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# Arens Regularity Of Banach Module Actions And The Strongly Irregular Property

Authors:
Journal of mathematics and computer science 13 (2014), 41-46
Arens Regularity of Banach Module Actions and the Strongly Irregular Property
Abotaleb Sheikhali1, Abdolmotaleb Sheikhali2, Neda Akhlaghi3
1Department of Mathematics, Kharazmi University, Tehran, Iran
2Department of Mathematics, Damghan University, Damghan, Iran
3Department of Mathematics, Kharazmi University, Tehran, Iran
Article history:
Accepted August 2014
Available online September 2014
Abstract
Let ,, be normed spaces. We show that, if is reﬂexive, then some extensions andadjointsof
the bounded bilinear map :× are Arens regular. Also the left strongly irregular propertyis
equivalent to the right strongly irregular property. We show that the right module action 2
: 󰇛+1󰇜×
() factors, where A is a Banach algebra.
Keywords: Arens regular, module action, derivation, topological center, factor.
2010 Mathematics Subject Classiﬁcation. 4620,4625.
1. Introductionand Preliminaries
Arens showed in [1] that a bounded bilinear map :× on normed spaces, has two natural
diﬀerent extensions ,  from  ×into . When these extensions are equal, is
saidto be Arens regular. Throughout the article, we identify a normed space with its canonical image
in thesecond dual.
Let ,, be normed spaces and:× be a bounded bilinear mapping. The natural
extensions of are as follows:
A. Sheikhali, A. Sheikhali, N. Akhlaghi/ J. Math. Computer Sci. 13 (2014), 41-46
42
i): × , given by󰇛,󰇜,=,(,) where X, y Y, ( is saidthe
ii) :  × , given by󰇛,󰇜,=,(, x) where X,  , .
iii):  × , given by 󰇛,󰇜,=,󰇛,󰇜where  ,
, .
Let :× be the ﬂip of deﬁned by 󰇛,󰇜=(,), for every Xand y Y. Thenis
a bounded bilinear map and it may extends as above to : × . In general, the
mapping  : × is not equal to . When these extensions are equal, then is
Arens regular. If the multiplication of a Banach algebra enjoys this property, then itself is
calledArens regular. The ﬁrst and the second Arens products are denoted by, respectively.
One may deﬁne similarly the mappings :  ×  and :  × and
the higher rank adjoints. Consider the nets ()X and ()Y converge to   and
 in the topologies, respectively, then
(,) = lim
lim
(,)
and
 (,) = lim
lim
(,)
so Arens regularity of is equivalent to the following
lim
lim
,(,)= lim
lim
,(,)
if the limits exit for each . The map  is the unique extension of such that
 󰇛,󰇜:  is continuous for each  and
 󰇛,󰇜:  is continuous for each .
The left topological center of is deﬁned by
󰇛󰇜= { : 󰇛,󰇜:  is continuous}.
Since  : ×  is the unique extension of such that the
map  󰇛,󰇜:  is continuous for each  , we can set
󰇛󰇜= { :󰇛,󰇜=  󰇛,󰇜, ( )}.
The right topological center of may therefore be deﬁned as
󰇛󰇜= { :  󰇛,󰇜:  is continuous}.
Again since the map
 󰇛,󰇜:  is continuous for each  , we can set
󰇛󰇜= { :󰇛,󰇜= 󰇛,󰇜, ( )}.
A bounded bilinear mapping is Arens regular if and only if󰇛󰇜=, or equivalently󰇛󰇜=.
It is clear that 󰇛󰇜. If () = then the map is said to be left strongly irregular. Also
() and if () = then the map is said to be right strongly irregular. A bounded
bilinearmapping × is said to factor if it is onto. Let be a Banach algebra, be
aBanachspace and 1 × be a bounded bilinear map (1 is said the left module action of
A. Sheikhali, A. Sheikhali, N. Akhlaghi/ J. Math. Computer Sci. 13 (2014), 41-46
43
on ).If 1(,) = 1(,1(,)), for each , , , then the pair (1,) is said to be a
left Banach
module. A right Banach module (,2) can be deﬁned similarly. A triple (1,,2) is said
tobe a Banach module if (1,) and (,2) are left and right Banach modules, respectively,
and1(,2(,)) = 2(1(,), ) for each , , . Let (1,,2) be a Banach
module. Abounded linear mapping is said to be a derivation if () = (). +
.(), for each, .
