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arXiv:1912.03294v1 [math.FA] 6 Dec 2019
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SOME PROPERTIES OF BOUNDED TRI-LINEAR
MAPS
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Abotaleb Sheikhali ·Ali Ebadian ·
Kazem Haghnejad Azar
Received: date / Accepted: date
Abstract Let X, Y, Z and Wbe normed spaces and f:X×Y×Z−→ W
be a bounded tri-linear mapping. In this Article, we define the topological
centers for bounded tri-linear mapping and we invistagate thier properties.
We study the relationships between weakly compactenss of bounded linear
mappings and regularity of bounded tri-linear mappings. For both bounded
tri-linear mappings fand g, let ffactors through g, we present necessary and
suficient condition such that the extensions of ffactors through extensions of
g. Also we establish relations between regularity and factorization property of
bounded tri-linear mappings.
Keywords Arens product ·Module action ·Factors ·Topological center ·
Tri-linear mappings.
Mathematics Subject Classification (2010) MSC 46H25 ·MSC 46H20 ·
MSC 46L06
1 Introduction
Let X, Y, Z and Wbe normed spaces and f:X×Y×Z−→ Wbe a bounded
tri-linear mapping. The natural extensions of fare as following:
1. f∗:W∗×X×Y−→ Z∗, given by hf∗(w∗, x, y), zi=hw∗, f (x, y, z)iwhere
x∈X, y ∈Y, z ∈Z, w∗∈W∗.
The map f∗is a bounded tri-linear mapping and is said the adjoint of f.
A. Sheikhali
Department of Mathematics, Payame Noor University (PNU), Tehran, Iran.
E-mail: Abotaleb.sheikhali.20@gmail.com
A. Ebadian
E-mail: Ebadian.ali@gmail.com
K. Haghnejad Azar
E-mail: Haghnejad@aut.ac.ir
2 Abotaleb Sheikhali et al.
2. f∗∗ = (f∗)∗:Z∗∗ ×W∗×X−→ Y∗, given by hf∗∗(z∗∗, w∗, x), yi=
hz∗∗, f ∗(w∗, x, y) where x∈X, y ∈Y, z ∗∗ ∈Z∗∗, w∗∈W∗.
3. f∗∗∗ = (f∗∗)∗:Y∗∗ ×Z∗∗ ×W∗−→ X∗, given by hf∗∗∗(y∗∗, z ∗∗, w∗), xi=
hy∗∗, f ∗∗(z∗∗, w∗, x)iwhere x∈X, y∗∗ ∈Y∗∗, z ∗∗ ∈Z∗∗, w∗∈W∗.
4. f∗∗∗∗ = (f∗∗∗)∗:X∗∗ ×Y∗∗ ×Z∗∗ −→ W∗∗, given by hf∗∗∗∗ (x∗∗ , y∗∗ , z∗∗),
w∗i=hx∗∗, f ∗∗∗(y∗∗ , z ∗∗, w∗)iwhere x∗∗ ∈X∗∗, y∗∗ ∈Y∗∗ , z ∗∗ ∈Z∗∗, w∗∈
W∗.
Now let fr:Z×Y×X−→ Wbe the flip of fdefined by fr(z, y, x) = f(x, y , z),
for every x∈X, y ∈Yand z∈Z. Then fris a bounded tri-linear map and it
may extends as above to fr∗∗∗∗ :Z∗∗ ×Y∗∗ ×X∗∗ −→ W∗∗. When f∗∗∗∗ and
fr∗∗∗∗rare equal, then fis called regular. Regularity of fis equvalent to the
following
w∗−lim
αw∗−lim
βw∗−lim
γf(xα, yβ, zγ) = w∗−lim
γw∗−lim
βw∗−lim
αf(xα, yβ, zγ),
where {xα},{yβ}and {zγ}are nets in X, Y and Zwhich converge to x∗∗ ∈
X∗∗, y∗∗ ∈Y∗∗ and z∗∗ ∈Z∗∗ in the w∗−topologies, respectively. For a
bounded tri-linear map f:X×Y×Z−→ W, if from X, Y or Zat least
two reflexive then f is regular.
A bounded bilinear(res tri-linear) mapping m:X×Y−→ Z(res f:
X×Y×Z−→ W) is said to be factor if it is surjective, that is f(X×Y) = Z(res
f(X×Y×Z) = W). For a good source of information on this subject, we
refer the reader to [5].
For a discussion of Arens regularity for Banach algebras and bounded bi-
linear maps, see [1], [2], [11], [12] and [17]. For example, every C∗−algebra is
Arens regular, see [4]. Also L1(G) is Arens regular if and only if G is finite,[18].
The left topological center of mmay be defined as following
Zl(m) = {x∗∗ ∈X∗∗ :y∗∗ −→ m∗∗∗(x∗∗, y ∗∗)is weak∗−to −weak∗−
continuous}.
Also the right topological center of mas
Zr(m) = {y∗∗ ∈Y∗∗ :x∗∗ −→ mr∗∗∗r(x∗∗, y∗∗ )is weak∗−to −weak∗−
continuous}.
