On Homological Properties of Some Module Derivations on Banach Algebras

In recent years, lots of papers have been published on module amenability. In this paper, our main aim is to study the homological properties of various module derivations and prove some results about module amenability. So this paper continous a line investigation in [3], [4] for Banach algebras.


Introduction
In noncommutative rings, the notion of a derivation was extended to a (σ, τ )derivation, a right (left) derivation, a Jordan derivation, a Lie derivation, and a central derivation etc.The properties of various derivations were discussed in many papers with respect to the ring structures.Later, these derivations were generalized by M. Bresar [5] and A. Nakajima [14].Furthermore, some properties of derivations and generalized derivations, such as homological or categorical properties, have been obtained on several algebras, especially on Banach algebras.
In recent years, lots of papers have been published on amenability and certain kinds of amenability of Banach algebras, related with the concept of derivation.A Banach algebra A is called module amenable (as a U -module where U is a Banach algebra) if for any commutative Banach A-U -module X, each module derivation D : A → X * is inner, equivalently if H(A, X * ) = {0} where H(A, X * ) is the 2000 Mathematics Subject Classification: Primary 46H25; Secondary 20M18 first relative (to U ) cohomology group of A with coefficients in X * .The module cohomology of certain Banach algebras is studied in [15] (see for details; [1,7,8,11,12]).Our main objective is to fill the gap between the homological properties of various module derivations and module amenability.
This paper is closely connected to the earlier works [3] and [4].Building upon ideas from [14] and [9], we shall give similar results and leave some open questions.
The paper is organized as follows.
In section 2, we give the main definitions about module derivations and basic facts on amenability and introduce the notations needed later.
In section 3, we give a necessary and sufficient condition for Z B (A, X) to be isomorphic to Z G (A, X).In the following two sections, we discuss the functorial and opposite properties of module derivations.
Finally, we extend the previous results to module (σ, τ )-derivations and give some open questions.In the cases where the proofs are similar to the classical case, the results are stated without proof.

Some Structures
Let U be a Banach algebra.A Banach space X which is also an U -bimodule is called a Banach U -bimodule if there exists a constant K > 0 such that for each α ∈ U and x ∈ X.
Throughout this paper, as in [2], A and U are Banach algebras such that A is a Banach U -bimodule with the compatible actions, as follows: Let X be a Banach A-bimodule and a Banach U -bimodule with the compatible actions, that is; and similarly for the right or two-sided actions.Then, we say that Throughout this paper, by a A-U -module, we shall always mean a commutative Banach A-U -module.
Note that in general, A is not a Banach A-U -module because A does not satisfy the compatibility condition a(α • b) = (a • α)b for α ∈ U and a, b ∈ A. But when A is a commutative U -bimodule and acts on itself by algebra multiplication from both sides, then it is also a Banach A-U -module.
If X is a (commutative) Banach A-U -module, then so is X * , where the actions of A and U on X * are defined as follows: Let X ⊗Y denote the projective tensor product of two Banach spaces X and Y .We consider the module projective tensor product A ⊗ U A which is a Banach A-U -module with canonical actions.Let I be the closed linear span of Also consider the bounded linear map w : A ⊗A → A defined by w(a⊗b) = ab and the closed ideal J of A generated by w(I).
It follows immediately that I and J are both A-submodules and U -submodules of A ⊗A and A, respectively, so the module projective tensor product A ⊗ U A ∼ = (A ⊗A)/I and the quotient Banach algebra A/J are both Banach A-modules and Banach U -modules.Also A/J is a Banach A-U -module with the compatible actions when A acts on A/J canonically.
The following proposition, which is proved in [16,Proposition 4.1], characterizes the A ⊗ U A op -U -module homomorphisms where A op is the opposite of A. By a left essential A-module X we mean a left Banach A-module X such that the linear span of A • X = {a • x | a ∈ A, x ∈ X} is dense in X.Right essential A-modules and two-sided essential A-bimodules are defined similarly.Proposition 2.1.Let A be a commutative Banach U -module, and let X, Y be A-U -modules.

