The Lipschitz operators algebras Lα(K, A) and lα(K, A) are developed here further, and we study their amenability and weak amenability of these algebras

Tomus 45 (2009), 213–222

ON THE LIPSCHITZ OPERATOR ALGEBRAS

A. Ebadian and A. A. Shokri

Abstract. In a recent paper by H. X. Cao, J. H. Zhang and Z. B. Xu an α-Lipschitz operator from a compact metric space into a Banach spaceA is defined and characterized in a natural way in the sence thatF :K→A is aα-Lipschitz operator if and only if for eachσ∈X^∗the mappingσ◦F is aα-Lipschitz function. The Lipschitz operators algebras L^α(K, A) and l^α(K, A) are developed here further, and we study their amenability and weak amenability of these algebras. Moreover, we prove an interesting result thatL^α(K, A) andl^α(K, A) are isometrically isomorphic toL^α(K) ˇ⊗Aand l^α(K) ˇ⊗Arespectively. Also we study homomorphisms on theL^α_A(X, B).

1. Introduction

Let (K, d) be compact metric space with at least two elements and (X,k · k) be a Banach space over the scalar fieldF(= R or C). For a constantα >0 and an operator T:K→X, set

(1) Lα(T) := sup

s6=t

kT(t)−T(s)k d(s, t)^α , which is called the Lipschitz constant ofT. Define

Tα(x, y) =T(x)−T(y)

d(x, y)^α , x6=y L^α(K, X) ={T:K→X:L_α(T)<∞}

and

l^α(K, X) ={T:K→X:kTα(x, y)k →0 as d(x, y)→0}.

The elements ofL^α(K, X) andl^α(K, X) are called big and little Lipschitz operators, respectively [1].

Let C(K, X) be the set of all continuous operators fromK intoX and for each T ∈C(K, X), define

kTk∞= sup

x∈K

kT(x)k. ForS,T inC(K, X) andλin F, define

(S+T)(x) =S(x) +T(x), (λT)(x) =λT(x), (x∈X).

2000Mathematics Subject Classification: primary 47B48; secondary 46J10.

Key words and phrases: Lipschitz algebras, amenability, homomorphism.

Received October 7, 2008. Editor A. Pultr.

It is easy to see that (C(K, X), k · k∞) becomes a Banach space over F and L^α(K, X) is a linear subspace ofC(K, X). For each elementT ofL^α(K, X), define kTk_α=L_α(T) +kTk_∞.

In their papers [3, 4], Cao, Zhang and Xu proved that (L^α(K, X),k · k_α) is a Banach space over F andl^α(K, X) is a closed linear subspace of (L^α(K, X),k · k_α).

Now, let (A,k · k) be a unital Banach algebra with unite. In this paper, we show that (L^α(K, A),k·kα) is a Banach algebra under pointwise and scalar multiplication andl^α(K, A) is a closed linear subalgebra of (L^α(K, A),k · kα) and study many aspects of these algebras. The spaces L^α(K, A) and l^α(K, A) are called big and little Lipschitz operators algebras. Note that Lipschitz operators algebras are, in fact, extensions of Lipschitz algebras. Sherbert [11, 12], Weaver [13, 14], Honary and Mahyar [7], Johnson [8, 9], Alimohammadi and Ebadian [1], Ebadian [6], Bade, Curtis and Dales [2], studied some properties of Lipschitz algebras. We will study (weak) amenability of Lipschitz operators algebras. Also we study homomorphisms on theL^α_A(X, B).

2. Characterizations of Lipschitz operators algebras

In this section, let (K, d) be a compact metric space which has at least two elements and (A,k · k) to denote a unital Banach algebra over the scalar field F (= R or C).

Theorem 2.1. (L^α(K, A),k · k_α) is a Banach algebra overFand l^α(K, A) is a closed linear subspace of (L^α(K, A),k · kα).

