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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
E-mail address:Abotaleb.sheikhali.20@gmail.com
2Department of Mathematics, Damghan University, Damghan, Iran
E-mail address:Abdolmotaleb.math88@gmail.com
3Department of Mathematics, Kharazmi University, Tehran, Iran
E-mail address:Neda.akhlaghi1365@gmail.com
Article history:
Received July 2014
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
adjoint of ).
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 reflexive 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=21
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(21)
(,)
= ,1(21 ) (,) = , 2(22)
(,)
= , 2(22)
(,) = , 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(21 )
(,)
=, 2(21)
(,) = , 2(21 )
(,)
=,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.
