Journal of Applied Sciences

Year: 2009 | Volume: 9 | Issue: 8 | Page No.: 1482-1488
DOI: 10.3923/jas.2009.1482.1488

n-Approximately Weak Amenability of Banach Algebras

H. Najafi and T. Yazdanpanah

Abstract: We introduce new notions of approximate amenability for a Banach algebra A. A Banach algebra A is n-approximately weakly amenable, for n ∈ N, if every continuous derivation from A into the n-th dual space A⁽ⁿ⁾ is approximately inner. First we examine the relation between m-approximately weak amenability and n-approximately weak amenability for distinct m,n ∈ N. Then we investigate (2n+1)-approximately weak amenability of module extension Banach algebras. Finally, we give an example of a Banach algebra that is 1-approximately weakly amenable but not 3-approximately weakly amenable.

Keywords: module extension Banach algebras, Banach algebras, amenability, n-weak amenability and approximate amenability

INTRODUCTION

Let A be a Banach algebra and X a Banach A-bimodule. A derivation from A into X is a bounded linear map satisfying:

For each x ∈ X we denote by ad_x the derivation D(a) = ax-xa for all a ∈ A, which is called inner derivation. We denote by Z¹(A, X) the space of all derivations from A into X and by B¹(A,X) the space of all inner derivations from A into X. The first cohomology group of A and X which is denoted by H¹(A,X), is the quotient space Z¹(A, X)/B¹(A, X). A Banach algebra A is amenable if H¹(A, X*) = 0 for each A-bimodule X (X* is the dual space of X which is an A-bimodule as usual). The concept of amenability for a Banach algebra A, introduced by Johnson (1972). The Banach algebra A is weakly amenable if H¹(A, A*) = 0. Ghahramani and Loy (2004) and Dales et al. (1998) introduced several modifications of this notion. We recall the definitions in definitions 1 and 2, below:

Definition 1: A Banach algebra A is called approximately amenable if for each A-bimodule X and for each derivation D: A→X* there is a net (x_α) ⊆ X such that for all a ∈ A.

Definition 2: A Banach algebra A is called n-weakly amenable if H¹(A,A⁽ⁿ⁾) = 0, where, A⁽ⁿ⁾ is the n-th dual space of A.

Dales et al. (1998) investigated the relation between m-weak amenability and n-weak amenability for distinct m,n ∈ N. They obtained important results on Banach algebras and they characterized large classes of them. Ghahramani and Loy (2004) extensively studied approximate amenability of Banach algebras and they opened a new research field on amenability.

Many other researchers have followed these studies and worked on this topic. For example, Dales et al. (2006) investigated this topic on Banach sequence algebras. Lashkarizadeh and Samea (2005) studied the approximate amenability for large classes of semiegroup algebras. Ghahramani and Loy (2004) developed valuable results and gave new proofs for the characterization of amenability for Beurling algebras. Also, Choi et al. (2008) developed the recent research and as a result, they solved Johnson’s (1972) problem which states that for any locally compact group G, the group algebra L¹(G) is n-weakly amenable for each n ∈ N.

The contribution of this study is defining a new notion of amenability which helps to characterize Banach algebras. Such as amenability, n-weak amenability and approximate amenability, present definition determines differences between two Banach algebras.

In this study, we compose two definitions 1 and 2 together and define n-approximately weak amenability and we determine the relations between m-approximately weak amenability and n-approximately weak amenability for distinct m, n ∈ N (N denotes the set of all positive integers). Then we investigate (2n+1)-approximate weak amenability of module extension Banach algebras. Finally, we give a counter example which shows that approximately weak amenability does not imply 3-approximately weak amenability.

n-APPROXIMATE WEAK AMENABILITY

Definition 3: Let A be a Banach algebra and n ∈ N. Then A is n-approximately weakly amenable, for n ∈ N, if every continuous derivation from A into the n-th dual space A⁽ⁿ⁾ is approximately inner. We say that A is approximately weakly amenable if A is 1-approximately weakly amenable.

Proposition 1: Let A be a Banach algebra and n ∈ N. Assume that A is (n+2)-approximately weakly amenable. Then A is n-approximately weakly amenable.

Proof: Let D: A→A⁽ⁿ⁾ be a bounded derivation. Then D: A→A⁽ⁿ⁺²⁾ is a bounded derivation, so there exists a net (Λ_α) ⊆ A⁽ⁿ⁺²⁾ such that Let P: A⁽ⁿ⁺²⁾ →A⁽ⁿ⁾ be the canonical projection and λ_α = P(Λ_α). Then and so D is approximately inner.

