let $a$ be a $c^*$-algebra and $e$ be a left hilbert $a$-module. in this paper we define a product on $e$ that making it into a banach algebra and show that under the certain conditions $e$  is arens regular. we also study the relationship between derivations of $a$ and $e$.

In this paper, we study the concept of weakly almost periodic functions on Fréchet algebras. For a algebra [Formula: see text], show that text]. We also text] is Arens regular if and only both are regular. Finally, for sequence algebras prove regualr each

and similarly for the right topological centre Z(r)(A′′). The algebra A is said to be Arens regular if Z(`)(A′′) = Z(r)(A′′) = A′′ and strongly Arens irregular if Z(`)(A′′) = Z(r)(A′′) = A. For example, every C∗-algebra is Arens regular [2]. There has been a great deal of study of these two algebras, especially in the case where A is the group algebra L(G) for a locally compact group G. Results...

A Banach algebra is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols A⁎=WAP(A). To identify the opposite behaviour, Granirer called a extremely non-Arens regular (enAr, for short) quotient A⁎/WAP(A) contains closed subspace that has A⁎ as quotient. In this paper we propose simplification and quantification of concept. We say r-enAr, with r≥1, there an iso...

Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule with the left and right module actions $pi_ell: Atimes Xrightarrow X$ and $pi_r: Xtimes Arightarrow X$, respectively. In this paper, we  study  the topological centers of the left module action $pi_{ell_n}: Atimes X^{(n)}rightarrow X^{(n)}$ and the right module action $pi_{r_n}:X^{(n)}times Arightarrow X^{(n)}$,  which inherit from th...

Let $A$ be a $C^*$-algebra and $E$ be a left Hilbert $A$-module. In this paper we define a product on $E$ that making it into a Banach algebra and show that under the certain conditions $E$  is Arens regular. We also study the relationship between derivations of $A$ and $E$.

A complex algebra A is called ideally factored if Ia = Ca is a left ideal of A for all a ∈ A. In this article, we investigate some interesting properties of ideally factored algebras and show that these algebras are always Arens regular but never amenable. In addition, we investigate ρhomomorphisms and (ρ, τ)-derivations on ideally factored algebra.

In this paper, we study approximate identity properties, some propositions from Baker, Dales, Lau in general situations and establish relationships between the topological centers of module actions factorization properties with results group algebras. We consider under which sufficient necessary conditions Banach algebra $A\widehat{\otimes}B$ is Arens regular.

We discuss Arens regularity of the projective tensor product Banach algebras consisting Schatten class operators. More precisely, exploiting Ülger’s biregularity criterion, we prove that for any Hilbert space ℋ, Sp(ℋ)⊗γ Sq (ℋ) and B(S2(ℋ))⊗γ S2(ℋ) are not regular every pair 1≤p,q≤2; whereas, when ℋ is separable equipped with Schur product, show completely continuous S2(ℋ)⊗γ regular.

For Φ,Ψ ∈ A′′, define 〈Φ Ψ, λ〉 = 〈Φ, Ψ · λ〉 (λ ∈ A′) , and similarly for ♦. Thus (A′′, ) and (A′′,♦) are Banach algebras each containingA as a closed subalgebra. The Banach algebra A is Arens regular if and ♦ coincide on A′′, and A is strongly Arens irregular if and ♦ coincide only on A. A subspace X of A′ is left-introverted if Φ · λ ∈ X whenever Φ ∈ A′′ and λ ∈ X . There has been a great deal...