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Bulletin of the Iranian Mathematical Society Vol. 35 No. 2 (2009 ), pp 25-36.

TOPOLOGICAL CENTERS OF CERTAIN BANACH

MODULE ACTIONS

S. BAROOTKOOB, S. MOHAMMADZADEH AND H.R.E. VISHKI*

Communicated by Hossein Eslamzadeh

Abstract. We study the topological centers of some speciﬁc ad-

joints of a Banach module action. Then, we investigate the Arens

regularity and strong irregularity of these actions.

1. Introduction and preliminaries

Based on the celebrated work of R. Arens [1], every bounded bilinear

map f:X ×Y → Z (on the normed spaces X,Yand Z) has two natural

but, in general, diﬀerent extensions f∗∗∗ and fr∗∗∗rfrom X∗∗ × Y∗∗ to

Z∗∗. Let us recall these notions with more detail.

For a bounded bilinear map f:X × Y → Z, we deﬁne the adjoint

f∗:Z∗× X → Y∗of fby

< f∗(z∗, x), y >=< z∗, f (x, y)>, (x∈ X , y ∈ Y and z∗∈ Z∗).

Continuing this process, we can deﬁne the second and the third adjoints

f∗∗ and f∗∗∗ of fby f∗∗ = (f∗)∗:Y∗∗ × Z∗→ X ∗and f∗∗∗ = (f∗∗)∗:

X∗∗ × Y∗∗ → Z∗∗ , respectively; and so on for the higher rank adjoints

of f. One can verify that f∗∗∗ is the unique extension of fwhich is

MSC(2000): 46H20, 46H25.

Keywords: Arens product, bounded bilinear map, Banach module action, topological center,

second dual.

Received: 15 July 2008, Accepted: 17 October 2008.

∗Corresponding author.

c

2009 Iranian Mathematical Society.

25

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26 Barootkoob, Mohammadzadeh and Vishki

w∗−separately continuous on X × Y∗∗. We deﬁne the left topological

center Z`(f) of fby

Z`(f) = {x∗∗ ∈ X ∗∗;y∗∗ −→ f∗∗∗(x∗∗, y ∗∗) is w∗−w∗−continuous}

={x∗∗ ∈ X ∗∗;f∗∗∗ (x∗∗ , y∗∗) = fr∗∗∗r(x∗∗, y ∗∗) for every y∗∗ ∈ Y∗∗ }.

We also denote by frthe ﬂip map of f, that is, the bounded bilinear map

fr:Y × X −→ Z deﬁned by fr(y, x) = f(x, y) (x∈ X , y ∈ Y).If we

repeat the latter process with frinstead of f, we come to the bounded

bilinear map fr∗∗∗r:X∗∗ × Y∗∗ → Z∗∗ , that is, the unique extension of

fwhich is w∗−separately continuous on X∗∗ × Y. We also deﬁne the

right topological center Zr(f) of fby

Zr(f) = {y∗∗ ∈ Y∗∗;x∗∗ −→ fr∗∗∗r(x∗∗ , y ∗∗) is w∗−w∗−continuous}

={y∗∗ ∈ Y∗∗;f∗∗∗(x∗∗ , y ∗∗) = fr∗∗∗r(x∗∗, y∗∗) for every x∗∗ ∈ X ∗∗ }.

From these observations, we have Zr(f) = Z`(fr).

An standard argument may also be used to interpret f∗∗∗ and fr∗∗∗rby

the following iterative limit processes,

f∗∗∗(x∗∗ , y∗∗) = w∗−lim

αlim

βf(xα, yβ),and

fr∗∗∗r(x∗∗, y∗∗) = w∗−lim

βlim

αf(xα, yβ),

where {xα}and {yβ}are nets in Xand Ythat converge, in w∗−topologies,

to x∗∗ and y∗∗, respectively.

A bounded bilinear mapping fis said to be (Arens) regular if f∗∗∗ =

fr∗∗∗r. This happens if and only if Z`(f) = X∗∗, or equivalently Zr(f) =

Y∗∗.The map fis said to be left (respectively, right) strongly irregular

if Z`(f) = X(respectively, Zr(f) = Y).

It is worthwhile mentioning that in the case where πis the multi-

plication of a Banach algebra A, then π∗∗∗ and πr∗∗∗rare actually the

ﬁrst and second Arens products, which will be denoted by and ♦,

respectively. We also say that Ais (Arens) regular, left strongly irreg-

ular or right strongly irregular if the multiplication πof Aenjoys the

corresponding property.

