Regularity of bounded tri-linear maps and the fourth adjiont of a tri-derivation

Abstract

In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if $f:X\times Y\times Z\longrightarrow W $ is a bounded tri-linear mapping and $h:W\longrightarrow S$ is a bounded linear mapping, then $f$ is regular if and only if $hof$ is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint $D^{****}$ of a tri-derivation $D$ is again tri-derivation.

Full text

Global Analysis and Discrete Mathematics
Volume 5, Issue 1, pp. 51–65
ISSN: 2476-5341
Regularity of Bounded Tri-Linear Maps and the Fourth
Adjoint of a Tri-Derivation
Abotaleb Sheikhali∗
·Ali Ebadian ·
Kazem Haghnejad Azar
Received: 24 February 2020 / Accepted: 3 June 2020
Abstract In this Article, we give a simple criterion for the regularity of a
tri-linear mapping. We provide if f:X×Y×Z−→ Wis a bounded tri-linear
mapping and h:W−→ Sis a bounded linear mapping, then fis regular if
and only if hof is regular. We also shall give some necessary and sufficient
conditions such that the fourth adjoint D∗∗∗∗ of a tri-derivation Dis again
tri-derivation.
Keywords Fourth adjoint ·Regular ·Tri-derivation ·Tri-linear
Mathematics Subject Classification (2010) 46H25 ·46H20 ·47B47 ·
16W25
1 Introduction and preliminaries
Richard Arens showed in [3] that a bounded bilinear map m:X×Y−→ Z
on normed spaces, has two natural different extensions m∗∗∗,mr∗∗∗rfrom
X∗∗×Y∗∗ into Z∗∗ . When these extensions are equal, mis called Arens regular.
A Banach algebra Ais said to be Arens regular, if its product π(a, b) = ab
∗Corresponding author
A. Sheikhali
Department of Mathematics, Payame Noor University (PNU), Tehran, Iran.
E-mail: Abotaleb.sheikhali.20@gmail.com
A. Ebadian
Faculty of Science, Department of Mathematics, Urmia University, Urmia, Iran.
E-mail: Ebadian.ali@gmail.com
K. Haghnejad Azar
Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil,
Iran.
E-mail: Haghnejad@aut.ac.ir
c
2020 Damghan University. All rights reserved. http://gadm.du.ac.ir/

52 A. Sheikhali et al.
considered as a bilinear mapping π:A×A−→ Ais Arens regular. The first
and second Arens products of A∗∗ by symbols and ♢respectively defined
by
a∗∗b∗∗ =π∗∗∗ (a∗∗ , b∗∗ ), a∗∗ ♢b∗∗ =πr∗∗∗r(a∗∗ , b∗∗ ).
Some characterizations for the Arens regularity of bounded bilinear map m
and Banach algebra Aare proved in [1], [2], [3], [4], [5], [9], [11], [14] and [15].
Suppose X, Y , Z, W and Sare normed spaces and f:X×Y×Z−→ Wis a
bounded tri-linear mapping. In this paper we first define regularity of fmap
and showing that fis regular if and only if f∗∗∗r∗(X∗∗, W ∗, Z )⊆Y∗and
f∗∗∗∗∗(W∗, X ∗∗, Y ∗∗ )⊆Z∗. Also we show that for a bounded tri-linear map
f:X×Y×Z−→ Wand a bounded linear operator h:W−→ S,fis regular
if and only if hof is regular.
The natural extensions of fare as follows:
1. f∗:W∗×X×Y−→ Z∗, given by ⟨f∗(w∗, x, y), z⟩=⟨w∗, f (x, y, z)⟩where
x∈X, y ∈Y, z ∈Z, w∗∈W∗(f∗is said the adjoint of fand is a
bounded tri-linear map).
2. f∗∗ = (f∗)∗:Z∗∗ ×W∗×X−→ Y∗, given by ⟨f∗∗(z∗∗, w∗, x), y⟩=
⟨z∗∗, f ∗(w∗, x, y) where x∈X, y ∈Y, z∗∗ ∈Z∗∗, w∗∈W∗.
3. f∗∗∗ = (f∗∗)∗:Y∗∗ ×Z∗∗ ×W∗−→ X∗, given by ⟨f∗∗∗(y∗∗ , z ∗∗, w∗), x⟩=
⟨y∗∗, f ∗∗(z∗∗ , w∗, x)⟩where x∈X, y∗∗ ∈Y∗∗, z ∗∗ ∈Z∗∗, w∗∈W∗.
4. f∗∗∗∗ = (f∗∗∗)∗:X∗∗ ×Y∗∗ ×Z∗∗ −→ W∗∗ , given by ⟨f∗∗∗∗(x∗∗ , y ∗∗, z∗∗ )
, w∗⟩=⟨x∗∗ , f ∗∗∗(y∗∗, z ∗∗, w∗)⟩where 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 said to be regular. For bounded tri-linear maps, we
have naturally six different Aron-Berner extensions to the bidual spaces based
on six different elements in S3 and compeletly regularity should be defined
accordingly to the equality of all these six Aron-Berner extensions. See [12].
Suppose Ais a Banach algebra and π1:A×X−→ Xis a bounded bi-
linear map. The pair (π1, X) is said to be a left Banach A−module when
π1(π1(a, b), x) = π1(a, π1(b, x)), for each a, b ∈Aand x∈X. A right Banach
A−module may is defined similarly. Let π2:X×A−→ Xbe a bounded
bilinear map. The pair (X, π2) is said to be a right Banach A−module if
π2(x, π2(a, b)) = π2(π2(x, a), b). A triple (π1, X, π2) is said to be a Banach
A−module if (X, π1) and (X, π2) are left and right Banach A−modules, re-
spectively, and π1(a, π2(x, b)) = π2(π1(a, x), b). Let (π1, X, π2) be a Banach
A−module. Then (πr∗r
2, X∗, π∗
1) is the dual Banach A−module of (π1, X, π2).
A bounded linear mapping D1:A−→ X∗is said to be a derivation if for
each a, b ∈A
D1(π(a, b)) = π∗
1(D1(a), b) + πr∗r
2(a, D1(b)).

