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mmr 2020, 6(2): 235-242 |
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Shikh Ali A, Haghnejad azar K, Abadian A. On the Properties of the Arens Regularity of Bounded Bilinear Mappings . mmr 2020; 6 (2) :235-242
URL: http://mmr.khu.ac.ir/article-1-2863-en.html
URL: http://mmr.khu.ac.ir/article-1-2863-en.html
1- Departement of Mathematics , Ebadian.ali@gmail.com
2- Department of mathematics, Faculty of Science, Urmia University, Urmia, Iran.
2- Department of mathematics, Faculty of Science, Urmia University, Urmia, Iran.
Abstract: (2433 Views)
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.
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.
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
Received: 2018/10/22 | Revised: 2021/01/9 | Accepted: 2020/05/11 | Published: 2020/08/22 | ePublished: 2020/08/22
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