The subject of regularity of bounded bilinear mappings and Banach

module actions have been investigated in [3], [6], [7] and [9]. In [7],

Eshaghi Gordji and Fillali gave several signiﬁcant results related to the

topological centers of Banach module actions. In [9], the authors have

obtained a criterion for the regularity of f, from which they gave several

results related to the regularity of Banach module actions with some

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Topological centers of certain Banach module actions 27

applications to the second adjoint of a derivation. For a good and rich

source of information on this subject, we refer the reader to the Memoire

in [5]. We also shall mostly follow [4] as a general reference on Banach

algebras.

The remainder of the paper is organized as follows. In Section 1,

we study the left topological centers of πr∗

1and π∗

2, where (π1,X) and

(X, π2) are approximately unital left and right Banach A−modules, re-

spectively. We show that πr∗

1and π∗

2are permanently left strongly irreg-

ular; see Theorem 2.2 below. This result improves some results of [6], [9]

and [7]. For instance, it covers [6, Proposition 4.5], [9, Proposition 3.6]

and [7, Corollary 2.4] as well. In Section 2, we shall characterize the

right topological centers of πr∗

1and π∗

2(see Theorem 3.4 below). We

apply this fact to determine the topological centers of πr∗rand π∗for

the multiplication πof a Banach algebra with a bounded approximate

identity. Section 3 is devoted to investigation of relationships between

the regularity of πr∗

1,π∗

2,π1,π2and π, in the case where (π1,X) and

(X, π2) enjoy some factorization properties and are not necessarily ap-

proximately unital.

As already done, throughout the paper we usually identify (an element

of) a normed space with its canonical image in its second dual.

2. Left strong irregularity of certain adjoints of a Banach

module action

Let Abe a Banach algebra, Xbe a Banach space and π1:A×X −→ X

be a bounded bilinear mapping. Then, the pair (π1,X) is said to be a

left Banach A−module if

π1(ab, x) = π1(a, π1(b, x)); (a, b ∈ A, x ∈ X ).

A right Banach A−module (X, π2) can be deﬁned similarly. A triple

(π1,X, π2) is said to be a Banach A−module if (π1,X) and (X, π2) are

left and right Banach A−modules, respectively, and

π1(a, π2(x, b)) = π2(π1(a, x), b); (a, b ∈ A, x ∈ X ).

Let (π1,X) and (X, π2) be left and right Banach A−modules, respec-

tively. Then, one may verify that (π∗∗∗

1,X∗∗) and (X∗∗ , π∗∗∗

2) are left and

right Banach (A∗∗,)−modules, respectively. Similarly, (πr∗∗∗r

1,X∗∗)

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28 Barootkoob, Mohammadzadeh and Vishki

and (X∗∗, πr∗∗∗r

2) are left and right Banach (A∗∗,♦)−modules, respec-

tively.

In the case where Aenjoys a bounded left approximate identity, we say

that (π1,X) is approximately unital if the involved bounded left approx-

imate identity of Ais that of (π1,X). The notion of being approximately

unital can be deﬁned similarly for a right Banach A−module (X, π2). It

can be easily veriﬁed that a left Banach A−module (π1,X) is approx-

imately unital if the left module action (πr∗∗∗r

1,X∗∗) is unital; that is,

πr∗∗∗r

1(e∗∗, x∗∗ ) = x∗∗, for every x∗∗ ∈ X ∗∗, in which e∗∗ is a w∗−cluster

point of the involved left approximate identity of (π1,X). A similar fact

is valid for the right Banach A−module (X, π2). We summarize these

observations in the next result.

Proposition 2.1. Let (π1,X)and (X, π2)be left and right Banach

A−modules, respectively. Then,

(i) (π1,X)is approximately unital if (πr∗∗∗r

1,X∗∗)is unital.

(ii) (X, π2)is approximately unital if (X∗∗, π∗∗∗

2)is unital.

It should be remarked that in contrast to the situation occurring

for (πr∗∗∗r

1,X∗∗) and (X∗∗ , π∗∗∗

2) in the above result, (π∗∗∗

1,X∗∗) and

(X∗∗, πr∗∗∗r

2) are not necessarily unital, in general. For instance, let

A=K(c0) be the Banach algebra of compact operators on the sequence

space c0. Then, Aenjoys a bounded approximate identity and a direct

veriﬁcation reveals that for the multiplication π1=πof A, (π∗∗∗

1,A∗∗)

is not unital. Also, a similar argument shows that for the reversed

multiplication π2=πron A, (A∗∗, πr∗∗∗r

2) is not unital; for more details,

see [8, Example 2.5].