Regularity and the Fourth Adjoint 53
A bounded bilinear map D2:A×A−→ X(or X∗) is called a bi-derivation, if
for each a, b, c and d∈A
D2(π(a, b), c) = π1(a, D2(b, c)) + π2(D2(a, c), b),
D2(a, π(b, c)) = π1(b, D2(a, c)) + π2(D2(a, b), c).
Let D1:A−→ A∗be a derivation. Dales, Rodriguez and Velasco, in [7] showed
that D∗∗
1: (A∗∗,)−→ A∗∗∗ is a derivation if and only if πr∗∗∗∗(D∗∗
1(A∗∗), A∗∗ )
⊆A∗. In [13], S. Mohamadzadeh and H. Vishki extends this and showed
that second adjont D∗∗
1: (A∗∗,)−→ A∗∗∗ is a derivation if and only if
π∗∗∗∗
2(D∗∗
1(A∗∗), X ∗∗)⊆A∗and which D∗∗
1: (A∗∗,♢)−→ A∗∗∗ is a derivation
if and only if πr∗∗∗∗
1(D∗∗
1(A∗∗), X ∗∗)⊆A∗.
A. Erfanian Attar et al, provide condition such that the third adjoint D∗∗∗
2
of a bi-derivation D2:A×A−→ X(or X∗) is again a bi-derivation, see [8]. For
a Banach A−module (π1, X, π2), the fourth adjoint D∗∗∗∗ of a tri-derivation
D:A×A×A−→ X∗is trivially a tri-linear extension of D. A problem
which is of interest is under what conditions we need that D∗∗∗∗ is again a
tri-derivation. In section 4 we will extend above mentioned result. A bounded
trilinear mapping f:X×Y×Z−→ Wis said to factor if it is surjective, that
is f(X×Y×Z) = W.
Throughout the article, we usually identify a normed space with its canon-
ical image in its second dual.
2 Regularity of bounded tri-linear maps
Theorem 1 Let f:X×Y×Z−→ Wbe a bounded tri-linear map. Then f
is regular if and only if
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.
Proof. For every w∗∈W∗we have
⟨f∗∗∗∗(x∗∗ , y∗∗, z∗∗), w∗⟩=⟨x∗∗ , f ∗∗∗(y∗∗ , z∗∗, w∗)⟩
= lim
α⟨f∗∗∗(y∗∗ , z ∗∗, w∗), xα⟩= lim
α⟨y∗∗, f ∗∗(z∗∗ , w∗, xα)⟩
= lim
αlim
β⟨f∗∗(z∗∗ , w∗, xα), yβ⟩= lim
αlim
β⟨z∗∗, f ∗(w∗, xα, yβ)⟩
= lim
αlim
βlim
γ⟨f∗(w∗, xα, yβ), zγ⟩= lim
αlim
βlim
γ⟨f(xα, yβ, zγ), w∗⟩.
Therefore f∗∗∗∗(x∗∗ , y∗∗, z∗∗ ) = w∗−lim
αw∗−lim
βw∗−lim
γf(xα, yβ, zγ). In the
other hands fr∗∗∗∗r(x∗∗, y∗∗, z∗∗) = w∗−lim
γw∗−lim
βw∗−lim
αf(xα, yβ, zγ),
and proof follows.

54 A. Sheikhali et al.
In the following theorem, we provide a criterion concerning to the regularity
of a bounded tri-linear map.
Theorem 2 For a bounded tri-linear map f:X×Y×Z−→ Wthe following
statements are equivalent:
1. fis regular.
2. f∗∗∗∗∗(W∗∗∗ , X ∗∗, Y ∗∗ ) = fr∗∗∗∗∗∗∗r(W∗∗∗ , X ∗∗, Y ∗∗).
3. f∗∗∗r∗(X∗∗, W ∗, Z)⊆Y∗and f∗∗∗∗∗ (W∗, X ∗∗, Y ∗∗ )⊆Z∗.
Proof. (1) ⇒(2), if fis regular, then f∗∗∗∗ =fr∗∗∗∗r. For every x∗∗ ∈
X∗∗, y∗∗ ∈Y∗∗, z∗∗ ∈Z∗∗ and w∗∗∗ ∈W∗∗∗ we have
⟨f∗∗∗∗∗(w∗∗∗ , x∗∗ , y∗∗), z∗∗⟩=⟨w∗∗∗, f ∗∗∗∗(x∗∗ , y ∗∗, z∗∗ )⟩
=⟨w∗∗∗, f r∗∗∗∗r(x∗∗, y∗∗, z∗∗ )⟩=⟨fr∗∗∗∗∗∗∗r(w∗∗∗ , x∗∗ , y ∗∗), z∗∗ ⟩.
as claimed.
(2) ⇒(1), let f∗∗∗∗∗ =fr∗∗∗∗∗∗∗r, then for every w∗∈W∗,
⟨fr∗∗∗∗r(x∗∗, y∗∗, z∗∗), w∗⟩=⟨fr∗∗∗∗∗∗∗r(w∗, x∗∗ , y∗∗), z∗∗⟩
=⟨f∗∗∗∗∗(w∗, x∗∗ , y∗∗), z∗∗⟩=⟨f∗∗∗∗(x∗∗ , y ∗∗, z∗∗ ), w∗⟩.
It follows that fis regular.
(1) ⇒(3), assume that fis regular and x∗∗ ∈X∗∗, y ∗∗ ∈Y∗∗, z ∈Z, w∗∈
W∗. Then we have
⟨f∗∗∗r∗(x∗∗, w∗, z), y∗∗ ⟩=⟨f∗∗∗∗ (x∗∗ , y∗∗, z), w∗⟩
=⟨fr∗∗∗∗r(x∗∗, y∗∗, z), w∗⟩=⟨fr∗∗(x∗∗ , w∗, z), y∗∗ ⟩.
Therefore f∗∗∗r∗(x∗∗, w∗, z) = fr∗∗(x∗∗ , w ∗, z)∈Y∗. So f∗∗∗r∗(X∗∗, W ∗, Z )⊆
Y∗. A similar argument shows that f∗∗∗∗∗(w∗, x∗∗, y∗∗) = fr∗∗∗r(w∗, x∗∗ , y∗∗ )∈
Z∗. Thus f∗∗∗∗∗(W∗, X ∗∗, Y ∗∗ )⊆Z∗, as claimed.
(3) ⇒(1), let {xα},{yβ}and {zγ}are nets in X, Y and Zwhich converge
to x∗∗, y∗∗ and z∗∗ in the w∗−topologies, respectively. For every w∗∈W∗we
have
⟨fr∗∗∗∗r(x∗∗, y∗∗, z∗∗), w∗⟩= lim
γlim
βlim
α⟨f(xα, yβ, zγ), w∗⟩
= lim
γlim
βlim
α⟨f∗∗∗(yβ, zγ, w∗), xα⟩= lim
γlim
β⟨x∗∗, f ∗∗∗(yβ, zγ, w∗)
= lim
γlim
β⟨x∗∗, f ∗∗∗r(w∗, zγ, yβ)⟩= lim
γlim
β⟨f∗∗∗r∗(x∗∗, w∗, zγ), yβ⟩
= lim
γ⟨f∗∗∗r∗(x∗∗, w∗, zγ), y∗∗⟩= lim
γ⟨x∗∗, f ∗∗∗r(w∗, zγ, y∗∗ )⟩
= lim
γ⟨x∗∗, f ∗∗∗(y∗∗ , zγ, w∗)⟩= lim
γ⟨f∗∗∗∗(x∗∗ , y∗∗, zγ), w∗⟩
= lim
γ⟨f∗∗∗∗∗(w∗, x∗∗ , y∗∗), zγ⟩=f∗∗∗∗∗ (w∗, x∗∗ , y∗∗ ), z ∗∗⟩
=⟨f∗∗∗∗(x∗∗ , y∗∗, z∗∗), w∗⟩.
It follows that fis regular and this completes the proof.