2. Arens regularity of bounded bilinear maps
Remark 2.1. Let be a bounded bilinear map from × into .  means that the number of
starsis 3 for every .
Let be a bounded bilinear map and , , . If is Arens regular then for every ,
(,), =(,), = (,), =(,),
Therefore, is Arens regular. Now let is Arens regular, for every  , , ,
 (,), = (,), =(,), =(,),
Hence is Arense regular if and only if is Arens regular.
Lemma 2.2. If :× is Arens regular and is a reﬂexive space, then and  are
Arens regular for every .
Proof. First, we show that  is Arens regular for an arbitrary . Then we show that  =
 .By [7,  2.1], for every  , , , we have
(,), =,(,)
= ,(,, )
= ,(,)
=󰇛,󰇜,
=  (,), .
It follows that  is Arens regular. This completes the proof of Arens regularity of . Now if
is Arens regular then we show that  is Arens regula. we should show
(1)  = 
Since is Arens regular,
(2) () = ()
so it is enough to show that
A. Sheikhali, A. Sheikhali, N. Akhlaghi/ J. Math. Computer Sci. 13 (2014), 41-46
44
(3)  = 
 is Arens regular, therefore =, so from the Arens regularity of ,we have
 = . Therefore = . From the Arens regularity of ,
 =. Now by 󰇟6,  2.1󰇠, for every  , ,
,we have
 (, ), =,(,)
=, (,)
=, (,)
= (,), .
Therefore equation (3) holds and  is Arens regular. Hence is Arens regular, for every .
Lemma 2.3. Let :× be is a bounded bilinear map. If is reﬂexive, then and every
adjoint and every ﬂip map of such that its domain contains ,,, . .. is Arens regular.
Proof. First we show that if is reﬂexive, then the result holds. (,)  and is
reﬂexive, therefore
(,) (,) .
Now by [7,  2.1], is Arens regular. Therefore is Arens regular, so the result holds.
Lemma 2.4. If is reﬂexive and the bounded bilinear map  factors, then and every adjoint map
and every ﬂip map of it is Arens regular.
Proof. If is reﬂexive space then by  2.3, and are Arens regular and by[7,  2.2]
itis equivalent that (,) and  factors, therefore  and it is equivalent
that is reﬂexive. Now for every adjoint map or every ﬂip map, or , or ,is contained in a part
ofits domain. Since these spaces are all reﬂexive, therefore by  2.3 the result holds.
Theorem 2.5. Let be is reﬂexive and let  factors. Then is left strongly irregular if and only if it is
right strongly irregular.
Proof.By  2.3 is Arens regular. From the deﬁnition, is Arens regular if and only if() = .
is reﬂexive therefore () = , i.e. is left strongly irregular, therefore is Arens regular if and only
is left strongly irregular.On the other hand by  2.4, is also reﬂexive, therefore by deﬁnition
A. Sheikhali, A. Sheikhali, N. Akhlaghi/ J. Math. Computer Sci. 13 (2014), 41-46
45
of topological centers, is Arens regular if and only if() = , since is reexive so () = ,
thus is right strongly irregular. therefore is Arens regular if and only is right strongly irregular. It
follows that f is left strongly irregular if and only if right strongly irregular.
3. Module action
In [5] Eshaghi Gordji and Filali show that left module action of a Banach algebra on () factors.
Now let 2 be the right module action of on (). Thus 2 maps () × into()and 2
maps
󰇛+1󰇜×󰇛󰇜 into , for every 1.Also1=21
 and 2=1 1
such that 󰇛0󰇜=,=
10=20. In the next theorem we show that the right module action factors.
Theorem 3.1. Let be a Banach algebra.
