*1- Damghan University , ramezanpour@du.ac.ir*

2- Damghan University

**Abstract:** (886 Views)

** Introduction**
The notion of weak amenability for commutative Banach algebras was introduced and studied for the first time by Bade, Curtis and Dales. Johnson extended this concept to the non commutative case and showed that group algebras of all locally compact groups are weakly amenable. A Banach algebra *A* is called weakly amenable if every continuous derivation *D*:*A*→*A** is inner.
It is often useful to restrict one's attention to derivations *D*:*A*→*A** satisfying the property *D**a**c*+*D**c**a*=0 for all *a*,*c*∈*A*. Such derivations are called cyclic. Clearly inner derivations are cyclic. A Banach algebra is called cyclic amenable if every continuous cyclic derivations *D*:*A*→*A**is inner. This notion was presented by Gronbaek. He investigated the hereditary properties of this concept, found some relations between cyclic amenability of a Banach algebra and the trace extension property of its ideals.
Ghahramani and Loy introduced several approximate notions of amenability by requiring that all bounded derivations from a given Banach algebra *A* into certain Banach *A*-bimodules to be approximately inner. In the same paper and the subsequent one, the authors showed the distinction between each of these concepts and the corresponding classical notions and investigated properties of algebras in each of these new classes. Motivated by this notions, Esslamzadeh and Shojaee defined the concept of approximate cyclic amenability for Banach algebras and investigated the hereditary properties for this new notion.
**Periliminaries**
Let *A* be a Banach algebra and let *X* be an *A*-bimodule. Then the l1-direct sum *A*×*X* under the multiplication
*a*,*x**b*,*y*=*ab*,*ay*+*xb* (*a*,*b*∈*A*,*x*,*y*∈*X*),
is a Banach algebra called the module extension of *A* by *X* and denoted by *A*⊕*X*. The class of module extension Banach algebras contains a wide variety of Banach algebras includes a triangular Banach algebra Tri(*A*,*X*,*B*). Every triangular Banach algebra Tri(*A*,*X*,*B*).can be identified with the module extension Banach algebra (*A*×*B*)⊕*X*.
On the other hand, for two Banach algebra *A* and *B* with ∆(*B*)≠∅ and for *θ*∈∆(*B*), the set of all non-zero multiplicative linear functionals on *B*, the *θ*-Lau product *A*×*θ**B* is a Banach algebra which is defined as the l1-direct sum *A*×*B* equipped with the algebra multiplication
*a*1,*b*1*a*2,*b*2=*a*1*a*2+*θ**b*2*a*1+*θ**b*1*a*2,*b*1*b*2 *a*1,*a*2∈*A**,**b*1,*b*2∈*B*.

This type of product was introduced by Lau for certain class of Banach algebras known as Lau algebras and was extended by Sangani Monfared for arbitrary Banach algebras. The unitization *A*♯ of *A* can be regarded as the *ι*-Lau product *A*×*ι*C, where *ι*∈Δ(C) is the identity map.
This product provides not only new examples of Banach algebras by themselves, but it can also serve as a source of (counter) examples for various purposes in functional and harmonic analysis. From the homological algebra point of view *A*×*θ**B* is a strongly splitting Banach algebra extension of *B* by *A.* The Lau product of Banach algebras enjoys some properties that are not shared in general by arbitrary strongly splitting extensions. For instance, commutativity is not preserved by a generally strongly splitting extension. However, *A*×*θ**B* is commutative if and only if both *A* and *B* are commutative.
**Results and discussion**
Many basic properties of *A*♯, some notions of amenability and some homological properties are extended to *A**×**θ**B* by many authors. In particular, Ghaderi, Nasr-Isfahani and Nemati extended some results on (approximate) cyclic amenability of *A*♯, obtained by Esslamzadeh and Shojaee, to *A**×**θ**B*. They showed that if *A**2* is dense in *A* then the cyclic amenability *A**×**θ**B* is equivalent to the cyclic amenability of both *A* and *B*.
In this paper, by characterizing of cyclic derivations on Lau product *A**×**θ**B* and module extension *A⊕X*, we present general necessary and sufficient conditions for those to be (approximate) cyclic amenable. This not only provides new results on (approximate) cyclic amenability of these type of Banach algebras but also improves some main results in this topic. In particular we show that, under mild condition, the cyclic amenability of Tri*(**A**,**X**,**B**)* is equivalent to the cyclic amenability of the corner algebras *A* and *B*.
Type of Study:

S |
Subject:

Mat Received: 2020/03/16 | Revised: 2022/11/16 | Accepted: 2020/10/21 | Published: 2022/05/21 | ePublished: 2022/05/21