Regularity and the Fourth Adjoint 55
Corollary 1 For a bounded tri-linear map f:X×Y×Z−→ Wthe following
statements are equivalent:
1. fis regular.
2. fr∗∗∗∗∗r=f∗∗∗∗∗∗∗.
3. fr∗∗∗r∗(Z∗∗, W ∗, X)⊆Y∗and f∗∗∗∗∗(W∗, Z ∗∗, Y ∗∗ )⊆X∗.
Proof. The mapping fis regular if and only if fris regular. Therefore by
Theorem 2, the desired result is obtained.
Corollary 2 For a bounded tri-linear map f:X×Y×Z−→ W, if from
X, Y or Zat least two reflexive then fis regular.
Proof. Without having to enter the whole argument, let Yand Zare reflexive.
Since Yis reflexive, Y∗=Y∗∗∗. Therefore
f∗∗∗r∗(X∗∗, W ∗, Z∗∗ )⊆Y∗∗∗ =Y∗(2 −1)
In the other hands, since Zis the reflexive space, thus
f∗∗∗∗∗(W∗∗∗ , X ∗∗, Y ∗∗ )⊆Z∗∗∗ =Z∗(2 −2)
Now Using (2-1), (2-2) and Theorem 2, the result holds.
Corollary 3 Let bounded tri-linear map f:X×Y×Z−→ Wbe regular.
Then
1. If f∗∗∗r∗(X∗∗ , W ∗, Z)factors, then Yis reflexive space.
2. If f∗∗∗∗∗ (W∗, X ∗∗, Y ∗∗ )factors, then Zis reflexive space.
3. If f∗∗∗∗r∗(W∗, Z, Y )factors, then Xis reflexive space.
Proof. (1) Let fbe regular. It follows that f∗∗∗r∗(X∗∗, W ∗, Z )⊆Y∗. In the
other hands, f∗∗∗r∗(X∗∗, W ∗, Z) is factor. So for each y∗∗∗ ∈Y∗∗∗ there exist
x∗∗ ∈X∗∗, w∗∈W∗and z∈Zsuch that f∗∗∗r∗(x∗∗, w∗, z) = y∗∗∗ . Therefore
Y∗∗∗ ⊆Y∗.
(2) The proof similar to (1).
(3) Enough show that f∗∗∗∗r∗(W∗, Z, Y )⊆X∗whenever fis regular. For
every x∗∗ ∈X∗∗, y ∈Y, z ∈Zand w∗∈W∗we have
⟨f∗∗∗∗r∗(w∗, z, y), x∗∗⟩=⟨w∗, f ∗∗∗∗(x∗∗, y , z)⟩
=⟨fr∗∗∗∗r(x∗∗, y, z), w∗⟩=⟨fr∗(w∗, z , y), x∗∗⟩.
Therefore f∗∗∗∗r∗(w∗, z, y) = fr∗(w∗, z, y)∈X∗. The rest of proof has similar
argument such as (1).
Corollary 4 If IX,IYand IZare weakly compact identity mapping, then all
of them and all of their adjoints are regular.

56 A. Sheikhali et al.
Example 1 1. Let Gbe a compact group. Let 1 < p, q < ∞and 1
p+1
q= 1 + 1
r.
Then by [10, Sections 2.4 and 2.5], we conclude that L1(G)⋆Lp(G)⊂Lp(G)
and Lp(G)⋆Lq(G)⊂Lr(G) where (g ⋆h)(x) = Gg(y)h(y−1x)dy for x∈G.
Since the Banach spaces Lp(G) and Lq(G) are reflexive, thus by corollary
2 we conclude that the bounded tri-linear mapping
f:L1(G)×Lp(G)×Lq(G)−→ Lr(G)
defined by f(k, g, h) = (k ⋆ g )⋆ h, is regular for every k∈L1(G), g ∈Lp(G)
and h∈Lq(G).
2. Let Gbe a locally compact group. We know from [16] that L1(G) is regular
if and only if it is reflexive or Gis finite. It follows that for every finite
locally compact group G, by corollary 2, the bounded tri-linear mapping
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).
3. C∗−algebras are standard examples of Banach algebras that are Arens
regular, see[6]. We know that a C∗−algebra is reflexive if and only if it
is of finite dimension. Since if Ais a finite dimension C∗-algebra, then by
corollary 2, we conclude that the bounded tri-linear mapping f:A×A×
A−→ Ais regular.
4. Let Gbe a locally compact group and let M(G) be measure algebra of G,
see [10, Section 2.5]. Let the convolution for µ1, µ2∈M(G) defined by
ψd(µ1∗µ2) =   ψ(xy)dµ1(x)dµ2(y),(ψ∈C0(G)).
We have
ψd(µ1∗(µ2∗µ3)) =    ψ(xyz)dµ1(x)dµ2(y)dµ3(z)
=ψd((µ1∗µ2)∗µ3)
for µ1, µ2and µ3∈M(G). Therefore convolution is associative. Now we
define the bounded tri-linear mapping
f:M(G)×M(G)×M(G)−→ M(G)
by f(µ1, µ2, µ3) = ψd(µ1∗µ2∗µ3). If Gis finite, then fis regular.
3 Some results for regularity
Dales, Rodriguez-Palacios and Velasco in [7, Theorem 4.1], for a bonded bilin-
ear map m:X×Y−→ Zhave shown that, mr∗r∗∗∗ =m∗∗∗r∗rif and only if
both mand mr∗are Arens regular. Now in the following we study it in general
case.
Remark 1 In the next theorem, fnis n−th adjoint of ffor each n∈N.