) If has a left bounded approximate identity, then 2
factors for every positive even integer .
) If has a right bounded approximate identity, then 2
factors for odd positive even integer .
Proof.) We use the induction on . Let = 2 and () be a left bounded approximate identity in
with a cluster point  . Therefore for every   we have 22
(, ) = .
Let(
)be a net in with a cluster point  , so for every ,
22
(, ),  = ,22 (,) = , 11
(,)
= lim
,11 (,
) = lim
, 20
 (,
)
= lim
lim
20
(
,), = lim
lim

,20 (,)
= lim

, = ,.
Therefor for = 2,2 factors. Now suppose that the result holds for = 2 2. So,
2(2)
(, ), = ,2(2)(,) = , 1(21)
(,)
= ,1(21 ) (,) = , 2(22)
 (,)
= , 2(22)
 (,) = , 2(22)
(,)
= ,2(2 2) (,) = 2(22)
(,), 
thus 2(2)
factors.
) Again by induction. Let () be a right bounded approximate identity in with a cluster point
 . for = 1 it is enough to show that 21
(, ) =for every .
A. Sheikhali, A. Sheikhali, N. Akhlaghi/ J. Math. Computer Sci. 13 (2014), 41-46
46
21
(,),  = ,21 (,) = , 10
(,)= lim
,10 (, ) = , .
Now suppose that is true for= 2 1, then
2(2+1)
(,),  = ,2(2 +1) (,) = , 1(2)
(,)
= ,1(2)(,) = , 2(21 )
 (,)
=, 2(21)
 (,) = , 2(21 )
(,)
=,2(2 1) (,) = 2(2 1 )
(,), .
so the result holds.
Here is a new proof for the theorem [4.7.1]
Theorem 3.2. If is a reﬂexive space and is a derivation, then  is also a derivation.
. As is reﬂexive, by  2.3 the following module actions are Arens regular,
1 × , 2 ×
1
× , 2
 ×
Now the maps bellow are Arens regular by [7, 4.4],
 (,󰅾 )  ,  (,) 
References
[1] A. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc., 2 (1951), 839-848.
[2] S. Barootkoob, S. Mohamadzadeh and H.R.E Vishki, Topological Centers of Certain Banach Module
Action, Bulletinof the iranian Mathematical Society, Vol. 35 No. 2 (2009), 25-36.
[3] H. G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monographs 24
(Clarendon Press, Oxford,2000)
[4] H.G. Dales, A. Rodrigues-Palacios and M.V. Velasco, The second transpose of a derivation, J. London
Math. Soc. 64 (2) (2001) 707-721.
[5] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math. 181 (3) (2007) 237-
254.
[6]M. Momeni, T. Yazdanpanah, M. R. Mardanbeigi, Sigma Ideal Amenability of Banach Algebras,
Journal of mathematics and computer science, 8 (2014), 319-325
[7] S. Mohamadzadeh and H.R.E Vishki, Arens regularity of module actions and the second adjoint of a
derivation, Bull. Austral. Mat. Soc. 77 (2008) 465-476.
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In this paper, first 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 then we investigate those conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included.
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We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if A has a brai (blai), then the right (left) module action of A on A* is Arens regular if and only if A is reflexive. We find that Arens regularity is implied by the factorization of A* or A** when A is a left or a right ideal in A**. The Arens regularity and strong irregularity of A are related to those of the module actions of A on the nth dual A(n) of A. Banach algebras A for which Z(A**) = A but A ⊆ Zt(A* *) are found (here Z(A**) and Zt(A**) are the topological centres of A** with respect to the first and second Arens product, respectively). This also gives examples of Banach algebras such that A ⊆ Z(A**) ⊆ A**. Finally, the triangular Banach algebras T are used to find Banach algebras having the following properties: (i) T*T = TT* but Z(T**) ≠ Zt(T**); (ii) Z(T**) = Zt(T* *) and T*T = T* but TT* ≠ T*; (iii) Z(T**) = T but T is not weakly sequentially complete. The results (ii) and (iii) are new examples answering questions asked by Lau and Ülger.