The next result studies the left strong irregularity of πr∗

1and π∗

2, from

which we improve some older results.

Theorem 2.2. Let (π1,X)and (X, π2)be approximately unital left

and right Banach A−modules, respectively. Then, πr∗

1and π∗

2are left

strongly irregular; that is,

Z`(π1r∗) = X∗=Z`(π2∗).

Proof. Let (X, π2) be approximately unital, x∗∗∗ ∈Z`(π∗

2) and x∗∗ ∈

X∗∗.Then, using Proposition 2.1, there exists e∗∗ ∈ A∗∗ such that

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Topological centers of certain Banach module actions 29

π∗∗∗

2(x∗∗, e∗∗ ) = x∗∗. Now,

hx∗∗∗, x∗∗ i=hx∗∗∗, π∗∗∗

2(x∗∗, e∗∗ )i

=hπ∗∗∗∗

2(x∗∗∗, x∗∗ ), e∗∗i

=hπ∗r∗∗∗r

2(x∗∗∗, x∗∗ ), e∗∗i

=hx∗∗, π∗r∗∗

2(x∗∗∗, e∗∗ )i

implies that x∗∗∗ =π∗r∗∗

2(x∗∗∗, e∗∗ )∈ X ∗. Therefore, Z`(π∗

2) = X∗, as

required. The other equality needs a similar argument.

As an immediate consequence of Theorem 2.2, we deduce the next

result of [9], (which in turn is a generalization of [6, Proposition 4.5]; see

also [2, Theorem 4] and [10, Theorem 3.1].)

Corollary 2.3. ([9, Proposition 3.6]) Let (π1,X)and (X, π2)be approx-

imately unital left and right Banach A−modules, respectively. Then, the

following assertions are equivalent:

(i)πr∗

1is regular.

(ii)π∗

2is regular.

(iii)Xis reﬂexive.

3. The right topological centers of πr∗

1and π∗

2

Before we proceed to the main result of this section, we need to intro-

duce a set MXand examine some of its properties. For a normed space

X, let JX:X → X ∗∗ denote the canonical embedding of Xinto X∗∗,

with the second adjoint (JX)∗∗ :X∗∗ → X ∗∗∗∗. We deﬁne MXby

MX={x∗∗ ∈ X ∗∗ :JX∗∗ (x∗∗) = (JX)∗∗(x∗∗)}.

It is routine to verify that MXis a closed subspace of X∗∗ containing X.

It should be mentioned that MXmay lie strictly between Xand X∗∗;

as we shall see in Corollary 3.3, this is the case for X=c0. It would be

desirable to characterize those Xfor which X=MX. The next lemma

clears the equality MX=X∗∗.

Lemma 3.1. For a normed space X, the equality MX=X∗∗ holds if

and only if Xis reﬂexive.

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30 Barootkoob, Mohammadzadeh and Vishki

Proof. If Xis reﬂexive, then trivially MX=X∗∗. For the converse,

suppose that x∗∗ ∈X∗∗ and x∗∗∗ ∈ X ∗∗∗ . Then,

hx∗∗∗, x∗∗ i=hJX∗∗ (x∗∗), x∗∗∗i

=h(JX)∗∗(x∗∗ ), x∗∗∗i

=hx∗∗,(JX)∗(x∗∗∗ )i

=hJX∗((JX)∗(x∗∗∗)), x∗∗ i.

Therefore x∗∗∗ =JX∗((JX)∗(x∗∗∗)) ∈JX∗(X∗); that is, X∗is reﬂexive,

and so Xis reﬂexive.

Example 3.2. We show that c0(Mc0, where c0is the Banach space of

all sequences converging to zero. Indeed, a direct veriﬁcation shows that

c⊆Mc0,in which cis the Banach space of all convergent sequences. To

see this, one may use the direct sum decomposition,

`∞∗=c∗⊕c⊥,

to show that for every x∗∗ ∈c⊂`∞=c∗∗

0and x∗∗∗ ∈c∗∗∗ =`∞∗,

h(JX)∗∗(x∗∗ ), x∗∗∗i=hJX∗∗ (x∗∗), x∗∗∗i,

from which we deduce x∗∗ ∈Mc0,as claimed.