Proof. As we have already L^α(K, A) is a Banach space andl^α(K, A) is a closed linear subspace if it. Now let T,S∈L^α(K, A), and define

(T S)(t) =T(t)S(t) (t∈K). Then

kT Skα=kT Sk∞+L_α(T S)

≤ kTk_∞kSk_∞+ sup

t6=s

kT(t)S(t)−T(s)S(s)k d(t, s)^α

≤ kTk_∞kSk_∞+kTk_∞Lα(S) +kSk_∞Lα(T)

≤ kTk_∞+Lα(T)

kSk_∞+Lα(S)

=kTk_αkSk_α.

So that we see that (L^α(K, A),k · kα) is a Banach algebra andl^α(K, A) is a closed

linear subspace of (L^α(K, A),k · kα).

Theorem 2.2. Let (K, d)be a compact metric space. Then L^α(K, A)is uniformly dense in C(K, A).

Proof. Let f ∈C(K, A). Then for every σ∈A^∗ we haveσ◦f ∈C(K), so that there isg∈L^α(K) such thatkg−σ◦fk∞< ε. We define, the mapη: C→A by

η(λ) =λ·e. It is easy to see that η◦g∈L^α(K, A), and for everyσ∈A^∗, we have

|σ g(x)·e−f(x)

|=|g(x)−(σ◦f)(x)|< ε , (x∈K).

Therefore |σ(η◦g−f)(x)| < ε for every σ∈ A^∗ andx∈K. This implies that k(η◦g−f)(x)k< εfor everyx∈K. Therefore,kη◦g−fk_∞< εand the proof is

complete.

Remark 2.3. LetA, Bbe unital Banach algebras over F. Then the injective tensor A⊗Bˇ is a unital Banach algebra under normk · k, [10].

Theorem 2.4. L^α(K, A) ={F:K→A|σ◦F ∈L^α(K,C),(∀σ∈A^∗)}

Proof. Use the principle of Uniform Boundedness.

Lemma 2.5. Let (E1,k · k1),(E2,k · k2)be Banach spaces. Then for G∈E1⊗Eˇ 2

kGkε= sup

k id⊗φ

(G)k1 : φ∈E₂^∗, kφk ≤1 .

Proof. See [10].

Theorem 2.6. Let(K, d)be a compact metric space andAbe a unital commutative Banach algebra. ThenL^α(K, A)is isometrically isomorphic toL^α(K) ˇ⊗A.

Proof. It is straightforward to prove that the mappingV:L^α(K)×A→L^α(K, A) defined by

V(f, a) =f a (f ∈L^α(K), a∈A), (f a)(x) :=f(x)a (x∈K),

is bilinear. Therefore there exists a unique linear mapT:L^α(K) ˇ⊗A→L^α(K, A) such thatT(f⊗a) =V(f, a) =f a, [10]. For everyG∈L^α(K) ˇ⊗A, there ism∈N, f_j∈L^α(K) anda_j ∈A(1≤j≤m) such thatG=Pm

j=1f_j⊗a_j, so we have kGkε= sup

φ∈A^∗,kφk≤1

k(id⊗φ)(G)kα= sup

φ∈A^∗,kφk≤1

(id⊗φ)X^m

j=1

fj⊗aj

= sup

φ∈A^∗,kφk≤1

m

X

j=1

fjφ(aj)

_α= sup

φ∈A^∗,kφk≤1

h sup

x∈K

m

X

j=1

fj(x)φ(aj)

+ sup

x6=y

|Pm

j=1fj(x)φ(aj)−Pm

j=1fj(y)φ(aj)|

d^α(x, y)

i

= sup

φ∈A^∗,kφk≤1

h sup

x∈K

φX^m

j=1

fj(x)aj

+ sup

x6=y

φ Pm

j=1(fj(x)aj−fj(y)aj) d^α(x, y)

i

≤ sup

φ∈A^∗,kφk≤1

h sup

x∈K

kφk

m

X

j=1

f_j(x)a_j

+ sup

x6=y

kφkkPm

j=1fj(x)aj−Pm

j=1fj(y)ajk d^α(x, y)

i

≤sup

x∈K

m

X

j=1

fj(x)aj

+ sup

x6=y

kPm

j=1f_j(x)a_j−Pm

j=1f_j(y)a_jk d^α(x, y)

=

m

X

j=1

fjaj

∞+pα

X^m

j=1

fjaj

=

m

X

j=1

fjaj

_α=

TX^m

j=1

fj⊗aj

_α

=kT Gkα =⇒ kGkε≤ kT Gkα.