Proposition 2: Let A be a Banach algebra and n ∈ N. Suppose that A is (2n-1)-approximately weakly amenable. Then A² is dense in A.

Proof: By proposition 1, it is sufficient to prove the proposition for case n = 1. Let Φ ∈ A* and Φ|A² = 0. Then D: A→A* with D(a) = Φ(a)Φ is a bounded derivation. Thus there exists a net (Φ_α) ⊆ A* such that

Then for a ∈ A we have:

Thus Φ = 0 and so A² is dense in A.

Let A be a non-unital Banach algebra. We denote by A^# the unitization of A which is A^# = A ⊕ C with the product:

It is obvious that A^# is a Banach algebra as well. We denote by e* the bounded linear functional on A^# which is zero on A and e*(1) = 1. By these notations we have the following identifications:

The module actions of A^# on A^#(2n) are same az the multiplication on A^# and so A⁽²ⁿ⁾ is a submodule of A^#(2n). The module actions of A^# on A^#(2n+1) are as follows:

and so A⁽²ⁿ⁺¹⁾ is not a submodule of A^#(2n+1).

Theorem 1: Let A be a non-unital Banach algebra and n ∈ N.

Proof

The following example shows that (2n+1)-weak amenability and (2n+1)-approximately weak amenability are two different notions of amenability.

Example 1: Let M_k denote the algebra of kxk matrices over C (the space of complex numberc), with norm:

It is easy to see that M_k⁽ⁿ⁾ ≅M_k for each n ∈ N. Set A_n = (M₂n)^# and

Where:

By example 6.2 of Ghahramani and Loy (2004) and proposition 1.2 of Dales et al. (1998), A is (2n+1)-approximately weakly amenable (approximately amenable) but it is not (2n+1)-weakly amenable.

We denote by ο and ◊ the first and second Arens product on A**, respectively. The Banach algebra A is called Arens regular if ο = ◊ (Palmer, 1994).

Proposition 3: Let A be an Arens regular Banach algebra and let every D ∈ Z¹(A**, A**) be approximately inner. Then A is 2-approximately weakly amenable.

Proof: Let D ∈ Z¹(A, A**), then by theorem 1.9 of Dales et al. (1998) there exists such that where, â denotes the canonical image of a ∈ A in A**. Thus there exists a net (Φ_α) ⊆ A** such that

So that, for a ∈ A, we have and so A is 2-approximately weakly amenable.

Theorem 2: Let A be a Banach algebra such that A^(2n–2) is Arens regular for each n ∈ N, (A⁽⁰⁾ = A). If every D ∈ Z¹(A⁽²ⁿ⁾, A⁽²ⁿ⁾) is approximately inner, then A is 2n-approximately weakly amenable for each n ∈ N.

Proof: The case n = 1 is proposition 3. Assume that every Banach algebra with the stated properties is 2k-approximately weakly amenable. We show that A is (2k+2)-approximately weakly amenable. Let D ∈ Z¹(A, A^(2k+2)) and let P:A^(2k+4)→A^(2k+2) be the canonical projection. By propositions 1.7 and 1.8 of Dales et al. (1998), we have D** ∈ Z¹(A**, A^(2k+4)) (D** is the second dual of D) and P is an A**-bimodule homomorphism. Let By applying A** instead of A, since A** is 2k-approximately weakly amenable, there exists a net such that

Therefore, for each a ∈ A and so A is (2k+2)-approximately weakly amenable.

It has been discussed by Dales et al. (1998) that any commutative, weakly amenable Banach algebra (or equivalently commutative, approximately weakly amenable Banach algebra) is n-weak amenable for each n ∈ N.

The next theorem is the partial result for general case of the earlier fact in special case. First, we note that (A⁽²ⁿ⁺²⁾, ο) is the second dual of (A⁽²ⁿ⁾, ο) for n ∈ N. Also, (A^(2k), ο) is a subalgebra of (A⁽²ⁿ⁾, ο) for k, n ∈ N and k≤n.

Theorem 3: Let A be an approximately weakly amenable Banach algebra such that A is an ideal in (A**, ο). Then A is (2n+1)-approximately weakly amenable for each n ∈ N.