Corollary 3.3. c0(Mc0(c∗∗

0.

The next result, being the main one in this section, characterizes the

right topological centers of πr∗

1and π∗

2.

Theorem 3.4. Let (π1,X)and (X, π2)be approximately unital left and

right Banach A−modules, respectively. Then,

Zr(πr∗

1) = MX=Zr(π∗

2).

Proof. We shall only prove MX=Zr(π∗

2); the other equality needs

a similar argument. Let x∗∗ ∈MX. Then, for every x∗∗∗ ∈ X ∗∗∗ ,

a∗∗ ∈ A∗∗ and bounded nets {xα}⊆X,{x∗

β}⊆X∗,w∗−converging to

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Topological centers of certain Banach module actions 31

x∗∗ and x∗∗∗, respectively, we have,

hπ∗∗∗∗

2(x∗∗∗, x∗∗ ), a∗∗i=hπ∗∗∗∗∗

2(a∗∗, x∗∗∗ ), x∗∗i

=hJX∗∗ (x∗∗), π∗∗∗∗∗

2(a∗∗, x∗∗∗ )i

=h(JX)∗∗(x∗∗ ), π∗∗∗∗∗

2(a∗∗, x∗∗∗ )i

=hx∗∗,(JX)∗(π∗∗∗∗∗

2(a∗∗, x∗∗∗ ))i

= lim

αh(JX)∗(π∗∗∗∗∗

2(a∗∗, x∗∗∗ )), xαi

= lim

αhπ∗∗∗∗∗

2(a∗∗, x∗∗∗ ), JX(xα)i

= lim

αhπ∗∗∗∗

2(x∗∗∗, xα), a∗∗ i

= lim

αhx∗∗∗, π∗∗∗

2(xα, a∗∗)i

= lim

αlim

βhπ∗∗∗

2(xα, a∗∗), x∗

βi

= lim

αlim

βhπ∗∗

2(a∗∗, x∗

β), xαi

= lim

αlim

βha∗∗, π∗

2(x∗

β, xα)i

=hπ∗r∗∗∗r

2(x∗∗∗, x∗∗ ), a∗∗i.

We thus have x∗∗ ∈Zr(π∗

2); that is, MX⊆Zr(π∗

2).To prove the re-

verse inclusion, let x∗∗ ∈Zr(π∗

2). As (X, π2) is approximately unital,

by Proposition 2.1, there exists a bounded right approximate identity

{eλ} ⊆ A for (X, π2) with e∗∗ ∈ A∗∗ as a w∗−cluster point of {eλ}such

that π∗∗∗

2(x∗∗, e∗∗ ) = x∗∗. Let x∗∗∗ ∈X∗∗∗,{xα} ⊆ X and {x∗

β} ⊆ X ∗

be bounded nets that are w∗−convergent to x∗∗ and x∗∗∗, respectively.

Then,

hJX∗∗ (x∗∗), x∗∗∗ i=hx∗∗∗, x∗∗i

=hx∗∗∗, π∗∗∗

2(x∗∗, e∗∗ )i

=hπ∗∗∗∗

2(x∗∗∗, x∗∗ ), e∗∗i

=hπ∗r∗∗∗r

2(x∗∗∗, x∗∗ ), e∗∗i

= lim

αlim

βlim

γhπ∗

2(x∗

β, xα), eγi

= lim

αlim

βlim

γhx∗

β, π2(xα, eγ)i

= lim

αlim

βhx∗

β, xαi

= lim

αlim

βhJX(xα), x∗

βi

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32 Barootkoob, Mohammadzadeh and Vishki

= lim

αhx∗∗∗, JX(xα)i

= lim

αh(JX)∗(x∗∗∗), xαi

=hx∗∗,(JX)∗(x∗∗∗ )i

=h(JX)∗∗(x∗∗ ), x∗∗∗i.

Therefore, x∗∗ ∈MX, completing the proof of the equality MX=

Zr(π∗

2).

As a rapid consequence of theorems 2.2 and 3.4, in the next result we

determine the topological centres of module actions of Aon A∗.

Proposition 3.5 (See [7, Theorem 2.1, Corollaries 2.1 and 2.4]).For

the multiplication πof a Banach algebra Awith a bounded approximate

identity, we have,

Z`(πr∗r) = MA=Zr(π∗) and Zr(πr∗r) = A∗=Z`(π∗).

In particular, πr∗ris regular if and only if π∗is regular if and only if A

is reﬂexive.

