Topological centers of certain banach module actions

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.

Full text

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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