Some Maps
Let A and B be Banach algebras and Banach U -bimodules with compatible actions, a U -module map is a bounded map h : Note that h is bounded if there exists M > 0 such that h(a) ≤ M a , for each a ∈ A.Here h is not necessarily linear, so it is not necessarily a U -module homomorphism.
We denote by Hom U (A, B) the metric space of all multiplicative U -module maps from A into B, with the metric derived from the usual linear operator norm .on L U (A, B); the set of all bounded linear operators from A into B, and denote Hom U (A, A) by Hom(A).
Let A and U be as above and X be a Banach A-U -module.A module derivation The set of all module derivations from A to X is denoted by Z U (A, X) (abb.Z(A, X)).Note that D is not necessarily linear, but still its boundedness implies its norm continuity since D preserves subtraction.
Let f and g be module derivations from A to X and α ∈ U , so are f + g and αf .Since X is a Banach U -bimodule, we have that Z(A, X) is a Banach U -bimodule.
For x ∈ X, define a map by D is a module derivation.Module derivations of this kind are called inner and denoted by InnZ(A, X).(In the amenability papers, generally, the notation B(A, X) is used for the set of inner module derivations from A to X, instead of InnZ(A, X)).
We consider the quotient space H(A, X) = Z(A, X)/InnZ(A, X) which call the first relative (to U ) cohomology group of A with coefficients in X. Hence A is module amenable if and only if H(A, X * ) = {0}, for each A-U -module X.

A Jordan module derivation
The set of all Jordan module derivations from A to X is denoted by JZ(A, X).
The U -module map f : A → X is said to be Lie module derivation if the identity holds for all a, b ∈ A. The set of all Lie module derivations is denoted by LieZ(A,X).

By a Bresar generalized module derivation (f, D), we mean
where D is a module derivation on A. We denote by Z B (A, X) is the set of Bresar generalized module derivation from A to X.
and (αf 1 , αD 1 ) are also Bresar generalized module derivations and hence, Z B (A, X) is a Banach U -bimodule.For x, y ∈ X, a U -module map satisfies the identity We also consider the quotient space H B (A, X) = Z B (A, X)/InnZ B (A, X), called the first generalized cohomology group from A into X (as in [13]).Then, A is said to be generalized module amenable (resp.weakly generalized module amenable for all a ∈ A.Here D is a Jordan module derivation.We denote the set of Bresar generalized Jordan module derivations from A to X by JZ B (A, X).
The U -module map f : A → X is said to be Bresar generalized Lie module derivation if the identity holds for all a, b ∈ A.Here D is a Lie module derivation.We denote the set by LieZ B (A, X).
For a U -module map f : A → X and an element x ∈ X, a pair (f, x) is called a generalized module derivation in the sense of Nakajima, if We denote the set of this type of generalized module derivations by Z G (A, X).This is also a Banach U -bimodule for a Banach A-U -module X.
For x, y ∈ X, a U -module map f x,y : A → X is called generalized inner module derivation if for all a, b ∈ A. We denote this derivation by (f x,y , −x − y).
We also denote the first Nakajima generalized cohomology group of A with coefficients in X by the quotient space H G (A, X) = Z G (A, X)/InnZ G (A, X).Similar to other definitions of amenability of Banach algebras we say A is a Nakajima generalized module amenable (resp.Nakajima weakly generalized module amenable for all a ∈ A. We denote the set of generalized Jordan module derivations from A to X by JZ G (A, X).
holds for all a, b ∈ A and the set of generalized Lie module derivations from A to X can be denoted by LieZ G (A, X).
If x = 0, then these definitions lead to the conventional notions of generalized Jordan and Lie module derivations.
Throughout this paper we use the following notations for the above sets: Z(A, X), the set of module derivations, InnZ(A, X), the set of inner module derivations, JZ(A, X), the set of Jordan module derivations, LieZ(A, X), the set of Lie module derivations, Z B (A, X), the set of Bresar generalized module derivations, InnZ B (A, X), the set of Bresar generalized inner module derivations, JZ B (A, X), the set of Bresar generalized Jordan module derivations, LieZ B (A, X), the set of Bresar generalized Lie module derivations, Z G (A, X), the set of generalized module derivations, InnZ G (A, X), the set of generalized inner module derivations, JZ G (A, X), the set of generalized Jordan module derivations, LieZ G (A, X), the set of generalized Lie module derivations.
Furthermore, we need some extra new derivation sets: Let σ and τ be arbitrary elements of Hom U (A, A) (abb. These are called inner module (σ, τ )-derivations.
for all a, b ∈ A. We use the notion Z B (σ,τ ) (A, X) for all Bresar generalized module (σ, τ )derivations from A to X.If f satisfies the relation for all a ∈ A, then it is called a Bresar generalized Jordan module (σ, τ )-derivation.
For x, y ∈ X, the U -module map f x,y : A → X is called Bresar generalized inner module (σ, τ )-derivation if f x,y (a) = x • τ (a) + σ(a) • y holds for all a ∈ A.