In the following, we have a more illuminating example characterizing

the right topological centers of πr∗

1and π∗

2.

Example 3.6. Let Abe a Banach space such that A MA A∗∗ (such

as c0). Fix e∈ A and e∗∈ A∗such that ke∗k≤ 1 and he∗, ei= 1.Then,

the multiplication π1(a, b) = he∗, aibturns Ainto a Banach algebra with

eas a left identity; similarly, π2(a, b) = πr

1(a, b) = he∗, biaturns Ainto

a Banach algebra with eas a right identity (see [6, Example 4.7]). As

theorems 2.2 and 3.4 demonstrate, we have,

Z`(π1r∗) = A∗=Z`(π2∗) and Zr(π1r∗) = MA=Zr(π2∗).

Moreover, for π∗

1=πr∗

2:A∗×A → A∗one may verify that π∗

1(a∗, a) =

he∗, aia∗; and this equality reveals that π∗

1is regular. In other words,

Z`(π1∗) = A∗∗∗ and Zr(π1∗) = A∗∗.Therefore, π∗

1=πr∗

2is neither left

nor right strongly irregular. Note that neither (A, π1) nor (π2,A) is

approximately unital.

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Topological centers of certain Banach module actions 33

4. (Arens) regularity of factorizable Banach module actions

A bounded bilinear mapping f:X × Y → Z is said to factor if it

is onto, that is f(X × Y) = Z.As a consequence of the so-called Co-

hen’s Factorization Theorem (see for example [4]), every approximately

unital (left or right) Banach A−module (π1,X) or (X, π2) factors. More-

over, as Proposition 2.1 demonstrates, in this case both (πr∗∗∗r

1,X∗∗) and

(X∗∗, π∗∗∗

2) are unital and thus factor.

However, many natural occurring Banach modules which factor are

approximately unital; but this is not the case, in general. For instance,

one may refer to [11] (see also [4]), for a wide variety of Banach algebras

and Banach modules which enjoy some types of factorization properties

but are not approximately unital. Here, we present some miscellaneous

results on the regularity of πr∗

1and π∗

2for the case where the module

actions (π1,X) and (X, π2) are not necessarily approximately unital. It

should be remarked that in the case where the involved module actions

are approximately unital, then these results can be derived as straight

corollaries of our results given in the former sections.

Proposition 4.1. Let Abe Arens regular and let (π1,X)and (X, π2)

be left and right Banach A−modules, respectively.

(i) If (πr∗∗∗r

1,X∗∗)factors, then the regularity of πr∗

1implies that of

π1.

(ii) If (X∗∗, π∗∗∗

2)factors, then the regularity of π∗

2implies that of

π2.

Proof. We only give a proof for (ii). For each x∗∗ ∈ X ∗∗ there exist

y∗∗ ∈ X ∗∗ and b∗∗ ∈ A∗∗ such that x∗∗ =π∗∗∗

2(y∗∗, b∗∗ ). Let a∗∗ ∈

A∗∗ and let {aα} ⊆ A,{bβ} ⊆ A and {yγ} ⊆ X be bounded nets

w∗−converging to a∗∗,b∗∗ and y∗∗,respectively. Then, for each x∗∈ X ∗,

πr∗

2(x∗, aα) converges, in w∗−topology, to πr∗∗∗∗

2(x∗, a∗∗) and

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34 Barootkoob, Mohammadzadeh and Vishki

hπr∗∗∗r

2(x∗∗, a∗∗ ), x∗i=hπr∗∗∗∗

2(x∗, a∗∗), π∗∗∗

2(y∗∗, b∗∗ )i

=hπ∗∗∗∗

2(πr∗∗∗∗

2(x∗, a∗∗), y∗∗), b∗∗i

=hπ∗r∗∗∗r

2(πr∗∗∗∗

2(x∗, a∗∗), y∗∗), b∗∗i

= lim

γlim

αhπ∗

2(πr∗

2(x∗, aα), yγ), b∗∗i

= lim

γlim

αlim

βhπ∗

2(πr∗

2(x∗, aα), yγ), bβi

= lim

γlim

αlim

βhπr∗

2(x∗, aα), π2(yγ, bβ)i

= lim

γlim

αlim

βhx∗, π2(π2(yγ, bβ), aα)i

= lim

γlim

αlim

βhx∗, π2(yγ, bβaα)i

= lim

γlim

αlim

βhπ∗

2(x∗, yγ), bβaαi

= lim

γhπ∗

2(x∗, yγ), b∗∗♦a∗∗ )i

=hπ∗∗∗

2(y∗∗, b∗∗ ♦a∗∗), x∗i

=hπ∗∗∗

2(y∗∗, b∗∗ a∗∗), x∗i

=hπ∗∗∗

2(π∗∗∗

2(y∗∗, b∗∗ ), a∗∗), x∗i

=hπ∗∗∗

2(x∗∗, a∗∗ ), x∗i.