The subject of topological centers have been investigated in [6], [7] and [15].
In [13], Lau and Ulger gave several significant results related to the topological
centers of certain dual algebras. In [11], Authors extend some problems from
Arens regularity and Banach algebras to module actions. They also extend the
definitions of the left and right multiplier for module actions, see [10] and [12].
Let Abe a Banach algebra, and let π:A×A−→ Adenote the product
of A, so that π(a, b) = ab for every a, b ∈A. The Banach algebra Ais Arens
regular whenever the map πis Arens regular. The first and second Arens
products denoted by and ♦respectively and definded by
a∗∗b∗∗ =π∗∗∗ (a∗∗ , b∗∗ ), a∗∗♦b∗∗ =πr∗∗∗r(a∗∗ , b∗∗ ),(a∗∗, b∗∗ ∈A∗∗ ).
SOME PROPERTIES OF BOUNDED TRI-LINEAR MAPS 3
2 Module actions for bounded tri-linear maps
Let (π1, X, π2) be a Banach A−module and let π1:A×X−→ Xand
π2:X×A−→ Xbe the left and right module actions of Aon X, re-
spectively. If (π1, X ) (res (X, π2)) is a left (res right) Banach A−module of A
on X, then (X∗, π∗
1)(res (πr∗r
2, X∗)) is a right (res left) Banach A−module
and (πr∗r
2, X∗, π∗
1) is the dual Banach A−module of (π1, X, π2). We note
also that (π∗∗∗
1, X∗∗, π∗∗∗
2) is a Banach (A∗∗ ,)−module with module ac-
tions π∗∗∗
1:A∗∗ ×X∗∗ −→ X∗∗ and π∗∗∗
2:X∗∗ ×A∗∗ −→ X∗∗. Simi-
lary, (πr∗∗∗r
1, X∗∗, πr∗∗∗r
2) is a Banach (A∗∗ ,♦)−module with module actions
πr∗∗∗r
1:A∗∗ ×X∗∗ −→ X∗∗ and πr∗∗∗r
2:X∗∗ ×A∗∗ −→ X∗∗. If we continue du-
alizing we shall reach (π∗∗∗r∗r
2, X∗∗∗, π∗∗∗∗
1) and (πr∗∗∗∗r
2, X∗∗∗, πr∗∗∗r∗
1) are the
dual Banach (A∗∗,)−module and Banach (A∗∗,♦)−module of (π∗∗∗
1, X∗∗
, π∗∗∗
2) and (πr∗∗∗r
1,X∗∗, πr∗∗∗r
2), respectively. In [8], Eshaghi Gordji and Fillali
show that if a Banach algebra Ahas a bounded left (or right) approximate
identity, then the left (or right) module action of Aon A∗is Arens regular if
and only if Ais reflexive.
We commence with the following definition for bounded tri-linear mapping.
Definition 1 Let Xbe a Banach space, Abe a Banach algebra and let Ω1:
A×A×X−→ Xbe a bounded tri-linear map. Then the pair (Ω1, X ) is said
to be a left Banach A−module when
Ω1(π(a, b), π(c, d), x) = Ω1(a, b, Ω1(c, d, x)),
for each a, b, c, d ∈Aand x∈X. A right Banach A−module may be defined
similarly. Let Ω2:X×A×A−→ Xbe a bounded tri-linear map. Then the
pair (X, Ω2) is said to be a right Banach A−module when
Ω2(x, π(a, b), π(c, d)) = Ω2(Ω2(x, a, b), c, d).
A triple (Ω1, X, Ω2) is said to be a Banach A−module when (Ω1, X ) and
(X, Ω2) are left and right Banach A−modules respectively, also
Ω2(Ω1(a, b, x), c, d) = Ω1(a, b, Ω2(x, c, d)).
If (Ω1, X, Ω2) is a Banach A−module, then (Ωr∗r
2, X∗, Ω∗
1) is a Banach A−module.
It follows that,
1. the triple (Ω∗∗∗∗
1, X∗∗, Ω ∗∗∗∗
2) is a Banach (A∗∗,,)−module.
2. the triple (Ωr∗∗∗∗r
1, X∗∗, Ω r∗∗∗∗r
2) is a Banach (A∗∗,♦,♦)−module.
Theorem 1 Let a, b, c, d ∈A,x∗∈X∗,x∗∗ ∈X∗∗ and b∗∗, c∗∗ ∈A∗∗.Then
1. If (Ω1, X )is a left Banach A−module, then
Ω∗∗∗
1(b∗∗, Ω ∗∗∗∗
1(c, d, x∗∗), x∗) = π∗∗ (b∗∗ , Ω ∗∗∗
1(π∗∗∗(c, d), x∗∗ , x∗)),
4 Abotaleb Sheikhali et al.
2. If (X , Ω2)is a right Banach A−module, then
Ωr∗∗∗r
2(x∗, Ωr∗∗∗∗r
2(x∗∗, a, b), c∗∗ ) = πr∗∗ (c∗∗ , Ω r∗∗∗r
2(x∗, x∗∗, π∗∗∗(a, b)).