We use notation H
for all a, b ∈ A.
Finally, we introduce the similar notations in the sense of Nakajima: holds for all a, b ∈ A and some x ∈ X.This module derivation is denoted by (f, x) σ,τ (abb.(f, x)).

We say that
for all a, b ∈ A and we denote this module derivation by (f x,y , −x − y).
τ for all a, b ∈ A. We denote by LieM ull σ,τ (A, X) the set of all Lie left module (σ, τ )-multipliers from A to X.
If f and g are left module multipliers in all types and α ∈ U , then f + g and αf are also left module multipliers in all types, hence all the above special sets are Banach U -bimodules.
At the end of this section, we want to give a well-known lemma which will be used several times in the next sections in our paper: be a commutative diagram of R-modules and R-module homomorphisms, with exact rows, the followings hold: (i) If α 1 is an epimorphism and α 2 , α 4 are monomorphisms, then α 3 is a monomorphism; (ii) If α 5 is a monomorphism and α 2 , α 4 are epimorphisms, then α 3 is a monomorphism

Homological Properties of Generalized Module Derivations and Generalized Module (σ, τ )-Derivations
In this section, we first discuss the relation between the U -bimodules Z B (A, X) and Z G (A, X).Now, we give some elementary lemmas which show the relation between module derivations and our generalized module derivations.

generalized module derivation. It means that the notions of generalized module derivations of Nakajima and Bresar coincide when A contains an identity element.
Proof: We only need to check the boundedness of the map d = f + l x : A → X.Since f is a U -module map and X is a A-U -bimodule, then we get for each a ∈ A. This means that the map f + l x is bounded.The other parts of the proof can be done easily.✷ Remark: Throughout this paper, the most important thing which we have to check is the boundedness of the maps (for U -module maps).In the next parts of the paper, we have omitted the boundedness of the maps (because all of them are done similarly).
Corollary 3.2.The following sequence of U -modules Z G (A, X) and Z(A, X) is exact and splitting: Our aim is to give necessary and sufficient condition for Z B (A, X) to be isomorphic to Z G (A, X) as a Banach U -bimodule when A does not have a unit element.Theorem 3.3.Suppose that Φ : Z G (A, X) → Z B (A, X) and ψ : X → Mull(A, X) are U -module morphisms such that Φ((f, x)) = (f, f + l x ) and ψ(x) = l x .Then Φ is a U -module isomorphism if and only if ψ is a U -module isomorphism.
Proof: We have the following split exact sequence of Banach U -bimodules: where i 0 , i 1 , i are the canonical module injections and Proof: All maps in the above diagram are U -module maps, and the commutativity of the diagram is easily seen.If ψ 2 (f x,y , −x−y) = 0, then we see that f x+y,0 = f x,y .Thus Kerψ 2 = Imψ 1 .The other part is clear by Corollary 3.2 using the definitions of ϕ 1 and ϕ 2 .✷ Afterwards, we generalize above theorems and corollaries to the module (σ, τ )derivations.All results in this part are similarly proved to the corresponding results in the previous parts, so we omit the proofs.Theorem 3.10.Suppose that {x ∈ X | Ax = 0} = 0 and σ is surjective.For the Banach U -bimodules Z (σ,τ ) (A, X) and Z G (σ,τ ) (A, X), the following sequence is split exact: in M is commutative.Hence we can say that Φ is a natural transformation of functors.

