On the Properties of the Arens Regularity of Bounded Bilinear Mappings 

Authors
Abotalb Shikh Ali 1
Kazem Haghnejad azar
Ali Abadian 2
1 Departement of Mathematics
2 Department of mathematics, Faculty of Science, Urmia University, Urmia, Iran.
10.52547/mmr.6.2.235
Abstract
Introduction

Let , and be Banach spaces and be a bilinear mapping. In 1951 Arens found two extension for as and from into . The mapping is the unique extension of such that from into is continuous for every , but the mapping is not in general continuous from into unless . Thus for all the mapping is continuous if and only if is Arens regular. Regarding as a Banach , the operation extends to and defined on . These extensions are known, respectively, as the first (left) and the second (right) Arens products, and with each of them, the second dual space becomes a Banach algebra.

Material and methods

The constructions of the two Arens multiplications in lead us to definition of topological centers for with respect to both Arens multiplications. The topological centers of Banach algebras, module actions and applications of them were introduced and discussed in some manuscripts. It is known that the multiplication map of every non-reflexive, -algebra is Arens regular. In this paper, we extend some problems from Banach algebras to the general criterion on module actions and bilinear mapping with some applications in group algebras.

Results and discussion

We will investigate on the Arens regularity of bounded bilinear mappings and we show that a bounded bilinear mapping is Arens regular if and only if the linear map with is weakly compact, so we prove a theorem that establish the relationships between Arens regularity and weakly compactness properties for any bounded bilinear mappings. We also study on the Arens regularity and weakly compact property of bounded bilinear mapping and we have analogous results to that of Dalse, lger and Arikan. For Banach algebras, we establish the relationships between Arens regularity and reflexivity.



Conclusion

The following conclusions were drawn from this research.

if and only if the bilinear mapping is Arens regular.
A bounded bilinear mapping is Arens regular if and only if the linear map with is weakly compact.
if and only if the bilinear mapping is Arens regular.
Assume that has approximate identity. Then is Arens regular if and only if is reflexive../files/site1/files/62/9Abstract.pdf




Keywords
arens product
Arens regularity
Banach algebra
bilinear map
second dual
weakly compact
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Mathematical Research
Volume 6, Issue 2 - Serial Number 13
Summer 2020
Pages 235-242