Now let γ >0 be arbitrary, such thatkT Gkα> γ. ThenkPm

j=1f_ja_jkα> γ, and so we have

m

X

j=1

fjaj

_∞+pα

X^m

j=1

fjaj

> γ

⇒sup

x∈K

m

X

j=1

f_j(x)a_j + sup

x6=y

kPm

j=1fj(x)aj−Pm

j=1fj(y)ajk d^α(x, y) > γ

⇒ sup

φ∈A^∗,kφk≤1

h sup

x∈K

m

X

j=1

f_j(x)φ(a_j)

+ sup

x6=y

|Pm

j=1f_j(x)φ(a_j)−Pm

j=1f_j(y)φ(a_j)|

d^α(x, y)

i

> γ

⇒ sup

φ∈A^∗,kφk≤1

h

m

X

j=1

fjφ(aj) ∞+pα

X^m

j=1

fjφ(aj)i

> γ

⇒ sup

φ∈A^∗,kφk≤1

hk(id⊗φ)X^m

j=1

f_j⊗a_j

_∞+p_α

(id⊗φ)X^m

j=1

f_j⊗a_ji

> γ

⇒ sup

φ∈A^∗,kφk≤1

(id⊗φ)X^m

j=1

f_j⊗a_{j α}> γ

⇒

m

X

j=1

fj⊗aj

_ε> γ ⇒ kGkε> γ .

Sinceγ >0 is arbitrary, then we havekT Gkα≤ kGkε. ThereforekT Gkα=kGkε, and this implies thatT is a linear isometry map. So T one-one and continuous map. Now, we show that T is a onto map. For this, we show that the range of T, RT is a closed and dense subset of L^α(K, A). It is easy to see that RT

is closed. Let f ∈ L^α(K, A) and γ > 0. There exist a₁, . . . , a_n ∈ A such that X :=f(K)⊂Sn

i=1B(a_i, γ). SetU_j=f⁻¹(B(a_j, γ)) wherej= 1, . . . , n. Then there

exist f₁, . . . , f_n ∈L^α(K, A) and σ∈A^∗such that supp(f_j)⊂U_j forj = 1, . . . , n andσ◦(f₁+. . .+f_n) = 1. For everyx∈K we have,

f(x)− (σ◦f1)a1+· · ·+ (σ◦fn)an

(x)

=

f(x) (σ◦f1)(x) +· · ·+ (σ◦fn)(x)

− (σ◦f₁)(x)a₁+· · ·+ (σ◦f_n)(x)a_n

=

(σ◦f₁)(x) f(x)−a₁

+· · ·+ (σ◦f_n)(x) f(x)−a_n

≤

n

X

i=1

(σ◦fi)(x)

f(x)−ai

< γ , since suppf_j⊂U_j. Therefore,

f− (σ◦f₁)a₁+· · ·+ (σ◦f_n)a_{n α}< γ . This implies that

f−

n

X

i=1

T(σ◦fi⊗ai) α

< γ .

We conclude that ¯RT =L^α(K, A). SoRT =L^α(K, A), sinceRT is closed. Hence T is a onto map. Also by product• onL^α(K) ˇ⊗A

(f⊗a)•(g⊗b) =f g⊗ab (f, g∈L^α(K), a, b∈A),

clearlyT is homomorphism.