Opposite Properties
In this section, we first consider the universal problem for generalized module derivations.
Let A be a Banach algebra.Define a new algebra A op with underlying set consisting of elements {a o | a ∈ A}.Addition in A op coincides with addition in A and multiplication in A op is the map where ba is the product in A. Moreover, this map is linear.Endow A op with the Banach space structure a o = a .Then A op is again a Banach algebra, called the opposite of A. If A is a commutative Banach U -bimodule, then it can be easily verified that A op is also a commutative Banach U -bimodule.
Furthermore, A ⊗ U A op is a Banach algebra with the following product rule: Let A be a Banach algebra and a Banach U -bimodule, and let X be a Banach A-U -module.Then X is a left Banach A ⊗ U A op -module by the action defined by If A has an identity element 1, then the map is a module derivation and δ A has the following universal mapping property: For every left Banach A ⊗ U A op -module X and every module derivation d : A → X, there exists a unique A ⊗ U A op -module map h : I → X such that d = hδ A .Using Corollary 4.3, we get the following theorem.Theorem 4.7.Let A be a Banach algebra and a Banach U -bimodule with identity element 1, and let X be a Banach A-U -module and (f, x) : A → X be a generalized module derivation.Then the map gδ A : A → I ⊕ (A ⊗ U A op ) defined by gδ A (a) = (δ A (a), −1 ⊗ 1 • ) gives a generalized module derivation (gδ A , (0, 1 ⊗ 1 • )) and there exists a unique left A ⊗ U A op -module map h f : I ⊕ (A ⊗ U A op ) → X such that f = h f (gδ A ) and h f ((0, 1 ⊗ 1 • ) = x.This induces the map which is a U -module isomorphism.
Proof: Since X is a left Banach A ⊗ U A op -module, by Corollary 4.3 and the above universal mapping property, we have a chain of U -module isomorphisms of Banach U -bimodules By these U -module isomorphisms, a generalized module derivation (f, x) corresponds to the map gives a generalized module derivation (gδ A , (0, 1)) and there exists a unique Umodule map h f : I/I 2 ⊕ A → Xsuch that f = h f (gδ A ) and h f ((0, 1)) = x.This induces the module map ξ x : Hom A I/I 2 ⊕ (A, X) → Z G (A, X), α → (αgδ A , α(0, 1)) which is a U -module isomorphism.
LieM ull(A, X) = {f | f : A → X, module map and f [a, b] = [−f (b), a] for all a, b ∈ A} which is called the set of Lie left module multipliers.
module map and f (ab) = f (a) • τ (b) for all a, b ∈ A} and these maps are called left module (σ, τ )-multipliers.
0 and the map σ : A → A is surjective, then d is determined by f uniquely.(ii)If D : A → X is a module (σ, τ )-derivation, then for any nonzero element x ∈ X, the map (f = D+l x τ , −x) : A → X is a generalized module (σ, τ )-derivation such that f = D and D associates to f .(iii) If A contains a unit element and (f, D) : A → X is a Bresar generalized module derivation, then (f, −f (1)) : A → X is a generalized module (σ, τ )derivation.