Proof: Let Aⁿ be the linear span of Since, A is an ideal in (A**, ο), the operators L_a and R_a: A→A (the left and right multiplication, respectively) are weakly compact for each a ∈ A. Thus L_a⁽²ⁿ⁾ and R_a⁽²ⁿ⁾: A⁽²ⁿ⁾→A⁽²ⁿ⁾ (the n-th dual operators of L_a and R_a) are weakly compact. Now we have A.A⁽²ⁿ⁾, A⁽²ⁿ⁾ A ⊆ A^(2n-2) for n ∈ N and so Aⁿ.A⁽²ⁿ⁾, A⁽²ⁿ⁾.Aⁿ ⊆ A. Since, A⁽²ⁿ⁺¹⁾ = A* ⊕ A^⊥ and A is approximately weakly amenable, it suffices to show that every derivation D: A→A^⊥ is approximately inner. For such derivation D and for a, b ∈ Aⁿ and Ψ ∈ A⁽²ⁿ⁾ we have:

Since, Ψ.a and b.Ψ ∈ A. Thus D(ab) = 0 and so D|_A2n = 0. Since, A²ⁿ is dense in A, by proposition 2, we have D = 0. Therefore, A is (2n+1)-approximately weakly amenable.

Corollary 1: Let A be a Banach algebra such that A is an ideal in (A**, ο). Then the following are equivalent:

Dales et al. (1998) proved that every C*-algebra is n-weakly amenable for each n ∈ N, so obviously every C*-algebra is n-approximately weakly amenable for each n ∈ N.

(2n+1)-APPROXIMATELY WEAK AMENABILITY OF A ⊕ X

Let A be a Banach algebra and X be a Banach A-bimodule. Then the module extension Banach algebra corresponding to A and X is A ⊕ X, the l₁-direct sum of A and X with the algebra product defined as follows:

We investigate (2n+1)-approximately weak amenability of module extension Banach algebra A ⊕ X. As it has been discussed by Zhang (2002), according to Dales et al. (1998), X** is a Banach A**-bimodule, where, A** is equipped with the first Arens product. The module actions are defined as follows:

For x ∈ X, f ∈ X*, Φ ∈ X** and u ∈ A**, define Φf, fx ∈ A* and uf ∈ X** by

Then, for Φ ∈ X** and u ∈ A**, define uΦ, Φu ∈ X** by

These give the left and right A**-module actions on X**. Also, the definition for uf with u ∈ A** and f ∈ X* gives a left Banach A**-module action on X*. When u = a ∈ A, all the above A**-module actions agree with the A-module actions on the corresponding dual modules X* and X**.

Viewing A⁽²ⁿ⁾ as a new A and X⁽²ⁿ⁾ as a new X, the preceding procedure will successively define X⁽²ⁿ⁺²⁾ as a Banach A⁽²ⁿ⁺²⁾-bimodule. The first Arens product is consistently assumed on each A⁽²ⁿ⁾. Now suppose that the bimodule action of A⁽²ⁿ⁾ on X⁽²ⁿ⁾ has been defined, where, n≥1. Then in a natural way, X^(2n+k), k≥1, is a Banach A⁽²ⁿ⁾-bimodule with the module multiplications uΛ and Λu ∈ X^(2n+k), for Λ ∈ X^(2n+k) and u ∈ A⁽²ⁿ⁾, defined by:

If u = a ∈ A, these module actions coincide with A-module actions on X^(2n+k). Then, for F ∈ X⁽²ⁿ⁺¹⁾ and Φ ∈ X⁽²ⁿ⁺²⁾, define FΦ, ΦF ∈ A⁽²ⁿ⁺¹⁾ by:

For a Banach space Y and an element y ∈ Y denote by the image of y in Y** under the canonical mapping. When F ∈ X⁽²ⁿ⁺¹⁾ and Φ ∈ X⁽²ⁿ⁾, we denote by FΦ and by ΦF. It is easy to check that:

By using the canonical image of F or Φ in the appropriate 2l-th dual space of the space that it belongs to, we can then signify a meaning for FΦ and ΦF for every F ∈ X⁽²ⁿ⁺¹⁾ and Φ ∈ X^(2m); they are elements of A^(2k+1), where, k = max {m-1, n}. Now for μ ∈ A⁽²ⁿ⁺²⁾ and F ∈ X⁽²ⁿ⁺¹⁾, we define μF ∈ X⁽²ⁿ⁺¹⁾ by:

This actually defines a left Banach A⁽²ⁿ⁺²⁾-module action on X⁽²ⁿ⁺¹⁾.

Finally, for μ ∈ A⁽²ⁿ⁺²⁾ and Φ ∈ X⁽²ⁿ⁺²⁾, define μΦ, Φμ ∈ X⁽²ⁿ⁺²⁾ by:

These actually define the A⁽²ⁿ⁺²⁾-module actions on X⁽²ⁿ⁺²⁾.