Proof (1) Since the pair (Ω1, X ) is a left Banach A−module, thus for every
x∈Xwe have
hΩ∗
1(x∗, π (a, b), π(c, d)), xi=hx∗, Ω1(π(a, b), π(c, d), x)i
=hx∗, Ω1(a, b, Ω1(c, d, x))i=hΩ∗
1(x∗, a, b), Ω1(c, d, x)i
=hΩ∗
1(Ω∗
1(x∗, a, b), c, d), xi.
Hence Ω∗
1(x∗, π(a, b), π(c, d)) = Ω∗
1(Ω∗
1(x∗, a, b), c, d) and this implies that
hπ∗(Ω∗∗∗
1(π∗∗∗(c, d), x∗∗ , x∗), a), bi=hΩ∗∗∗
1(π∗∗∗(c, d), x∗∗ , x∗), π(a, b)i
=hπ∗∗∗(c, d), Ω ∗∗
1(x∗∗, x∗, π(a, b))i=hc, π∗∗ (d, Ω∗∗
1(x∗∗, x∗, π(a, b)))i
=hd, π∗(Ω∗∗
1(x∗∗, x∗, π(a, b)), c)i=hΩ∗∗
1(x∗∗, x∗, π(a, b)), π(c, d)i
=hx∗∗, Ω ∗
1(x∗, π(a, b), π(c, d))i=hx∗∗ , Ω∗
1(Ω∗
1(x∗, a, b), c, d)i
=hΩ∗∗
1(x∗∗, Ω ∗
1(x∗, a, b), c), di=hΩ∗∗∗
1(d, x∗∗, Ω ∗
1(x∗, a, b)), ci
=hΩ∗∗∗
1(c, d, x∗∗), Ω ∗
1(x∗, a, b)i=hΩ∗∗
1(Ω∗∗∗
1(c, d, x∗∗), x∗, a), bi.
Thus π∗(Ω∗∗∗
1(π∗∗∗(c, d), x∗∗ , x∗), a) = Ω∗∗
1(Ω∗∗∗
1(c, d, x∗∗), x∗, a). Finally, we
have
hΩ∗∗∗
1(b∗∗, Ω∗∗∗∗
1(c, d, x∗∗), x∗), ai=hb∗∗ , Ω∗∗
1(Ω∗∗∗∗
1(c, d, x∗∗), x∗, ai
=hb∗∗, π∗(Ω∗∗∗
1(π∗∗∗(c, d), x∗∗ , x∗), a)i
=hπ∗∗(b∗∗ , Ω ∗∗∗
1(π∗∗∗(c, d), x∗∗ , x∗)), ai.
The proof is complete. A similar argument applies for (2).
3 Topological centers of bounded tri-linear maps
In this section, we shall investigate the topological centers of bounded tri-linear
maps. The main definition of this section is in the following.
Definition 2 Let f:X×Y×Z−→ Wbe a bounded tri-linear map. We
define the topological centers of fby
Z1
l(f) = {x∗∗ ∈X∗∗|y∗∗ −→ f∗∗∗∗(x∗∗ , y∗∗, z∗∗)is weak∗−to −weak∗−
continuous},
Z2
l(f) = {x∗∗ ∈X∗∗|z∗∗ −→ f∗∗∗∗(x∗∗ , y∗∗ , z ∗∗)is weak∗−to −weak∗−
continuous},
Z1
r(f) = {z∗∗ ∈Z∗∗|x∗∗ −→ fr∗∗∗∗r(x∗∗, y∗∗, z ∗∗)is weak∗−to −weak∗−
continuous},
Z2
r(f) = {z∗∗ ∈Z∗∗|y∗∗ −→ fr∗∗∗∗r(x∗∗, y ∗∗, z∗∗)is weak∗−to −weak∗−
continuous},
Z1
c(f) = {y∗∗ ∈Y∗∗|x∗∗ −→ fr∗∗∗∗r(x∗∗, y ∗∗, z∗∗)is weak∗−to −weak∗−
continuous}.
Z2
c(f) = {y∗∗ ∈Y∗∗|z∗∗ −→ f∗∗∗∗(x∗∗ , y ∗∗, z∗∗)is weak∗−to −weak∗−
continuous}.
SOME PROPERTIES OF BOUNDED TRI-LINEAR MAPS 5
For a bounded tri-linear map f:X×Y×Z−→ W, We have
1. The map f∗∗∗∗ is the unique extension of fsuch that x∗∗ −→ f∗∗∗∗(x∗∗ , y∗∗
, z∗∗ ) is weak∗−weak∗continuous for each y∗∗ ∈Y∗∗ and z∗∗ ∈Z∗∗.
2. The map f∗∗∗∗ is the unique extension of fsuch that y∗∗ −→ f∗∗∗∗(x, y∗∗
, z∗∗ ) is weak∗−weak∗continuous for each x∈Xand z∗∗ ∈Z∗∗.