Regularity and the Fourth Adjoint 57
Theorem 3 If fand fr n are reular, then f4r nr =frnr4.
Proof. Since fis regular, so f4r=fr4. Therefore f4rn =fr(n+4). In the other
hands, regularity of frn follows that fr(n+4) =frnr4r. Thus frnr4r=f4rn
and this completes the proof.
Theorem 4 Let f:X×Y×Z−→ Wbe a bounded tri-linear mapping. Then
1. f∗∗∗∗r∗∗r=fr∗∗r∗∗∗∗ if and only if both fand fr∗∗ are regular.
2. f∗∗∗∗r∗∗∗r=fr∗∗∗r∗∗∗∗ if and only if both fand fr∗∗∗ are regular.
Proof. We prove only (1), the other part has the same argument. If both fand
fr∗∗ are regular, then by applying Theorem 3, for n= 2, f∗∗∗∗r∗∗r=fr∗∗r∗∗∗∗.
Conversely, suppose that f∗∗∗∗r∗∗r=fr∗∗r∗∗∗∗ . First we show that fis
regular. Let {zγ}is net in Zwhich converge to z∗∗ ∈Z∗∗ in the w∗−topologies.
Then for every x∗∗ ∈X∗∗, y∗∗ ∈Y∗∗ and w∗∈W∗we have
⟨f∗∗∗∗(x∗∗ , y∗∗, z∗∗), w∗⟩=⟨f∗∗∗∗r(z∗∗ , y∗∗, x∗∗ ), w∗⟩
=⟨f∗∗∗∗r∗∗r(z∗∗, w∗, x∗∗), y∗∗ ⟩=⟨fr∗∗r∗∗∗∗ (z∗∗ , w∗, x∗∗), y∗∗ ⟩
= lim
γ⟨y∗∗, f r∗∗r(zγ, w∗, x∗∗)⟩=⟨fr∗∗∗∗r(x∗∗ , y ∗∗, z∗∗ ), w∗⟩.
Therefore fis regular. Now we show that fr∗∗ is regular. Let {x∗∗
α}be net in
X∗∗ which converge to x∗∗∗∗ ∈X∗∗∗∗ in the w∗−topologies. Then for every
y∗∗ ∈Y∗∗, z ∗∗ ∈Z∗∗ and w∗∗∗ ∈W∗∗∗ we have
⟨fr∗∗r∗∗∗∗r(x∗∗∗∗, w∗∗∗, z∗∗ ), y ∗∗⟩=⟨fr∗∗r∗∗∗∗(z∗∗ , w∗∗∗ , x∗∗∗∗ ), y ∗∗⟩
=⟨f∗∗∗∗r∗∗r(z∗∗, w∗∗∗, x∗∗∗∗), y ∗∗⟩= lim
α⟨w∗∗∗, f ∗∗∗∗(x∗∗
α, y∗∗ , z∗∗)⟩
= lim
α⟨w∗∗∗, f r∗∗∗∗r(x∗∗
α, y∗∗ , z∗∗)⟩= lim
α⟨w∗∗∗, f r∗∗∗∗(z∗∗ , y∗∗, x∗∗
α)⟩
=⟨fr∗∗∗∗∗∗(x∗∗∗∗ , w∗∗∗, z∗∗ ), y ∗∗⟩.
It follows that fr∗∗ is regular and this completes the proof.
Arens has shown [3] that a bounded bilinear map mis regular if and only
if for each z∗∈Z∗, the bilinear form z∗om is regular. In the next theorem we
give an important characterization of regularity bounded tri-linear mappings.
Lemma 1 Suppose X, Y, Z, W and Sare normed spaces and f:X×Y×
Z−→ Wand h:W−→ Sare bounded tri-linear mapping and bounded linear
mapping, respectively. Then we have
1. h∗∗of ∗∗∗∗ = (hof )∗∗∗∗ .
2. h∗∗of r∗∗∗∗r= (hof )r∗∗∗∗r.
Proof. 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. For

58 A. Sheikhali et al.
each s∗∈S∗we have
⟨h∗∗of ∗∗∗∗ (x∗∗ , y∗∗, z∗∗), s∗⟩=⟨h∗∗(f∗∗∗∗ (x∗∗ , y ∗∗, z∗∗ )), s∗⟩
=⟨f∗∗∗∗(x∗∗ , y∗∗, z∗∗), h∗(s∗)⟩= lim
αlim
βlim
γ⟨h∗(s∗), f (xα, yβ, zγ)⟩
= lim
αlim
βlim
γ⟨s∗, h(f(xα, yβ, zγ))⟩= lim
αlim
βlim
γ⟨s∗, hof (xα, yβ, zγ)⟩
=⟨(hof)∗∗∗∗(x∗∗ , y∗∗, z∗∗), s∗⟩.
Hence h∗∗of ∗∗∗∗ (x∗∗ , y∗∗, z∗∗) = (hof )∗∗∗∗ (x∗∗ , y∗∗ , z ∗∗). A similar argument
applies for (2).
Theorem 5 Let f:X×Y×Z−→ Wand h:W−→ Sbe bounded tri-linear
mapping and bounded linear mapping, respectively. Then fis regular if and
only if hof is regular.
Proof. Assume that fis regular. Then for every x∗∗ ∈X∗∗, y∗∗ ∈Y∗∗ , z ∗∗ ∈
Z∗∗ and s∗∈S∗we have
⟨h∗∗(fr∗∗∗∗r(x∗∗ , y∗∗, z∗∗)), s∗⟩=⟨fr∗∗∗∗r(x∗∗, y ∗∗, z∗∗ ), h∗(s∗)⟩
=⟨f∗∗∗∗(x∗∗ , y∗∗, z∗∗), h∗(s∗)⟩=⟨h∗∗(f∗∗∗∗ (x∗∗ , y ∗∗, z∗∗ )), s∗⟩.
Therefore h∗∗of r∗∗∗∗r(x∗∗ , y∗∗, z∗∗) = h∗∗of ∗∗∗∗(x∗∗, y ∗∗, z∗∗ ) and by apply-
ing Lemma 1, we implies that
(hof)r∗∗∗∗r(x∗∗, y∗∗, z∗∗) = (hof )∗∗∗∗ (x∗∗ , y∗∗ , z ∗∗).
It follows that hof is regular.
For the converse, suppose that hof is regular. By contradiction, let fbe
not regular. Thus there exist x∗∗ ∈X∗∗, y∗∗ ∈Y∗∗ and z∗∗ ∈Z∗∗ such that
f∗∗∗∗(x∗∗ , y∗∗, z∗∗)=fr∗∗∗∗r(x∗∗, y ∗∗, z∗∗ ). Therefore we have
(hof)∗∗∗∗(x∗∗ , y∗∗, z∗∗) = w∗−lim
αw∗−lim
βw∗−lim
γ(hof)(xα, yβ, zγ)
= lim
αlim
βlim
γ⟨f(xα, yβ, zγ), h⟩=⟨f∗∗∗∗(x∗∗ , y∗∗, z∗∗), h⟩
=⟨fr∗∗∗∗r(x∗∗, y∗∗, z∗∗), h⟩= lim
γlim
βlim
α⟨f(xα, yβ, zγ), h⟩
=w∗−lim
γw∗−lim
βw∗−lim
α(hof)(xα, yβ, zγ)
= (hof)r∗∗∗∗r(x∗∗, y∗∗, z∗∗).
It follows that (hof)∗∗∗∗(x∗∗ , y∗∗ , z ∗∗)= (hof)r∗∗∗∗r(x∗∗, y∗∗, z∗∗ ).
Another interesting case of regularity is in the following.
Theorem 6 Let X, Y, Z, W and Sbe Banach spaces, f:X×Y×Z−→ W
be a bounded tri-linear mapping and x∈X, y ∈Y, z ∈Z. Then
1. Let g1:S×Y×Z−→ Wbe a bounded tri-linear mapping and let h1:
X−→ Sbe a bounded linear mapping such that f(x, y, z) = g1(h1(x), y, z).
If h1is weakly compact, then f∗∗∗∗r∗(W∗∗∗, Z∗∗ , Y ∗∗ )⊆X∗.

