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Available Online: http://saspjournals.com/sjpms 65
Scholars Journal of Physics, Mathematics and Statistics ISSN 2393-8056 (Print)
Abbreviated Key Title: Sch. J. Phys. Math. Stat. ISSN 2393-8064 (Online)
©Scholars Academic and Scientific Publishers (SAS Publishers)
(An International Publisher for Academic and Scientific Resources)
Arens Regularity of Bilinear Mapping and Reflexivity
Abotaleb Sheikhali, Nader Kanzi
E-mail address: Abotaleb.sheikhali.20@gmail.com, Nad.kanzi@gmail.com
Depatment of Mathematics, Payame Noor University (PNU), Tehran, Iran
*Corresponding author
Abotaleb Sheikhali
Article History
Received: 20.12.2017
Accepted: 20.01.2018
Published: 30.01.2018
DOI:
10.21276/sjpms.2018.5.1.3
Abstract: Let X, Y and Z be normed spaces. In this article we give a tool to
investigate Arens regularity of a bounded bilinear map f : X Y Z. Also, under
some assumptions on and , we give some new results to determine
reflexivity of the spaces.
Keywords: Arens regular, bilinear map, topological center, factor, second dual.
2010 Mathematics Subject Classification. 46H20, 46H25
INTRODUCTION AND PRELIMINARIES
Arens showed in [1] that a bounded bilinear map f : X Y Z on normed
spaces, has two natural different extensions , from into .
When these extensions are equal, f is said to be Arens regular. Throughout the article,
we identify a normed space with its canonical image in the second dual.
We denote by the topological dual of a normed space X. We write for
and so on. Let X, Y and Z be normed spaces and f : X × Y Z be a bounded bilinear
mapping. The natural extensions of fare as following:
(i) , give by where
, , ( is said the adjoint of ).
(ii) , give by
where , , .
(iii) , give by
where , , .
Let now : Y X Z be the flip of f defined by (y, x) = f (x, y), for every x X and y Y . Then is 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 f is Arens regular.
One may define similarly the mappings : and : and the higher rank
adjoints. Consider the nets and converge to and in the topologies,
respectively, then
and
So Arens regularity of is equivalent to the following
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
Since is the unique extension of such that the map is
continuous for each , we can set
.
The right topological center of f may therefore be defined as
Again since the map is continuous for each , we have
.
Abotaleb Sheikhali & Nader Kanzi.; Sch. J. Phys. Math. Stat., 2018; Vol-5; Issue-1 (Jan-Feb); pp-65-68
Available Online: http://saspjournals.com/sjpms 66
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 bilinear mapping f : X Y Z is said to factor if
it is onto.
Investigate Arens regularity of bounded bilinear maps
S. Mohammadzadeh and Vishki H.R proved in [6] acriterion concerning to the regularity of a bounded bilinear
map. They showed that is Arens regular if and only if In the section we provide the same
conditions of Arens regularity. First, we give a similar lemma to the
Lemma 2.1. For a bounded bilinear map from into the following statements are equivalent:
(i) is Arens regular;
(ii) ;
(iii) .
Proof. If (i) hold then is Arens regular. Therefor . For every and we
have
Therefore .
(ii)
(iii) Let and we have
(iii)
(i) Let For every and we have
.
It follows that is Arens regular and this completes the proof
Theorem 2.2. Bounded bilinear map from into is Arens regular if and only if
Proof. Let and be arbitrary. If is Arens regular Then Therefore
.
Conversely, suppose and let and be two nets that are converge to and
in the-topologies, respectivety. Then
Therefore is Arens regular and this completes the proof
Corollary 2.3. For a bounded bilinear map f : X Y Z, the following statements are equivalent:
(i) ;
(ii) and are Arens regular;
(iii)
Proof. The implication (i) (ii) follows from the fact that . Now Theorem
2.2 implies the Arens regularity of , or equivalenty From which
Abotaleb Sheikhali & Nader Kanzi.; Sch. J. Phys. Math. Stat., 2018; Vol-5; Issue-1 (Jan-Feb); pp-65-68
Available Online: http://saspjournals.com/sjpms 67
Therefore the Arens regularity of follows again by Theorem 2.2. Thus is Arens regular.