3. The map f∗∗∗∗ is the unique extension of fsuch that z∗∗ −→ f∗∗∗∗ (x, y, z∗∗ )
is weak∗−weak∗continuous for each x∈Xand y∈Y.
4. The map fr∗∗∗∗ris the unique extension of fsuch that z∗∗ −→ fr∗∗∗∗r(x∗∗
, y∗∗ , z∗∗ ) is weak∗−weak∗continuous for each x∗∗ ∈X∗∗ and y∗∗ ∈Y∗∗.
5. The map fr∗∗∗∗ris the unique extension of fsuch that x∗∗ −→ fr∗∗∗∗r(x∗∗
, y, z) is weak∗−weak∗continuous for each y∈Yand z∈Z.
6. The map fr∗∗∗∗ris the unique extension of fsuch that y∗∗ −→ fr∗∗∗∗r(x∗∗
, y∗∗ , z) is weak∗−weak∗continuous for each x∗∗ ∈X∗∗ and z∈Z.
As immediate consequences, we give the next Theorem.
Theorem 2 If f:X×Y×Z−→ Wis a bounded tri-linear map, then
X⊆Z1
l(f)and Z⊆Z2
r(f).
The mapping f∗∗∗∗ is the unique extension of fsuch that x∗∗ −→ f∗∗∗∗(x∗∗ , y∗∗
, z∗∗ ) from X∗∗ into W∗∗ is weak∗−to −weak∗continuous for every y∗∗ ∈Y∗∗
and z∗∗ ∈Z∗∗, hence the first right topological center of fmay be defined as
following
Z1
r(f) = {z∗∗ ∈Z∗∗|fr∗∗∗∗r(x∗∗ , y∗∗, z ∗∗) = f∗∗∗∗(x∗∗ , y∗∗ , z ∗∗), f or every
x∗∗ ∈X∗∗, y∗∗ ∈Y∗∗ }.
The mapping fr∗∗∗∗ris the unique extension of fsuch that z∗∗ −→
fr∗∗∗∗r(x∗∗, y∗∗, z ∗∗) from Z∗∗ into W∗∗ is weak∗−to −weak∗continuous
for every x∗∗ ∈X∗∗ and y∗∗ ∈Y∗∗, hence the second left topological center of
fmay be defined as following
Z2
l(f) = {x∗∗ ∈X∗∗|fr∗∗∗∗r(x∗∗ , y∗∗, z ∗∗) = f∗∗∗∗(x∗∗ , y∗∗ , z ∗∗), f or every
y∗∗ ∈Y∗∗, z ∗∗ ∈Z∗∗}.
It is clear that Z1
r(f) = Z2
l(fr) and Z1
r(fr) = Z2
l(f). Also fis regular if
and only if Z1
r(f) = Z∗∗ or Z2
l(f) = X∗∗. Let g:X×X×X−→ Xbe a
bounded tri-linear map, If gis regular, Then Z1
r(g) = Z2
l(g).
Example 1 Let Gbe a finite locally compact Hausdorff group. Then we have
f:L1(G)×L1(G)×L1(G)−→ L1(G)
defined by f(k, g, h) = k∗g∗h, is regular for every k, g and h∈L1(G). So
Z1
r(f) = Z2
l(f) = L1(G).
Definition 3 Let f:X×Y×Z−→ Wbe a bounded tri-linear map. Then
the map fis said to be first left (right) strongly irregular when Z1
l(f)⊆X
(Z1
r(f)⊆Z). The definition of second and third left (right) strongly irregular
are similar.
The proof of the following theorem is straightforward and we left its proof.
6 Abotaleb Sheikhali et al.
Theorem 3 Let Ybe a reflexive space and let f:X×Y×Z−→ Wbe a
bounded tri-linear map. Then
1. The map fis regular and first right strongly irregular if and only if Zis
reflexive.
2. The map fis regular and second left strongly irregular if and only if Xis
reflexive.
As immediate consequences of the Theorem 3 we have the next corollary.
Corollary 1 Let f:X×Y×Z−→ Wbe a bounded tri-linear map. If Xand
Y(or Zand Y) are reflexive spaces then fis regular.
Corollary 2 Let Abe a Banach algebra. If Abe reflexive, then
1. The bounded tri-linear map Ω1is regular, first and second left strongly
irregular.
2. The bounded tri-linear map Ω2is regular, first and second right strongly
irregular.
Corollary 3 Let m:X×X−→ Xbe a bounded bilinear map and let f:
X×X×X−→ Xbe a bounded tri-linear map. Then
1. If fis regular and first right (or second left) strongly irregular then mis
Arens regular.
2. If mis Arens regular and right (or left) strongly irregular then fis regular.
Example 2 Let Gbe an infinite, compact Hausdorff group and let 1 < p < ∞.
We know from [9, pp 54], that Lp(G)∗L1(G)⊂Lp(G) where
(k∗g)(x) = ZG
k(y)g(y−1x)dy, (x∈G, k ∈Lp(G), g ∈L1(G)).