Regularity and the Fourth Adjoint 59
2. Let g2:X×S×Z−→ Wbe a bounded tri-linear mapping and let h2:
Y−→ Sbe a bounded linear mapping such that f(x, y, z) = g2(x, h2(y), z).
If h2is weakly compact, then f∗∗∗r∗(X∗∗, W ∗, Z ∗∗)⊆Y∗.
3. Let g3:X×Y×S−→ Wbe a bounded tri-linear mapping and let h3:
Z−→ Sbe a bounded linear mapping such that f(x, y, z) = g3(x, y, h3(z)).
If h3is weakly compact, then f∗∗∗∗∗(W∗∗∗, X ∗∗, Y ∗∗ )⊆Z∗.
Proof. We prove only (1), the other parts have the same argument. For every
x∈X, y ∈Y, z ∈Zand w∗∈W∗we have
⟨f∗(w∗, x, y), z⟩=⟨w∗, f(x, y, z)⟩=⟨w∗, g1(h1(x), y , z)⟩=⟨g∗
1(w∗, h1(x), y), z⟩.
Therefore f∗(w∗, x, y) = g∗
1(w∗, h1(x), y), and implies that for every z∗∗ ∈Z∗∗,
⟨f∗∗(z∗∗ , w∗, x), y⟩=⟨z∗∗, f ∗(w∗, x, y)⟩
=⟨z∗∗, g∗
1(w∗, h1(x), y)⟩=⟨g∗∗
1(z∗∗, w∗, h1(x)), y⟩.
So f∗∗(z∗∗ , w∗, x) = g∗∗
1(z∗∗, w∗, h1(x)) and implies that for every y∗∗ ∈Y∗∗,
⟨f∗∗∗(y∗∗ , z ∗∗, w∗), x⟩=⟨y∗∗, f ∗∗(z∗∗ , w∗, x)⟩=⟨y∗∗ , g ∗∗
1(z∗∗, w∗, h1(x))⟩
=⟨g∗∗∗
1(y∗∗, z ∗∗, w∗), h1(x)⟩=⟨h∗
1(g∗∗∗
1(y∗∗, z ∗∗, w∗)), x⟩.
Thus f∗∗∗(y∗∗, z ∗∗, w∗) = h∗
1(g∗∗∗
1(y∗∗, z ∗∗, w∗)) and implies that for every
x∗∗ ∈X∗∗,
⟨f∗∗∗∗(x∗∗ , y∗∗, z∗∗ ), w ∗⟩=⟨x∗∗, f ∗∗∗(y∗∗, z∗∗ , w ∗)⟩
=⟨x∗∗, h∗
1(g∗∗∗
1(y∗∗, z ∗∗, w∗))⟩=⟨h∗∗
1(x∗∗),(g∗∗∗
1(y∗∗, z ∗∗, w∗)⟩
=⟨g∗∗∗∗
1(h∗∗
1(x∗∗), y∗∗, z∗∗), w∗⟩.
Therefore for every w∗∗∗ ∈W∗∗∗ we have
⟨f∗∗∗∗r∗(w∗∗∗, z ∗∗, y∗∗), x∗∗ ⟩=⟨w∗∗∗ , f ∗∗∗∗r(z∗∗, y ∗∗, x∗∗)⟩
=⟨w∗∗∗, f ∗∗∗∗(x∗∗ , y∗∗, z∗∗ )⟩=⟨w∗∗∗ , g ∗∗∗∗
1(h∗∗
1(x∗∗), y∗∗, z∗∗)⟩
=⟨w∗∗∗, g∗∗∗∗r
1(z∗∗, y∗∗, h∗∗
1(x∗∗))⟩=⟨g∗∗∗∗r∗
1(w∗∗∗, z ∗∗, y∗∗), h∗∗
1(x∗∗)⟩
=⟨h∗∗∗
1(g∗∗∗∗r∗
1(w∗∗∗, z ∗∗, y∗∗)), x∗∗ ⟩.
Therefore f∗∗∗∗r∗(w∗∗∗, z ∗∗, y∗∗) = h∗∗∗
1(g∗∗∗∗r∗
1(w∗∗∗, z ∗∗, y∗∗)). The weak com-
pactness of h1implies that of h∗
1, from which we have h∗∗∗
1(S∗∗∗)⊆X∗. Thus
h∗∗∗
1(g∗∗∗∗r∗
1(w∗∗∗, z ∗∗, y∗∗)) ∈X∗and this completes the proof.
This theorem, combined with Theorem 2, yields the next result.
Corollary 5 With the assumptions Theorem 6, if h2and h3are weakly com-
pact, then fis regular.

