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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

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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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