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
Type of Study:
Research Paper |
Subject:
alg Received: 2018/10/22 | Revised: 2021/01/9 | Accepted: 2020/05/11 | Published: 2020/08/22 | ePublished: 2020/08/22