60 A. Sheikhali et al.
Proof. Both h2and h3are weakly compact, so by Theorem 6 we have
f∗∗∗r∗(X∗∗, W ∗, Z∗∗ )⊆Y∗, f ∗∗∗∗∗ (W∗∗∗ , X ∗∗, Y ∗∗ )⊆Z∗.
In particular
f∗∗∗r∗(X∗∗, W ∗, Z)⊆Y∗, f ∗∗∗∗∗ (W∗, X ∗∗, Y ∗∗ )⊆Z∗.
Now by Theorem 2, fis regular.
The converse of previous result is not true in general sense as following
corollary.
Corollary 6 With the assumptions Theorem 6, if fis regular and both g∗∗∗r∗
2
and g∗∗∗∗∗
3are factors, then h2and h3are weakly compact.
Proof. Since f∗∗∗r∗(X∗∗, W ∗, Z ∗∗) = h∗∗∗
2(g∗∗∗r∗
2(X∗∗, W ∗, Z∗∗ )), so h∗∗∗
2(g∗∗∗r∗
2
(X∗∗, W ∗, Z∗∗ )) ⊆Y∗. In the other hands g∗∗∗r∗
2is factors, so implies that
h∗∗∗
2(S∗∗∗)⊆Y∗. Therefore h∗
2is weakly compact and implies that h2is weakly
compact. The other part has the same argument for h3.
4 The fourth adjoint of a tri-derivation
Definition 1 Let (π1, X , π2) be a Banach A−module. A bounded tri-linear
mapping D:A×A×A−→ Xis said to be a tri-derivation when
1. D(π(a, d), b, c) = π2(D(a, b, c), d) + π1(a, D(d, b, c)),
2. D(a, π(b, d), c) = π2(D(a, b, c), d) + π1(b, D(a, d, c)),
3. D(a, b, π(c, d)) = π2(D(a, b, c), d) + π1(c, D(a, b, d)),
for each a, b, c, d ∈A. If (π1, X, π2) is a Banach A−module, then (πr∗r
2, X∗, π∗
1)
is the dual Banach A−module of (π1, X, π2). Therefore a bounded tri-linear
mapping D:A×A×A−→ X∗is a tri-derivation when
1. D(π(a, d), b, c) = π∗
1(D(a, b, c), d) + πr∗r
2(a, D(d, b, c)),
2. D(a, π(b, d), c) = π∗
1(D(a, b, c), d) + πr∗r
2(b, D(a, d, c)),
3. D(a, b, π(c, d)) = π∗
1(D(a, b, c), d) + πr∗r
2(c, D(a, b, d)).
It can also be written, a bounded tri-linear mapping D:A×A×A−→ Ais
said to be a tri-derivation when
1. D(π(a, d), b, c) = π(D(a, b, c), d) + π(a, D(d, b, c)),
2. D(a, π(b, d), c) = π(D(a, b, c), d) + π(b, D(a, d, c)),
3. D(a, b, π(c, d)) = π(D(a, b, c), d) + π(c, D(a, b, d)).
Example 2 Let Abe a Banach algebra, for any a, b ∈Athe symbol [a, b] =
ab −ba stands for multiplicative commutator of aand b. Let Mn×n(C) be the
Banach algebra of all n×nmatrix and A={x y
0 0∈Mn×n(C)|x, y ∈C}.
Then Ais Banach algebra with the norm
∥a∥= (Σi,j |αij |2)1
2,(a= (αij )∈A).

Regularity and the Fourth Adjoint 61
We define D:A×A×A−→ Ato be the bounded tri-linear map given by
D(a, b, c) = [0 1
0 0, abc],(a, b, c ∈A).
Then for a=x1y1
0 0 , b =x2y2
0 0 , c =x3y3
0 0 and d=x4y4
0 0 ∈Awe
have
D(π(a, d), b, c) = D(x1x4x1y4
0 0 ,x2y2
0 0 ,x3y3
0 0 )
= [0 1
0 0,x1x2x3x4x1x2x4y3
0 0 ] = 0−x1x2x3x4
0 0 
=0−x1x2x3
0 0 x4y4
0 0 +x1y1
0 0 0−x2x3x4
0 0 
= (0 0
0 0−0x1x2x3
0 0 )x4y4
0 0 +x1y1
0 0 (0 0
0 0−0x2x3x4
0 0 )
= (0 1
0 0x1x2x3x1x2y3
0 0 −x1x2x3x1x2y3
0 0 0 1
0 0)x4y4
0 0 
+x1y1
0 0 (0 1
0 0x2x3x4x2x4y3
0 0 −x2x3x4x2x4y3
0 0 0 1
0 0)
= [0 1
0 0,x1x2x3x1x2y3
0 0 ]x4y4
0 0 
+x1y1
0 0 [0 1
0 0,x2x3x4x2x4y3
0 0 ]
= [0 1
0 0,x1y1
0 0 x2y2
0 0 x3y3
0 0 ]x4y4
0 0 
+x1y1
0 0 [0 1
0 0,x4y4
0 0 x2y2
0 0 x3y3
0 0 ]
=D(x1y1
0 0 ,x2y2
0 0 ,x3y3
0 0 )x4y4
0 0 
+x1y1
0 0 D(x4y4
0 0 ,x2y2
0 0 ,x3y3
0 0 )
=π(D(a, b, c), d) + π(a, D(d, b, c)).
Similarly, we have D(a, π(b, d), c) = π(D(a, b, c), d)+π(b, D(a, d, c)) and D(a, b,
π(c, d)) = π(D(a, b, c), d) + π(c, D(a, b, d)). Thus Dis tri-derivation.
Now, we provide a necessary and sufficient condition such that the fourth
adjoint D∗∗∗∗ of a tri-derivation D:A×A×A−→ Xis again a tri-derivation.
For the fourth adjoint D∗∗∗∗ of a tri-derivation D:A×A×A−→ X, we are

