بررسی و مطالعه مفهومσ--میانگین پذیری دوری روی جبرهای باناخ.

Introduction The notion of amenability for Banach algebras was first introduced and initiated by Johnson [3]. Afterwards, Gronbaek [2] studied the concept of cyclic amenability and it‘s hereditary properties of Banach algebras. Many Authors study and investigate the several notions of σ-amenability, for more details refer to [1], [5], [6], [8] and [9]. In the throughout of this paper we assume that A is a Banach algebra, I is a closed two-sided ideal in A, σ is a continuous homomorphism on A and X is an A-bimodule Banach. The linear map D: A→X is called derivation if D(ab) = D(a)∙b + a∙D(b) for all a,bϵA. Moreover, the linear mapping D: A → X is called a σ-derivation if D(ab) = D(a)∙σ(b) + σ(a)∙D(b) for each a,bϵA. A linear map D : A → X is called σ-inner derivation if there exists x₁∈X such that D = adσ_x1, where adσ_x1(a) = σ(a).x₁-x₁.σ(a), for all a∈A. The Banach algebra A is called σ-amenable if every σ-derivation D : A → X^∗ is σ-inner. This notion was first introduced and studied by Moslehian and Motlagh [9]. Recall that the derivation D:A→A^∗ is cyclic if <D(a), b>+<D(b), a>=0, (a,bϵA). A is cyclic amenable if every cyclic derivation D:A→A^∗ is inner. So, by using these notion we introduced the new notions σ-cyclic derivation and σ-cyclic amenability for Banach algebras. We say that the σ-derivation D : A → A^∗ is σ-cyclic if ⟨D(a), σ(b)⟩ + ⟨D(b), σ(a)⟩ = 0, (a,bϵA). Also, the Banach algebra A is σ-cyclic amenable if every σ-cyclic derivation D:A→A^∗ is σ-inner. The closed two-sided ideal I in A has the trace extension property if for each λϵI^∗ with a∙λ=λ∙a (aϵA), there exists ΛϵA^∗ such that Λ|I = λ and Λ∙a=a∙Λ, for all aϵA. Every homomorphism σ on A can be extended to a homomorphism σ̂: A/I → A/I suth that for aϵA defined by σ̂(a+I)=σ(a)+I. Next, we present the results obtained in this paper. We showed that let A/I be σ̂-cyclic amenable. Then I has the trace extension property. Also, Suppose that σ has dense range on A, I has the trace extension property and Ais σ-cyclic. Then A I is σ̂-cyclic amenable. Recall that the Banch algebra A is essential, if A̅̅² = A. We showed that every σ-cyclic amenable Banach algebra is essential. If σ(I) ^⊆ I, we can restrict σ to I and denoted it by σ_I :I → I. It is clear that σ_I is continuous homomorphism on I. Thus, if σ(I) ^⊆ I, I is σ_I -cyclic amenable and A/I is σ̂-cyclic amenable, then A is σ-cyclic amenable. For the unitization of Banach algebra A i.e. A^#=A ^⊕ ℂe, and for every continuous homomorphism σ on A there exists a continuous homomorphism σ^# : A^# → A^# such that defined by σ^# (a, λe)=σ(a)+λe, (aϵA, λϵℂ). Therefore, with above notes, A is σ-cyclic amenable if and only if A^# be σ^# -cyclic amenable. Finally, for two Banach algebras A and B and nonzero multiplicative linear functional θ on B, the Cartesian space A×B with norm and multiplication ||(a,b)||=||a||+||b||, (a,b)(a’,b’)=(aa’+θ(b)a’+θ(b’)a, bb’), is a Banach algebra and denoted by A×_𝜃𝐵. If σ and τ are homomorphisms on A and B respectively, then (σ, τ) denoted by <(σ, τ),(a, b)>=(σ(a), τ(b)) is a homomorphism on A×_𝜃𝐵 if and only if θ∘τ=θ. By using this we showed that A×_𝜃𝐵 is (σ, τ)-cyclic amenable if and only if A is σ-cyclic amenable and B is τ-cyclic amenable. [ABSTRACT FROM AUTHOR]