It is easy to see that (A ⊕ X)⁽²ⁿ⁺¹⁾ can be identified with A⁽²ⁿ⁺¹⁾ ⊕ ₄ X⁽²ⁿ⁺¹⁾, the l_∞-direct sum of A⁽²ⁿ⁺¹⁾ and X⁽²ⁿ⁺¹⁾. Also, the (A ⊕ X)-bimodule actions on A⁽²ⁿ⁺¹⁾ ⊕ ₄X⁽²ⁿ⁺¹⁾ are as follows:

where, a ∈ A, x ∈ X, F ∈ A⁽²ⁿ⁺¹⁾ and G ∈ X⁽²ⁿ⁺¹⁾.

It is assume that A is a Banach algebra, X is a Banach A-bimodule and A ⊕ X is their corresponding module extension Banach algebra.

Lemma 1: Let T:X→A⁽²ⁿ⁺¹⁾ be a continuous A-bimodule homomorphism. Then defined by is a bounded derivation. The derivation is approximately inner if and only if there exists a net (F_α) ⊆ X⁽²ⁿ⁺¹⁾ such that 12158793 598 9797 7979

for all a ∈ A and x ∈ X.

Proof: It is routinely checked that is a bounded derivation. Let be approximately inner. Then there are nets (G_α) ⊆ A⁽²ⁿ⁺¹⁾ and (F_α) ⊆ A⁽²ⁿ⁺¹⁾ such that

Thus and so Since, ((a, 0)) = (0, 0), we have and so Therefore, For the converse let such a net (F_α) exists. Then

Therefore, is approximately inner.

Lemma 2: Let D: A→X⁽²ⁿ⁺¹⁾ be a bounded derivation and D⁽²ⁿ⁺¹⁾ be the (2n+1)-th dual operator of it. Then : A ⊕ X→(A ⊕ X)⁽²ⁿ⁺¹⁾ defined by ((a, x)) = (-D⁽²ⁿ⁺¹⁾ (x),D(a)), for all a ∈ A and x ∈ X, is a bounded derivation. Moreover, if is approximately inner, then so is D. Also, if D is approximately inner, then there is a net of bounded derivations for all α and for all a ∈ A and is inner.

Proof: By lemma 3.4 of Zhang (2002), is a bounded derivation. If is approximately inner, then there are nets (G_α) ⊆ A⁽²ⁿ⁺¹⁾ and (F_α) ⊆ X⁽²ⁿ⁺¹⁾ such
that:

Thus,

Therefore, for all a ∈ A and so D is approximately inner. For the converse let for a ∈ A and (F_α) ⊆ X⁽²ⁿ⁺¹⁾. Let T_α: X→A⁽²ⁿ⁺¹⁾ be defined by:

and let be defined by:

Then for a ∈ A and x ∈ X we have and

Thus is inner and so is the net as required.

Lemma 3: Let D: A→A⁽²ⁿ⁺¹⁾, n≥0, be a bounded derivation. Then defined by is a bounded derivation. Moreover, is approximately inner if and only if D is approximately inner.

Proof: It is routine to check that is a bounded derivation. Now let D be approximately inner. Thus for some net and for all a ∈ A. Then

and so is approximately inner. Conversely, let be approximately inner. Thus there exist nets and such that

Since,

and so

Lemma 4: Let T: X→X⁽²ⁿ⁺¹⁾, n≥0, be a continuous A-bimodule homomorphism, satisfying xT(y)+T(x)y = 0 for all x,y ∈ X. Then defined by is a bounded derivation. Moreover, is approximately inner if and only if T = 0.

Proof: It is routine to check that is a bounded derivation. Let be approximately inner. Thus for some nets

Since, we have
Therefore, T = 0.

Theorem 4: For n≥0, the module extension Banach algebra A ⊕ X is (2n+1)-approximately weakly amenable if and only if the following conditions hold:

Proof: Let A ⊕ X be (2n+1)-approximately weakly amenable. Then by lemmas 2 and 3, A is (2n+1)-approximately weakly amenable and every derivation from A into X⁽²ⁿ⁺¹⁾ is approximately inner. Furthermore, lemma 1 gives condition (iii) and lemma 4 gives condition (iv). For the converse, let:

be the canonical projections and let and are the canonical inclusion maps. Obviously, P₁ and P₂ are A-bimodule homomorphisms and is an algebra homomorphism. Let be a bounded derivation. Then is a bounded derivation and so and are bounded derivations. Thus they are approximately inner by conditions (i) and (ii). Therefore, is approximately inner. By lemmas 2-4:

is a bounded derivation and there is a net of bounded derivations such that for all α and for all a ∈ A and

is inner. We have

Let:

Then, for all α, is a bounded derivation that satisfies for all a ∈ A. Moreover:

and

Clearly is a continuous A-bimodule homomorphism. By condition (iii), for each α, there exists a net such that:

for a ∈ A and for x ∈ X. On the other hand:

Thus

Therefore, by condition (iv). Thus we have

So, is approximately inner. Thus

is approximately inner.