Furthermore T is open map, for this purpose, let τ and τ⁰ be topologies on L^α(K) ˇ⊗AandL^α(K, A) respectively. LetU ∈τ, we show thatT(U)∈τ⁰. Letpbe a limit point inL^α(K, A)\T(U). Then there exists a sequence{p_n}inL^α(K, A)\T(U) converges to p. SinceT is onto, there is a sequence{q_n}and qinL^α(K) ˇ⊗Asuch that T(qn) = pn and T q = p. Therefore T(qn) converges to p in L^α(K). Since qn∈L^α(K) ˇ⊗A, we can findm∈N,f_j⁽ⁿ⁾∈L^α(K) anda⁽ⁿ⁾_j ∈Asuch that whenever 1≤j≤mwe have

(1) T(q_n) =

m

X

j=1

f_j⁽ⁿ⁾a⁽ⁿ⁾_j .

Also, sinceq∈L^α(K) ˇ⊗A there existr∈N,gi∈L^α(K) andbi∈Asuch that

(2) p=T(q) =

r

X

i=1

g_ib_i.

Since kT(qn)−pkα → 0 as n → ∞, for every positive number γ there exists a positive integerN such that

(3)

m

X

j=1

f_j⁽ⁿ⁾a⁽ⁿ⁾_j −

r

X

i=1

g_ib_{i α}< γ ,

whenn≥N. By applying (3), we have sup

(x∈K)

m

X

j=1

f_j⁽ⁿ⁾(x)a⁽ⁿ⁾_j −

r

X

i=1

g_i(x)b_i + sup

(x6=y)

1 d(x, y)^α

×

m

X

j=1

f_j⁽ⁿ⁾(x)a⁽ⁿ⁾_j −

r

X

i=1

g_i(x)b_i−

m

X

j=1

f_j⁽ⁿ⁾(y)a⁽ⁿ⁾_j +

r

X

i=1

g_i(y)b_i < γ . Therefore ifσ∈A^∗ withkσk ≤1 then

sup

(x∈K)

m

X

j=1

f_j⁽ⁿ⁾(x)σ(a⁽ⁿ⁾_j )−

r

X

i=1

gi(x)σ(bi) + sup

(x6=y)

1 d(x, y)^α

×

m

X

j=1

f_j⁽ⁿ⁾(x)σ(a⁽ⁿ⁾_j )−

r

X

i=1

gi(x)σ(bi)

m

X

j=1

f_j⁽ⁿ⁾(y)σ(a⁽ⁿ⁾_j ) +

r

X

i=1

gi(y)σ(bi) < γ . This implies that

(4)

m

X

j=1

f_j⁽ⁿ⁾σ(a⁽ⁿ⁾_j )−

r

X

i=1

giσ(bi) _α< γ Now by using (4), for everyφ∈L^α(K)^∗ withkφkα≤1 we have,

φX^m

j=1

f_j⁽ⁿ⁾σ(a⁽ⁿ⁾_j )−

r

X

i=1

giσ(bi) < γ , hence

(5)

m

X

j=1

φ(f_j⁽ⁿ⁾)σ(a⁽ⁿ⁾_j )−

r

X

i=1

φ(gi)σ(bi) < γ , By (5), we conclude

(6) sup

m

X

j=1

φ(f_j⁽ⁿ⁾)σ(a⁽ⁿ⁾_j )−

r

X

i=1

φ(g_i)σ(b_i)

< γ , kσk ≤1, kφkα≤1. Thereforekqn−qk≤γand henceqn→qorqn→T⁻¹pinL^α(K) ˇ⊗A. This show that p∈T(U)^c.

Remark 2.7. By using the above theorem we can prove thatl^α(K, A)∼=l^α(K) ˇ⊗A.

3. (Weak) Amenability ofL^α(K, A)

Let Abe a Banach algebra and X be a BanachA-module over F. The linear map D: A →X is called an X-derivation onA, ifD(ab) = D(a)·b+a·D(b), for everya, b∈A. The set of all continuesX-derivations onA is a vector space over F which is denoted by Z¹(A, X). For each x ∈ X the map δx: A → X, defined byδx(a) =a·x−x·a, is a continuesX-derivation onA. TheX-derivation D:A→X is called an inner derivation onA if there exists anx∈X such that D=δ_x. The set of all innerX-derivations onA is a linear subspace ofZ¹(A, X)

which is denoted by B¹(A, X). The quotient space Z¹(A, X)/B¹(A, X) is deno- ted byH¹(A, X) and is called the first cohomology group ofAwith coefficients inX. Definition 3.1. The Banach algebra A over F is called amenable if for every Banach A-moduleX over F,H¹(A, X^∗) ={0}. The Banach algebra Aover F is called weakly amenable ifH¹(A, A^∗) ={0}.