62 A. Sheikhali et al.
faced with the case eight:
(case1) D∗∗∗∗ : (A∗∗,)×(A∗∗ ,)×(A∗∗ ,)−→ X∗∗ ,
(case2) D∗∗∗∗ : (A∗∗,♢)×(A∗∗ ,)×(A∗∗ ,)−→ X∗∗ ,
(case3) D∗∗∗∗ : (A∗∗,)×(A∗∗ ,♢)×(A∗∗ ,)−→ X∗∗ ,
(case4) D∗∗∗∗ : (A∗∗,)×(A∗∗ ,)×(A∗∗ ,♢)−→ X∗∗ ,
(case5) D∗∗∗∗ : (A∗∗,♢)×(A∗∗ ,♢)×(A∗∗ ,)−→ X∗∗ ,
(case6) D∗∗∗∗ : (A∗∗,♢)×(A∗∗ ,)×(A∗∗ ,♢)−→ X∗∗ ,
(case7) D∗∗∗∗ : (A∗∗,)×(A∗∗ ,♢)×(A∗∗ ,♢)−→ X∗∗ ,
(case8) D∗∗∗∗ : (A∗∗,♢)×(A∗∗ ,♢)×(A∗∗ ,♢)−→ X∗∗ .
In the following, we prove the state of case 1. The remaining state are proved
in the same way.
Theorem 7 Let (π1, X, π2)be a Banach A−module and D:A×A×A−→ X
be a tri-derivation. Then D∗∗∗∗ : (A∗∗,)×(A∗∗,)×(A∗∗ ,)−→ X∗∗ is a
tri-derivation if and only if
1. π∗∗r∗
2(D∗∗∗∗(A, A, A∗∗ ), X ∗)⊆A∗,
2. π∗∗∗∗
2(X∗, D∗∗∗∗ (A, A∗∗ , A∗∗ )) ⊆A∗,
3. D∗∗∗∗r∗(π∗∗∗∗
1(X∗, A∗∗), A∗∗ , A∗∗ )⊆A∗,
4. D∗∗∗∗∗∗(A∗∗ , π∗∗∗∗
1(X∗, A∗∗), A)⊆A∗,
5. D∗∗∗∗∗∗∗(A∗∗ , A∗∗ , π∗∗∗∗
1(X∗, A∗∗)) ⊆A∗.
Proof. Let D:A×A×A−→ Xbe a tri-derivation and (1),(2),(3),(4),(5)
holds. If {aα},{bβ},{cγ}and {dτ}are bounded nets in A, converging in
w∗−topology to a∗∗ , b∗∗ , c∗∗ and d∗∗ ∈A∗∗ respectively, in this case using (2),
we conclude that w∗−lim
αw∗−lim
τw∗−lim
βw∗−lim
γπ2(D(aα, bβ, cγ), dτ) =
π∗∗∗
2(D∗∗∗∗(a∗∗ , b∗∗ , c∗∗ ), d∗∗ ). Thus for every x∗∈X∗we have
⟨D∗∗∗∗(π∗∗∗ (a∗∗ , d∗∗ ), b∗∗ , c∗∗ ), x∗⟩
= lim
αlim
τlim
βlim
γ⟨x∗, D(π(aα, dτ), bβ, cγ)⟩
= lim
αlim
τlim
βlim
γ⟨x∗, π2(D(aα, bβ, cγ), dτ) + π1(aα, D(dτ, bβ, cγ))⟩
= lim
αlim
τlim
βlim
γ⟨x∗, π2(D(aα, bβ, cγ), dτ)⟩
+ lim
αlim
τlim
βlim
γ⟨x∗, π1(aα, D(dτ, bβ, cγ))⟩
=⟨x∗, π∗∗∗
2(D∗∗∗∗(a∗∗ , b∗∗ , c∗∗ ), d∗∗ )⟩+⟨x∗, π∗∗∗
1(a∗∗, D∗∗∗∗(d∗∗, b∗∗, c∗∗))⟩
=⟨π∗∗∗
2(D∗∗∗∗(a∗∗ , b∗∗ , c∗∗ ), d∗∗ ) + π∗∗∗
1(a∗∗, D∗∗∗∗(d∗∗, b∗∗ , c∗∗ )), x∗⟩.
Therefore
D∗∗∗∗(π∗∗∗ (a∗∗ , d∗∗ ), b∗∗ , c∗∗ )
=π∗∗∗
2(D∗∗∗∗(a∗∗ , b∗∗ , c∗∗ ), d∗∗ ) + π∗∗∗
1(a∗∗, D∗∗∗∗(d∗∗, b∗∗, c∗∗)).