On the other hand, the Banach space Lp(G) is reflexive, thus the bounded
tri-linear mapping
f:Lp(G)×L1(G)×Lp(G)−→ Lp(G)
defined by f(k, g, h) = (k∗g)∗h, is regular for every k, h ∈Lp(G) and
g∈L1(G). Therefore Z2
l(f) = Lp(G)∗∗ =Lp(G), thus fis second left strongly
irregular.
Theorem 4 Let Abe a Banach algebra. Then
1. If (Ω1, X )is a left Banach A−module and Ω∗∗∗
1, π∗∗∗ (A, A)are factors
then, Z1
l(Ω1)⊆Zl(π).
2. If (X , Ω2)is a right Banach A−module and Ωr∗∗∗r
2, π∗∗∗ (A, A)are factors
then, Z2
r(Ω2)⊆Zr(π).
SOME PROPERTIES OF BOUNDED TRI-LINEAR MAPS 7
Proof We prove only (1), the other one uses the same argument. Let a∗∗ ∈
Z1
l(Ω1), we show that a∗∗ ∈Zl(π). Let {b∗∗
α}be a net in A∗∗ which converge to
b∗∗ ∈A∗∗ in the w∗−topologies. We must show that π∗∗∗(a∗∗ , b∗∗
α) converge to
π∗∗∗(a∗∗ , b∗∗ ) in the w∗−topologies. Let a∗∈A∗, since Ω∗∗∗
1factors, so there
exist x∗∈X∗, x∗∗ ∈X∗∗ and c∗∗ ∈A∗∗ such that a∗=Ω∗∗∗
1(c∗∗, x∗∗ , x∗).
In the other hands π∗∗∗(A, A) factors, thus there exist c, d ∈Asuch that
π∗∗∗(c, d) = c∗∗ . Because a∗∗ ∈Z1
l(Ω1) thus Ω∗∗∗∗
1(a∗∗, b∗∗
α, x∗∗) converge to
Ω∗∗∗∗
1(a∗∗, b∗∗ , x∗∗ ) in the w∗−topologies.
In partiqular Ω∗∗∗∗
1(a∗∗, b∗∗
α, Ω∗∗∗∗
1(c, d, x∗∗)) converge to Ω∗∗∗∗
1(a∗∗, b∗∗ ,
Ω∗∗∗∗
1(c, d, x∗∗)) in the w∗−topologies. Now by Theorem 1, we have
lim
αhπ∗∗∗(a∗∗ , b∗∗
α), a∗i= lim
αhπ∗∗∗(a∗∗ , b∗∗
α), Ω∗∗∗
1(c∗∗, x∗∗ , x∗)i
= lim
αhπ∗∗∗(a∗∗ , b∗∗
α), Ω∗∗∗
1(π∗∗∗(c, d), x∗∗ , x∗)i
= lim
αha∗∗, π∗∗(b∗∗
α, Ω∗∗∗
1(π∗∗∗(c, d), x∗∗ , x∗))i
= lim
αha∗∗, Ω∗∗∗
1(b∗∗
α, Ω∗∗∗∗
1(c, d, x∗∗), x∗)i
= lim
αhΩ∗∗∗∗
1(a∗∗, b∗∗
α, Ω∗∗∗∗
1(c, d, x∗∗), x∗i
=hΩ∗∗∗∗
1(a∗∗, b∗∗ , Ω ∗∗∗∗
1(c, d, x∗∗), x∗i
=ha∗∗, Ω∗∗∗
1(b∗∗, Ω ∗∗∗∗
1(c, d, x∗∗), x∗)i
=ha∗∗, π∗∗(b∗∗ , Ω ∗∗∗
1(π∗∗∗(c, d), x∗∗ , x∗))i
=ha∗∗, π∗∗(b∗∗ , Ω ∗∗∗
1(c∗∗, x∗∗ , x∗))i
=ha∗∗, π∗∗(b∗∗ , a∗)i
=hπ∗∗∗(a∗∗ , b∗∗ ), a∗i.
Therefore π∗∗∗ (a∗∗, b∗∗
α) converge to π∗∗∗ (a∗∗ , b∗∗) in the w∗−topologies, as
required.
Theorem 5 Let Abe a Banach algebra and Ω:A×A×A−→ Abe a bounded
tri-linear mapping. Then for every a∈A, a∗∈A∗and a∗∗ ∈A∗∗,
1. If Ahas a bounded right approximate identity and bounded linear map
T:A∗−→ A∗given by T(a∗) = π∗∗ (a∗∗ , a∗)be weakly compactenss, then
Ωis regular.
2. If Ahas a bounded left approximate identity and bounded linear map T:
A−→ A∗given by T(a) = πr∗r∗(a∗∗ , a)be weakly compactenss, then Ωis
regular.
Proof We only prove (1). Let Tbe weakly compact, then T∗∗ (A∗∗∗ )⊆A∗. In
the other hand, a direct verification reveals that T∗∗(A∗∗∗) = π∗∗∗∗∗(A∗∗ , A∗∗∗ ).