The notion of amenability of Banach algebras were first introduced by B. E.

Johnson in 1972 [8]. Bade, Curtis and Dales [2], studied the (weak) amenability of Lipschitz algebras in 1987 [2]. In this section, we study the (weak) amenability of L^α(K, A).

For every Banach algebraB, let Φ_B be the space of maximal ideal ofB.

Definition 3.2. LetA be a commutative Banach algebra and letφ∈Φ_A∪ {0}.

The non-zero linear functionalD onA is called point derivation atφif D(ab) =φ(a)D(b) +φ(b)D(a), (a, b∈A).

Lemma 3.3. For each non-isolated point x ∈ K and σ ∈ A^∗, if the map φ:L^α(K, A)→Cis given by

φ(f) = (σ◦f)(x), f ∈L^α(K, A) then φ∈ΦL^α(K,A).

Proof. Obvious.

Let (K, d) be a fixed non-empty compact metric space, set

∆ =

(x, y)∈K×K:x=y , W =K×K−∆.

We now examine the amenability and weak amenability of Lipschitz operators algebras L^α(K, A) andl^α(K, A).

Theorem 3.4. Let (K, d)be an infinite compact metric space and takeα∈(0,1].

ThenL^α(K, A) is not weakly amenable.

Proof. Letxbe a non-isolated point inK. We define Wx:=

{(xn, yn)}^∞_n=1: (xn, yn)∈W, (xn, yn)→(x, x) asn→ ∞ . For the net w={(x_n, y_n)}^∞_n=1 in W_xandσ∈A^∗, we put

w(f) =(σ◦f)(xn)−(σ◦f)(yn)

d(x_n, y_n)^α , f ∈L^α(K, A) thenkw(f)k_∞≤ kσk kfkα. Hence,wis continues. Now set

D_w(f) = LIM w(f)

, f ∈L^α(K, A) ,

where LIM(·) is Banach limit [12]. We show that the linear mapD_w is a non-zero point derivation atφ, whichφis given by Lemma 6. We have,

Dw(f g) = LIM(w(f g))

= LIM(σ◦f g)(xn)−(σ◦f g)(yn) d(x_n, y_n)^α

= LIM 1

d(xn, yn)^α

σ◦ f(x_n)g(x_n)−f(x_n)g(y_n)

= LIM 1

d(x_n, y_n)^α

σ◦ f(xn) g(xn)−g(yn)

+g(yn) f(xn)−f(yn)

= (σ◦f)(x) LIM w(g)

+ (σ◦g)(x) LIM w(g)

=φ(f)Dw(g) +φ(g)Dw(f)

Therefore, by the continuityf,g and properties of Banach limit we concludeD_w is a non-zero, continues point derivation at φonL^α(K, A), an so by [5], L^α(K, A)

is not weakly amenable.

Corollary 3.5. L^α(K, A)is not amenable.

Theorem 3.6. LetK⊆C be an infinite compact set, and takeα∈(0,1). Then l^α(K, A)is not amenable.

Proof. Letx0∈K. We define

Mx₀ :={f ∈l^α(K, A) : (σ◦f)(x0) = 0 ∀σ∈A^∗}. Ifσ∈A^∗, then for eachf ∈M_x²

0 we have (σ◦f)(x)

d(x, x0)^2α −→0 as d(x, x0)−→0. For β ∈ (α,2α), set fβ(x) := η d(x, x0)^β

, x ∈ K where, the map η: C → A defined byη(λ) =λ·e. Thenfβ∈Mx₀ and{fβ+M_x²₀ : β ∈(α,2α)} is a linearly independent set in ^M_M^x2⁰

x0

becausex0 is non-isolated inK. ThereforeM_x²₀ has infinite codimension in Mx₀, and so Mx₀ 6= M_x²₀ then by [5] Mx₀ has not a bounded approximate identity, and sinceMx₀ is closed ideal inl^α(K, A), thenl^α(K, A) is

not amenable.

