψ(g)=0 for all g∈lip(K,α), by the density of PK in (lip(X,K,α),∥·∥α,K) and continuity of Ψ on lip(K,α). Let f∈lip(X,K,α). If g=f|K, then g∈lip(K,α), and so Ψ(g)=0. Therefore,
(3.13)Φ(f)=∫W(K)TX,K(f)dμ=∫W(K)TK,K(g)dμ|W(K)=Ψ(g)=0,
by (3.3). This completes the proof.By applying the above result, we show thatLip(X,K,1) is dense in (lip(X,K,α),∥·∥α,K).Bade et al. obtained the following result.Lemma 3.2 (see [4, Lemma  3.3]).
For each finite subsetE of X and each h∈Lip(X,α), there exists f∈Lip(X,1) with f|E=h|E and with ∥f∥α⩽2∥h∥α.We now generalize the above lemma by applying it and Tietze’s extension theorem as follows.Lemma 3.3.
For each finite subsetE of X and each h∈Lip(X,K,α), there exists f∈Lip(X,K,1) with f|E=h|E and with ∥f∥α,K⩽3∥h∥α,K.Proof.
LetE be a finite subset of X, and let h∈Lip(X,K,α). Define g=h|K. Then g∈Lip(K,α). By Lemma 3.2, there exists g0∈Lip(K,1) with g0|E∩K=g|E∩K and with ∥g0∥α⩽2∥g∥α. We now define the function g1:E∪K→ℂ by
(3.14)g1(x)={g0(x),x∈K,h(x),x∈E∖K.
Clearly, E∪K is a compact subset of X and g1∈C(E∪K). By Tietze’s extension theorem, there exists f∈C(X) such that f|E∪K=g1 and ∥f∥X=∥g1∥E∪K. It follows that f∈Lip(X,K,1) and f|E=h|E. Furthermore,
(3.15)‖f‖α,K=‖g1‖E∪K+pα(g0)⩽‖g1‖K+‖g1‖E∖K+Pα(g0)=‖g0‖K+‖h‖E∖K+pα(g0)=‖g0‖α+‖h‖E∖K⩽2‖g‖α+‖h‖E∖K=(‖g‖K+pα(g))+‖h‖E∖K⩽2‖h‖X+‖h‖E∖K+2Pα,K(h)⩽3‖h‖α,K.
This completes the proof.Theorem 3.4.
Lip(X,K,1) is dense in (lip(X,K,α),∥·∥α,K).Proof.
TakeP=Lip(X,K,1). Then P is a linear subspace of lip(X,K,α) and h∈P for all h∈C(X) with h|K=0. Let E be a finite subset of K and f∈lip(X,K,α). By Lemma 3.3, there exists g∈P with g|E=f|E and with ∥g∥α,K⩽3∥f∥α,K. Therefore, P is dense in (lip(X,K,α),∥·∥α,K), by Theorem 3.1.Corollary 3.5.
Lip(X,1) is dense in (lip(X,α),∥·∥α).Proof.
It is enough to takeK=X in Theorem 3.4.



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*Source: 187952-2012-04-19.xml*