Therefore, πr∗∗∗r

2(x∗∗, a∗∗ ) = π∗∗∗

2(x∗∗, a∗∗ ),for all a∗∗ ∈ A∗∗, x∗∗ ∈ X ∗∗ ,

which meaning that π2is regular.

Proposition 4.2. Let Abe a Banach algebra and (π1,X, π2)be a Ba-

nach A−module.

(i)If (πr∗∗∗r

1,X∗∗)factors, π1and πr∗

1are regular, then π2is regular.

(ii)If (X∗∗, π∗∗∗

2)factors, π2and π∗

2are regular, then π1is regular.

Proof. As the proof is similar to that of Proposition 4.1, we only give a

brief explanation for part (i). For each x∗∗ ∈ X ∗∗, there exist y∗∗ ∈ X ∗∗

and b∗∗ ∈ A∗∗ such that x∗∗ =πr∗∗∗r

1(b∗∗, y∗∗). Let a∗∗ ∈A∗∗ and let

{aα} ⊆ A,{bβ} ⊆ A and {yγ} ⊆ X be bounded nets, w∗−converging

to a∗∗,b∗∗ and y∗∗ ,respectively. Using the regularity of πr∗

1and π1, for

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Topological centers of certain Banach module actions 35

each x∗∈ X ∗,

hπr∗∗∗r

2(x∗∗, a∗∗ ), x∗i=hπr∗∗∗∗

2(x∗, a∗∗), πr∗∗∗

1(y∗∗, b∗∗ )i

=hπr∗∗∗∗

1(πr∗∗∗∗

2(x∗, a∗∗), y∗∗), b∗∗i

=hπr∗r∗∗∗r

1(πr∗∗∗∗

2(x∗, a∗∗), y∗∗), b∗∗i

= lim

γlim

αhπr∗

1(πr∗

2(x∗, aα), yγ), b∗∗i

= lim

γlim

αlim

βhπr∗

1(πr∗

2(x∗, aα), yγ), bβi

= lim

γlim

αlim

βhπr∗

2(x∗, aα), π1(bβ, yγ)i

= lim

γlim

αlim

βhx∗, π2(π1(bβ, yγ), aα)i

= lim

γlim

αlim

βhx∗, π1(bβ, π2(yγ, aα))i

=hπr∗∗∗r

1(b∗∗, π∗∗∗

2(y∗∗, a∗∗ )), x∗i

=hπ∗∗∗

1(b∗∗, π∗∗∗

2(y∗∗, a∗∗ )), x∗i

=hπ∗∗∗

2(x∗∗, a∗∗ ), x∗i.

Therefore, π2is regular, as required.

As an immediate consequence of propositions 4.1 and 4.2, we have

the next corollaries.

Corollary 4.3. Let (π1,X, π2)be a Banach A−module such that

(πr∗∗∗r

1,X∗∗)and (X∗∗ , π∗∗∗

2)factor. We have:

(i)If either πand πr∗

1or π2and π∗

2are regular, then so is π1.

(ii)If either πand π∗

2or π1and πr∗

1are regular, then so is π2.

Corollary 4.4. Let Abe (Arens) regular and let (π1,X, π2)be a Banach

A−module such that (πr∗∗∗r

1,X∗∗)and (X∗∗ , π∗∗∗

2)factor. If either π∗

2

or πr∗

1is regular, then both π1and π2are regular.

Acknowledgments

The useful comments of the anonymous referee are gratefully acknowl-

edged.

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36 Barootkoob, Mohammadzadeh and Vishki

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S. Barootkoob

Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159,

Mashhad 91775, IRAN.

Email: se-ba406@mail.um.ac.ir

S. Mohammadzadeh

Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O.Box 1159,

Mashhad 91775, IRAN.

Email: somohammadzad@yahoo.com

H.R.E. Vishki

Department of Pure Mathematics and Center of Excellence in Analysis on Alge-

braic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mash-

had 91775, IRAN.

Email: vishki@um.ac.ir

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