Thus π∗∗∗∗∗(A∗, A∗∗∗)⊆A∗. Now let a∗∗ , b∗∗ ∈A∗∗ and a∗∗∗ ∈A∗∗∗ too
{aα}and {a∗
β}are nets in Aand A∗which converge to a∗∗ and a∗∗∗ in the
w∗−topologies, respectively. Then we have
hπ∗r∗∗∗r(a∗∗∗, a∗∗ ), b∗∗ i=hπ∗r∗∗∗ (a∗∗ , a∗∗∗ ), b∗∗ i=ha∗∗ , π∗r∗∗(a∗∗∗ , b∗∗ )i
= lim
αhπ∗r∗∗(a∗∗∗ , b∗∗ ), aαi= lim
αha∗∗∗, π∗r∗(b∗∗, aα)i
8 Abotaleb Sheikhali et al.
= lim
αlim
βhπ∗r∗(b∗∗, aα), a∗
βi= lim
αlim
βhb∗∗, π∗r(aα, a∗
β)i
= lim
αlim
βhb∗∗, π∗(a∗
β, aα)i= lim
αlim
βhπ∗∗(b∗∗ , a∗
β), aαi
= lim
αlim
βhπ∗∗∗(aα, b∗∗ ), a∗
βi= lim
αha∗∗∗, π∗∗∗(aα, b∗∗ )i
= lim
αhπ∗∗∗∗(a∗∗∗ , aα), b∗∗ i= lim
αhπ∗∗∗∗∗(b∗∗ , a∗∗∗ ), aαi
=ha∗∗, π∗∗∗∗∗(b∗∗ , a∗∗∗ )i=hπ∗∗∗∗ (a∗∗∗ , a∗∗ ), b∗∗ i.
Therefore π∗is Arens regular. Now we implies that Ais reflexive by [8, The-
orem 2.1]. It follows that Ωis regular and this completes the proof.
4 Factors of bounded tri-linear mapping
We commence with the following definition.
Definition 4 Let X, Y, Z, S1, S2and S3be normed spaces, f:X×Y×Z−→
Wand g:S1×S2×S3−→ Wbe bounded tri-linear mappings. Then we
say that ffactors through gby bounded linear mappings h1:X−→ S1,
h2:Y−→ S2and h3:Z−→ S3, if f(x, y, z ) = g(h1(x), h2(y), h3(z)).
In the next result we provide a necessary and sufficient condition such that
first and second extension of a bounded tri-linear mapping ffactors through
first and second extension of a bouneded tri-linear mapping g, respectively.
Theorem 6 Let f:X×Y×Z−→ Wand g:S1×S2×S3−→ Wbe bounded
tri-linear mapping. Then
1. The map ffactors through gif and only if f∗∗∗∗ factors through g∗∗∗∗ ,
2. The map ffactors through gif and only if fr∗∗∗∗rfactors through gr∗∗∗∗r.
Proof (1) Let ffactors through gby bounded linear mappings h1:X−→
S1, h2:Y−→ S2and h3:Z−→ S3, then f(x, y, z ) = g(h1(x), h2(y), h3(z))
for every x∈X, y ∈Yand z∈Z. Let {xα},{yβ}and {zγ}be nets in
X, Y and Zwhich converge to x∗∗ ∈X∗∗ , y∗∗ ∈Y∗∗ and z∗∗ ∈Z∗∗ in the
w∗−topologies, respectively. Then for every w∗∈W∗we have
hf∗∗∗∗(x∗∗ , y∗∗, z ∗∗), w∗i= lim
αlim
βlim
γhw∗, f (xα, yβ, zγ)i
= lim
αlim
βlim
γhw∗, g(h1(xα), h2(yβ), h3(zγ))i
= lim
αlim
βlim
γhg∗(w∗, h1(xα), h2(yβ)), h3(zγ)i
= lim
αlim
βlim
γhh∗
3(g∗(w∗, h1(xα), h2(yβ))), zγi
= lim
αlim
βhz∗∗, h∗
3(g∗(w∗, h1(xα), h2(yβ)))i
= lim
αlim
βhh∗∗
3(z∗∗), g∗(w∗, h1(xα), h2(yβ))i
= lim
αlim
βhg∗∗(h∗∗
3(z∗∗), w∗, h1(xα)), h2(yβ)i
SOME PROPERTIES OF BOUNDED TRI-LINEAR MAPS 9
= lim
αlim
βhh∗
2(g∗∗(h∗∗
3(z∗∗), w∗, h1(xα))), yβi
= lim
αhy∗∗, h∗
2(g∗∗(h∗∗
3(z∗∗), w∗, h1(xα)))i
= lim
αhh∗∗
2(y∗∗), g∗∗(h∗∗
3(z∗∗), w∗, h1(xα))i
= lim
αhg∗∗∗(h∗∗
2(y∗∗), h∗∗
3(z∗∗), w∗), h1(xα)i
= lim
αhh∗
1(g∗∗∗(h∗∗
2(y∗∗), h∗∗
3(z∗∗), w∗)), xαi
=hx∗∗, h∗
1(g∗∗∗(h∗∗
2(y∗∗), h∗∗
3(z∗∗), w∗))i
=hh∗∗
1(x∗∗), g∗∗∗(h∗∗
2(y∗∗), h∗∗
3(z∗∗), w∗)i
=hg∗∗∗∗(h∗∗
1(x∗∗), h∗∗
2(y∗∗), h∗∗
3(z∗∗)), w∗i.