Theorem 3.7. Let(K, d)be a compact metric space andAbe a unital commutative Banach algebra. If ¹₂ < α <1, thenl^α(T, A)is not weakly amenable, whereT is unit circle in complex plane.

Proof. By Remark 2.7, we have l^α(T, A)∼=l^α(T) ˇ⊗A. Since by [5],l^α(T) is not weakly amenable, hencel^α(T, A) is not weakly amenable.

Corollary 3.8. LetA be a finite-dimensional weakly amenable Banach algebra. If 0< α < ¹₂, thenl^α(K, A)is weakly amenable.

Proof. By [10],l^α(K) ˆ⊗Ais weakly amenable. Now by [10], we have l^α(K) ˆ⊗A∼= l^α(K) ˇ⊗Aand this implies that l^α(K) ˇ⊗Ais weakly amenable and so l^α(K, A) is

weakly amenable.

4. Homomorphisms on theL^α_A(X, B)

Definition 4.1. Let (X, d) be a compact metric space inC,α∈(0,1], (B,k · k) be a commutative Banach algebra with unite, andB^∗ be the dual space ofB, define

A(X, B) =

f ∈C(X, B) : Λ◦f is analytic in interior ofX,Λ∈B^∗ L^α_A(X, B) =

f ∈L^α(X, B) : Λ◦f is analytic in interior ofX, Λ∈B^∗ l^α_A(X, B) =

f ∈l^α(X, B) : Λ◦f is analytic in interior of X, Λ∈B^∗ In this case, we have

L^α_A(X, B) =L^α(X, B)∩A(X, B) and

l^α_A(X, B) =l^α(X, B)∩A(X, B). SoL^α_A(X, B)∼=L^α_A(X) ˇ⊗B andl_A^α(X, B)∼=l_A^α(X) ˇ⊗B.

Theorem 4.2. Every characterχonL^α_A(X, B)(andl_A^α(X, B)) is of formχ=ψ◦δz

for some character ψ onB and some z∈X.

Proof. SinceL^α_A(X, B)∼=L^α_A(X) ˇ⊗B, letj:L^α(X)→L^α_A(X, B),h7→h⊗e, be the canonical embedding. Then there isz∈X such thatχ◦j is the evaluation in z, that isχ◦j=δz where δz(ϕ) =ϕ(z). Consider the ideal

I:=

f ∈L^α_A(X, B) :f(z) = 0 .

We will show thatI is contained in the kernel ofχ. Givenf ∈Iwe define, ϕ(ω) :=

(ω−z if ω6=z;

0 if ω=z.

and

g(ω) :=





 f(ω)

ω−z if ω6=z;

f⁰(z) if ω=z.

Sincef has a Taylor series expansion f(ω) =

∞

X

n=1

f⁽ⁿ⁾(z)

n! (ω−z)ⁿ

around z, it is easy to see that Λ◦g is holomorphic (Λ∈ B^∗), and hence g ∈ L^α_A(X, B). We have

χ(f) =χ j(ϕ)g

= (χ◦j)(ϕ)χ(g) =δ_z(ϕ)χ(g) =ϕ(z)χ(g) = 0.

The evaluation δ_z is an epimorphism and since kerδ_z=I ⊂kerχ, we obtain the desired factorization χ=ψ◦δ_z for some characterψ onB.

References

[1] Alimohammadi, D., Ebadian, A.,Headberg’s theorem in real Lipschitz algebras, Indian J.

Pure Appl. Math.32(2001), 1479–1493.

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Faculty of Basic Science, Science and Research Brunch Islamic Azad University (IAU), Tehran, Iran

E-mail:[email protected]

Department of Mathematics, Faculty of Science Urmia University, Urmia, Iran

E-mail:[email protected]