A COUNTER EXAMPLE

Here, motivated by Zhang (2002), we consider the case that the module action on one side of X is trivial. Then we give a counter example which shows the converse of proposition 1 is not correct.

We denote by X₀ and Y₀ the A-bimodules X with trivial right module action and Y with trivial left module action, respectively. By proposition 2, in case X = X₀ it is easy to see that conditions (iii) and (iv) of Theorem 4 are reduced as follow:

(iii)₀ for each continuous A-bimodule homomorphism T: X₀→A⁽²ⁿ⁺¹⁾, there is a net (F_α) ⊆ X₀⁽²ⁿ⁺¹⁾ such that for a ∈ A and
(iv)₀ AX₀ is dense in X₀.

Proposition 4: Let A be a (2n+1)-approximately weakly amenable Banach algebra with bounded approximate identity and let AA⁽²ⁿ⁾ = A⁽²ⁿ⁾. Then A ⊕ X₀ is (2n+1)-approximately weakly amenable if and only if AX₀ is dense in X₀.

Proof: By proposition 1.5 of Johnson (1972), condition (ii) in theorem 4 always hold. Let T: X₀→A⁽²ⁿ⁺¹⁾ be a continuous A-bimodule homomorphism. Then aF, T(x) = F, T(xa) = 0 for all a ∈ A and F ∈ A⁽²ⁿ⁾. Thus T(x)|_AA(2n) = 0 and so T = 0 since AA⁽²ⁿ⁾ = A⁽²ⁿ⁾. So condition (iii)₀ holds.

For n = 0, an easy application of Cohen Factorization Theorem (Bonsall and Duncan, 1973) implies that:

Corollary 2: Let A be an approximately weakly amenable Banach algebra with bounded approximate identity. Then A ⊕ X₀ is approximately weakly amenable if and only if AX₀ is dense in X₀.

A dual result to corollary 2 is as follows:

Corollary 3: Let A be an approximately weakly amenable Banach algebra with bounded approximate identity. Then A ⊕ ₀Y is approximately weakly amenable if and only if A₀Y is dense in ₀Y.

If X and Y are Banach A-bimodules, we denote by X ⊕ ₁Y the l₁-direct sum of X and Y.

Proposition 5: Suppose that A ⊕ Y and A ⊕ Y are approximately weakly amenable. Then the following are equivalent:

Proof: The proof is quite similar to the proof of lemma 7.1 of Zhang (2002) and so we omit it’s proof.

Corollary 4: The algebra A ⊕ (X ⊕ ₁Y) is approximately weakly amenable if and only if both A ⊕ X and A ⊕ Y are approximately weakly amenable and condition (ii) or (iii) in proposition 5 holds.

Proof: This is an immediate consequence of theorem 4 and proposition 5.

Let H be an infinite dimensional Hilbert space. We denote by B(H) and K(H) the space of all bounded linear operators and compact operators on H, respectively. By lemma 7.4 of Zhang (2002), there is an element a₀ ∈ B(H)\K(H) such that a₀ is not right invertible in B(H) and a₀K(H) is dense in K(H). For such element a₀ ∈ B(H)\K(H), a₀B(H) is a proper right ideal of B(H). Thus cl(a₀B(H)), the closure of a₀B(H), is also a proper right ideal of B(H). So there is 0≠Λ ∈ B(H)* such that Λ.a₀ = 0. Then ΛB(H)≠{0} is a right B(H)-submodule of B(H)*. Set

Example 2: By above notations, B(H) ⊕ (X₀ ⊕ _{1 0}Y) is (approximately) weakly amenable but not 3-approximately weakly amenable.

Proof: By example 7.5 (Zhang, 2002), B(H) ⊕ (X₀ ⊕ _{1 0}Y) is weakly amenable. Also, it is shown that B(H) ⊕ (X₀ ⊕ _{1 0}Y) fails condition (iv) theorem 2.1 (Zhang, 2002) for m = 1, which is condition (iv) of present theorem 4 for n = 1. Thus B(H) ⊕ (X₀ ⊕ _{1 0}Y) is not 3-approximately weakly amenable.

ACKNOWLEDGMENT

The authors would like to thank the Persian Gulf University Research Council for their financial support.

REFERENCES