Therefore f∗∗∗∗ factors through g∗∗∗∗ .
Conversely, suppose that f∗∗∗∗ factors through g∗∗∗∗, thus
f∗∗∗∗(x∗∗ , y∗∗ , z ∗∗) = g∗∗∗∗(h∗∗
1(x∗∗), h∗∗
2(y∗∗), h∗∗
3(z∗∗)),
in particular, for x∈X, y ∈Yand z∈Zwe have
f∗∗∗∗(x, y, z) = g∗∗∗∗ (h∗∗
1(x), h∗∗
2(y), h∗∗
3(z)).
Then for every w∗∈W∗we have
hw∗, f(x, y, z )i=hf∗(w∗, x, y), zi=hf∗∗(z , w∗, x), yi
=hf∗∗∗(y, z, w∗), xi=hf∗∗∗∗(x, y, z ), w∗i
=hg∗∗∗∗(h∗∗
1(x), h∗∗
2(y), h∗∗
3(z)), w∗i=hh∗∗
1(x), g∗∗∗ (h∗∗
2(y), h∗∗
3(z), w∗)i
=hx, h∗
1(g∗∗∗(h∗∗
2(y), h∗∗
3(z), w∗))i=hg∗∗∗(h∗∗
2(y), h∗∗
3(z), w∗), h1(x)i
=hh∗∗
2(y), g∗∗ (h∗∗
3(z), w∗, h1(x))i=hy, h∗
2(g∗∗(h∗∗
3(z), w∗, h1(x)))i
=hg∗∗(h∗∗
3(z), w∗, h1(x)), h2(y)i=hh∗∗
3(z), g∗(w∗, h1(x), h2(y))i
=hz, h∗
3(g∗(w∗, h1(x), h2(y)))i=hg∗(w∗, h1(x), h2(y)), h3(z)i
=hw∗, g(h1(x), h2(y)), h3(z))i.
It follows that ffactors through gand this completes the proof.
(2) The proof similar to (1).
Corollary 4 Let f:X×Y×Z−→ Wand g:S1×S2×S3−→ Wbe bounded
tri-linear map and let ffactors through g. If gis regular then fis also regular.
Proof Let gbe regular then g∗∗∗∗ =gr∗∗∗∗r. Since the ffactors through g
then for every x∗∗ ∈X∗∗, y ∗∗ ∈Y∗∗ and z∗∗ ∈Z∗∗ we have
f∗∗∗∗(x∗∗ , y∗∗, z ∗∗) = g∗∗∗∗(h∗∗
1(x∗∗), h∗∗
2(y∗∗), h∗∗
3(z∗∗))
=gr∗∗∗∗r(h∗∗
1(x∗∗), h∗∗
2(y∗∗), h∗∗
3(z∗∗))
=fr∗∗∗∗r(x∗∗, y∗∗, z ∗∗).
Therefore f∗∗∗∗ =fr∗∗∗∗r, as claimed.
10 Abotaleb Sheikhali et al.
5 Approximate identity and Factorization properties
Let Xbe a Banach space, Aand Bbe Banach algebras with bounded left
approximate identitis {eα}and {eβ}, respactively. Then a bounded tri-linear
mapping K1:A×B×X−→ Xis said to be left approximately unital if
w∗−lim
βw∗−lim
αK1(eα, eβ, x) = x,
and K1is said left unital if there exists e1∈Aand e2∈Bsuch that
K1(e1, e2, x) = x, for every x∈X.
Similarly, bounded tri-linear mapping K2:X×B×A−→ Xis said to be
right approximately unital if
w∗−lim
βw∗−lim
αK1(x, eβ, eα) = x,
and K2is also said to be right unital if K2(x, e2, e1) = x.
Lemma 1 Let Xbe a Banach space, Aand Bbe Banach algebras. Then
bounded tri-linear mapping
1. K1:A×B×X−→ Xis left approximately unital if and only if Kr∗∗∗∗r
1:
A∗∗ ×B∗∗ ×X∗∗ −→ X∗∗ is left unital.
2. K2:X×B×A−→ Xis right approximately unital if and only if K∗∗∗∗
2:
X∗∗ ×B∗∗ ×A∗∗ −→ X∗∗ is right unital.
