Arens regularity of module actions

Subject Areas : Statistics

MEHRDAD SHABANI SOLTANMORADI ¹ (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, CENTRAL, TEHRAN BRANCH, ISLAMIC AZAD UNIVERSITY, TEHRAN, IRAN)
DAVOOD EBRAHIMI BAGHA ² (PR OFESSOR, 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.

References:

[1] C. A. Akemann, “The dual space of an operator algebra”, Trans. Amer. Math, Soc. 126(1967), 286-302.

 

[2] R. Arens, “The adjoint of a bilinear operation”, Proc. Am. Math. Soc. 2-(1951), 839-848.

 

[3] J. W. Bunce and W. L. Paschke, “Derivations on a -algebra and its double dual”, J, Funct, Anal, 37(1980), 235-247.

 

[4] H. G. Dales, “Banach algebra and automatic continuity”, Clarendon, Oxford, 2000.

 

[5] H. G. Dales and A. T. M. Lan, “The second duals of Beurling algebras”, Men. Amer. Math. Soc. 177(836) (2005).

 

[6] H. G. Dales, A. Rodrigues-Palacios and M. V. Velasco, “The second transpose of a derivation”, J. London Math. Soc. 64(2001), 707-721.

 

[7] F. Gourdean, “Amenability and the secod dual of a Banach algebra”, Studia Math. 125 (1997), 75-81.

 

[8] S. Mohammad zadeh and H. R. E. Vishki “Arens regularity of module actions and the second adjoint of a derivation”, Bull. Austral. Math. Soc. 77(2008), 465-476.

 

[9] T. W. Palmer, “Banach algebras and the general theory of *- algebras”, Volume. 1, Cambridge University, (1994).