(ii) (iii) If is Arens regular. Then
Now if is Arens regular. Then we have
The equalities (2-1) and (2-2) together establish the assertion.
(iii) (i) First we show that . For every and
As lies in thus and the proof
Theorem 2.4. Let and A be normed spaces and is a bounded bilinear map. If
factor and is Arens regular. Then is Arens regular.
Proof. Let factor. Thus for every there exists and such that .
Suppose that and , and be bounded nets −converging to and
respectively. For every we have
It follows that is Arens regular
As an cosequnce of this theorem we have the following result:
Corollary 2.5. Let and be normed spaces and is a bounded bilinear map. If
factor and is Arens regular. Then is Arens regular.
Arens regularity and reflexivity
In this section, we show that with which assumptions left strongly irregular property is equivalent to the right
strongly irregular property.
Theorem 3.1. For a bounded bilinear map f : X Y Z,
Abotaleb Sheikhali & Nader Kanzi.; Sch. J. Phys. Math. Stat., 2018; Vol-5; Issue-1 (Jan-Feb); pp-65-68
Available Online: http://saspjournals.com/sjpms 68
(i) If factor then both and are Arens regular if and only if is reflexive.
(ii) If factor then both and are Arens regular if and only if is reflexive.
Proof. We only give a proof for (ii), A similar proof applies for (i). Let and are Arens regular by Corollary 2.3
. On the other hand factors, So . Therefore
Conversely, using is obvious
As an immediate consequnce of Theorem 3.1 and , we have the next Corollary.
Corollary 3.2. If one of the two following statement is assumed:
(i) and are Arens regular and factor;
(ii) and are Arens regular and factor;
Then every adjoint map and every flip map of is Arens regular.
Corollary 3.3. Let and are Arens regular and factor. Then is left strongly irregular if
and only it is right strongly irregular.
Proof. The follows by applying Theorem 3.1 and
If is reflexive. Then obviously bounded bilinear map from into is Arens regular. But from Arens regularity
does not imply the reflexivity of . The next Theorem, we use the Theorem 2.2 and show that if .
Then is reflexive.
Theorem 3.4. Let bounded bilinear map from into is Arens regular and let is a Banach space. If
for some . Then is reflexive.
Proof. Let define by for every . Obviously for
every . We have
Therefore = for every . Now Theorem 2.2 implies that . Since
thus is onto. Therefore from into is onto. Let so there exists
such that = = Thus is reflexive
REFERENCES
1. Arens A, The adjoint of a bilinear operation, Proc. Amer. Math. Soc, 2 (1951), 839-848.
2. Barootkoob S, Mohammadzadeh S and Vishki H.R, Topological centers of certain Banach module action, bulletin of
the Iranian Mathematical Society, Vol. 35 No. 2 (2009), 25-36.
3. Dales H.G, Banach algebras and automatic continuity, London Math. Soc. Monographs 24 (Clarendon Press,
Oxford,2000)
4. Dales H.G, Rodrigues-Palacios A, and Velasco M.V, The second transpose of a derivation, J. London Math. Soc.
64(2) (2001), 707-721.
5. Eshaghi Gordji M and Filali M, Arens regularity of module actions, Studia Math. 181 (3) (2007), 237-254.
6. Mohammadzadeh S and Vishki H.R, Arens regularity of module actions and the second adjoint of a derivation, Bull
Austral. Mat. Soc. 77 (2008), 465-476.
7. Ulger A, Weakly compact bilinear forms and Arens regularity, Proc. Amer. Math. Soc. 101 (1978), 697-701.
8. Sheikhali A, Sheikhali A, Akhlaghi N, Arens regularity of Banach module actions and the strongly irregular
property, J. Math. Computer Sci, 13 (2014), 41-46.