Proof We prove only (1), the other part have the same argument. Let K1
be a left approximately unital. Thus there exists bounded left approximate
identitys {eα} ⊆ Aand {eβ} ⊆ Bshuch that
w∗−lim
βw∗−lim
αK1(eα, eβ, x) = x,
for every x∈X. Let {eα}and {eβ}which converge to e∗∗
1∈A∗∗ and e∗∗
2∈B∗∗
in the w∗−topologies, respectively. In the other hand, for every x∗∗ ∈X∗∗, let
{xγ} ⊆ Xwhich converge to x∗∗ in the w∗−topologies, then we have
hKr∗∗∗∗r
1(e∗∗
1, e∗∗
2, x∗∗), x∗i=hKr∗∗∗∗
1(x∗∗, e∗∗
2, e∗∗
1), x∗i
=hx∗∗, K r∗∗∗
1(e∗∗
2, e∗∗
1, x∗)i
= lim
γhKr∗∗∗
1(e∗∗
2, e∗∗
1, x∗), xγi
= lim
γhe∗∗
2, Kr∗∗
1(e∗∗
1, x∗, xγ)i
= lim
γlim
βhKr∗∗
1(e∗∗
1, x∗, xγ), eβi
= lim
γlim
βhe∗∗
1, Kr∗
1(x∗, xγ, eβ)i
= lim
γlim
βlim
αhKr∗
1(x∗, xγ, eβ), eαi
= lim
γlim
βlim
αhx∗, Kr
1(xγ, eβ, eα)i
= lim
γlim
βlim
αhx∗, K1(eα, eβ, xγ)i
= lim
γhx∗, xγi=hx∗∗, x∗i.
SOME PROPERTIES OF BOUNDED TRI-LINEAR MAPS 11
Therefore Kr∗∗∗∗r
1(e∗∗
1, e∗∗
2, x∗∗) = x∗∗ . It follows that Kr∗∗∗∗r
1is left unital.
Conversely, suppose that Kr∗∗∗∗r
1is left unital. So there exists e∗∗
1∈A∗∗
and e∗∗
2∈b∗∗ shuch that Kr∗∗∗∗r
1(e∗∗
1, e∗∗
2, x∗∗) = x∗∗ for every x∗∗ ∈X∗∗ .
Now let {eα},{eβ}and {xγ}be nets in A, B and Xwhich converge to e∗∗
1, e∗∗
2
and x∗∗ in the w∗−topologies, respectively. Thus
w∗−lim
γw∗−lim
βw∗−lim
αK1(eα, eβ, xγ) = Kr∗∗∗∗r
1(e∗∗
1, e∗∗
2, x∗∗)
=x∗∗ =w∗−lim
γxγ.
Therefore K1is left approximately unital and this completes the proof.
Remark 1 It should be remarked that in contrast to the situation occurring for
Kr∗∗∗∗r
1and K∗∗∗∗
2in the above lemma, K∗∗∗∗
1and Kr∗∗∗∗r
2are not necessarily
left and right unital respectively, in general.
Theorem 7 Suppos X, S are Banach spaces and A, B are Banach algebras.
1. Let K1:A×B×X−→ Xbe left approximately unital and factors through
gr:A×B×S−→ Xfrom rigth by h:X−→ S. If his weakly compactenss,
then Xis reflexive.
2. Let K2:X×B×A−→ Xbe right approximately unital and factors through
gl:S×B×A−→ Xfrom left by h:X−→ S. If his weakly compactenss,
then Xis reflexive.
Proof We only give a proof for (1). Since K1is left approximately unital, thus
there exists e∗∗
1∈A∗∗ and e∗∗
2∈B∗∗ shut that
Kr∗∗∗∗r
1(e∗∗
1, e∗∗
2, x∗∗) = x∗∗ ,
for every x∗∗ ∈X∗∗. In the other hand, the bounded tri-linear mapping K1fac-
tors through grfrom rigth, so Theorem 6 follows that Kr∗∗∗∗r
1factors through
gr∗∗∗∗r
rfrom rigth. In other words
Kr∗∗∗∗r
1(e∗∗
1, e∗∗
2, x∗∗) = gr∗∗∗∗r
r(e∗∗
1, e∗∗
2, h∗∗
3(x∗∗)).
Then for every x∗∗∗ ∈X∗∗∗ we have
hx∗∗∗, x∗∗ i=hx∗∗∗ , K r∗∗∗∗r
1(e∗∗
1, e∗∗
2, x∗∗)i
=hx∗∗∗, g r∗∗∗∗r
r(e∗∗
1, e∗∗
2, h∗∗(x∗∗ ))i
=hgr∗∗∗∗r∗
r(x∗∗∗, e∗∗
1, e∗∗
2), h∗∗(x∗∗ )i
=hh∗∗∗(gr∗∗∗∗r∗
r(x∗∗∗, e∗∗
1, e∗∗
2)), x∗∗i.
Therefore x∗∗∗ =h∗∗∗(gr∗∗∗∗r∗
r(x∗∗∗, e∗∗
1, e∗∗
2)). The weak compactness of h
implies that of h∗from which we have h∗∗∗(S∗∗∗ )⊆X∗. In particular
h∗∗∗(gr∗∗∗∗r∗
r(x∗∗∗, e∗∗
1, e∗∗
2)) ⊆X∗,
that is, X∗is reflexive. So Xis reflexive.
12 Abotaleb Sheikhali et al.
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