Arens regularity of module actions

Subject Areas : Statistics

MEHRDAD SHABANI SOLTANMORADI ¹ , DAVOOD EBRAHIMI BAGHA ²

1 - DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, CENTRAL, TEHRAN BRANCH, ISLAMIC AZAD UNIVERSITY, TEHRAN, IRAN
2 - DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, CENTRAL, TEHRAN BRANCH, ISLAMIC AZAD UNIVERSITY, TEHRAN, IRAN

Received: 2019-07-09 Accepted : 2020-03-12 Published : 2020-08-22

Keywords: آرنز منظم, جبر باناخ, عملهای مدولی باناخ,

Abstract :

Let A be a Banach algebra, A’’ a Banach A-module. In this paper, we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions,and them investigate those conditions under which the second adjoint of a derivation into a dual Banach algebra module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included. A^(4) is a banach algebra with four Arens products. The bilinear map T is Arens regular when the equality T*** = T^( r***r ) . If T: A × A’’ → A’’ is multiplication left module on A , the following statements are equivalent , i:T is regular ii : T**** = T^(r****r) iii : T****( A’’’, A’’) ⊆ A’’’ iv : the linear map a → T*( a’’’, a) : A → A’’’ is weakly compact for every a’’’ ∈ A’’’. Also If module actions are regular, then every inner derivation D : A → A’’’ is weakly compact; moreover, D** : (A’’, □ ) → A^(5) and D** : (A’’, ⋄ ) → A^(5) are also inner derivation.

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