Regularity and the Fourth Adjoint 63
Applying (1) and (3) respectively, we can deduce that w∗−lim
αw∗−lim
βw∗−
lim
τw∗−lim
γπ2(D(aα, bβ, cγ), dτ) = π∗∗∗
2(D∗∗∗∗(a∗∗ , b∗∗ , c∗∗ ), d∗∗ ) and w∗−
lim
αw∗−lim
βw∗−lim
τw∗−lim
γπ1(bβ, D(aα, dτ, cγ)) = π∗∗∗
1(b∗∗, D∗∗∗∗(a∗∗, d∗∗ ,
c∗∗)).So in similar way, we can deduce that
D∗∗∗∗(a∗∗ , π∗∗∗(b∗∗ , d∗∗ ), c∗∗ )
=π∗∗∗
2(D∗∗∗∗(a∗∗ , b∗∗ , c∗∗ ), d∗∗ ) + π∗∗∗
1(b∗∗, D∗∗∗∗(a∗∗, d∗∗ , c∗∗ )).
Applying (4) and (5), we can write w∗−lim
αw∗−lim
βw∗−lim
γw∗−lim
τπ1(cγ, D(aα
, bβ, dτ)) = π∗∗∗
1(c∗∗, D∗∗∗∗(a∗∗, b∗∗ , d∗∗ )).Thus
D∗∗∗∗(a∗∗ , b∗∗ , π∗∗∗(c∗∗ , d∗∗ ))
=π∗∗∗
2(D∗∗∗∗(a∗∗ , b∗∗ , c∗∗ ), d∗∗ ) + π∗∗∗
1(c∗∗, D∗∗∗∗(a∗∗, b∗∗ , d∗∗ )).
By comparing equations (4.1), (4.2) and (4.3) follows that D∗∗∗∗ : (A∗∗,)×
(A∗∗,)×(A∗∗ ,)−→ X∗∗ is a tri-derivation.
For the converse, let Dand D∗∗∗∗ : (A∗∗,)×(A∗∗,)×(A∗∗ ,)−→ X∗∗
be tri-derivation. We have to show that (1), (2), (3), (4) and (5) hold. We shall
only prove (2) the others parts have similar argument. Fourth adjoint D∗∗∗∗
is tri-derivation, thus we have
D∗∗∗∗(π∗∗∗ (a, d∗∗ ), b∗∗ , c∗∗ ) = π∗∗∗
2(D∗∗∗∗(a, b∗∗ , c∗∗ ), d∗∗ )
+π∗∗∗
1(a, D∗∗∗∗ (d∗∗ , b∗∗ , c∗∗ )).
In the other hands, the mapping Dis tri-derivation, which follows that
D∗∗∗∗(π∗∗∗ (a, d∗∗ ), b∗∗ , c∗∗ ) = w∗−lim
τw∗−lim
βw∗−lim
γπ2(D(a, bβ, cγ), dτ)
+π∗∗∗
1(a, D∗∗∗∗ (d∗∗ , b∗∗ , c∗∗ )).
Therefore follows that
π∗∗∗
2(D∗∗∗∗(a, b∗∗ , c∗∗ ), d∗∗ )
=w∗−lim
τw∗−lim
βw∗−lim
γπ2(D(a, bβ, cγ), dτ).
So, for every d∗∗ ∈A∗∗ we have
⟨π∗∗∗∗
2(x∗, D∗∗∗∗ (a, b∗∗ , c∗∗ )), d∗∗ ⟩=⟨x∗, π∗∗∗
2(D∗∗∗∗(a, b∗∗ , c∗∗ ), d∗∗ )⟩
= lim
τlim
βlim
γ⟨x∗, π2(D(a, bβ, cγ), dτ)⟩= lim
τlim
βlim
γ⟨x∗, πr
2(dτ, D(a, bβ, cγ))⟩
= lim
τlim
βlim
γ⟨πr∗
2(x∗, dτ), D(a, bβ, cγ)⟩= lim
τlim
βlim
γ⟨D∗(πr∗
2(x∗, dτ), a, bβ), cγ⟩
= lim
τlim
β⟨c∗∗, D∗(πr∗
2(x∗, dτ), a, bβ)⟩= lim
τlim
β⟨D∗∗(c∗∗ , πr∗
2(x∗, dτ), a), bβ⟩
= lim
τ⟨b∗∗, D∗∗(c∗∗, π r∗
2(x∗, dτ), a)⟩= lim
τ⟨D∗∗∗(b∗∗ , c∗∗ , πr∗
2(x∗, dτ)), a⟩
= lim
τ⟨D∗∗∗∗(a, b∗∗ , c∗∗ ), πr∗
2(x∗, dτ)⟩= lim
τ⟨D∗∗∗∗(a, b∗∗ , c∗∗ ), πr∗r
2(dτ, x∗)⟩
= lim
τ⟨πr∗r∗
2(D∗∗∗∗(a, b∗∗ , c∗∗ ), dτ), x∗⟩= lim
τ⟨πr∗r∗∗
2(x∗, D∗∗∗∗ (a, b∗∗ , c∗∗ )), dτ⟩
=⟨πr∗r∗∗
2(x∗, D∗∗∗∗ (a, b∗∗ , c∗∗ )), d∗∗ ⟩.

64 A. Sheikhali et al.
As πr∗r∗∗
2(x∗, D∗∗∗∗ (a, b∗∗ , c∗∗ )) always lies in A∗, we have reached (2).
For case 2, fourth adjoint D∗∗∗∗ of tri-derivation D:A×A×A−→ Xis
a tri-derivation if and only if
1. π∗∗r∗
2(D∗∗∗∗(A∗∗ , A∗∗ , A∗∗ ), X ∗)⊆A∗,
2. D∗∗∗∗r∗(π∗∗∗∗
1(X∗, A∗∗), A∗∗ , A∗∗ )⊆A∗,
3. D∗∗∗∗∗∗(A∗∗ , π∗∗∗∗
1(X∗, A∗∗), A)⊆A∗,
4. D∗∗∗∗∗∗∗(A∗∗ , A∗∗ , π∗∗∗∗
1(X∗, A∗∗)) ⊆A∗.
For case 3, fourth adjoint D∗∗∗∗ of tri-derivation D:A×A×A−→ Xis a
tri-derivation if and only if
1. π∗∗∗∗
2(X∗, D∗∗∗∗ (A, A∗∗ , A∗∗ )) ⊆A∗,
2. D∗∗∗∗∗∗(A∗∗ , π∗∗∗∗
1(X∗, A∗∗), A)⊆A∗,
3. D∗∗∗∗∗∗∗(A∗∗ , A∗∗ , π∗∗∗∗
1(X∗, A∗∗)) ⊆A∗.
For case 4, fourth adjoint D∗∗∗∗ of tri-derivation D:A×A×A−→ Xis a
tri-derivation if and only if
1. π∗∗r∗
2(D∗∗∗∗(A, A, A∗∗ ), X ∗)⊆A∗,
2. π∗∗∗∗
2(X∗, D∗∗∗∗ (A, A∗∗ , A∗∗ )) ⊆A∗,
3. D∗∗∗∗r∗(π∗∗∗∗
1(X∗, A∗∗), A∗∗ , A∗∗ )⊆A∗,
4. D∗∗∗∗∗(π∗∗∗∗
1(X∗, A∗∗), A, A)⊆A∗,
5. D∗∗∗∗∗∗(A∗∗ , π∗∗∗∗
1(X∗, A∗∗), A)⊆A∗,
6. D∗∗∗∗∗∗∗(A∗∗ , A∗∗ , π∗∗∗∗
1(X∗, A∗∗)) ⊆A∗.
For case 5, fourth adjoint D∗∗∗∗ of tri-derivation D:A×A×A−→ Xis a
tri-derivation if and only if
1. π∗∗r∗
2(D∗∗∗∗(A∗∗ , A∗∗ , A∗∗ ), X ∗)⊆A∗,
2. π∗∗∗∗
2(X∗, D∗∗∗∗ (A, A∗∗ , A∗∗ )) ⊆A∗,
3. D∗∗∗∗r∗(π∗∗∗∗
1(X∗, A∗∗), A∗∗ , A∗∗ )⊆A∗,
4. D∗∗∗∗∗∗(A∗∗ , π∗∗∗∗
1(X∗, A∗∗), A)⊆A∗,
5. D∗∗∗∗∗∗∗(A∗∗ , A∗∗ , π∗∗∗∗
1(X∗, A∗∗)) ⊆A∗.
For case 6, fourth adjoint D∗∗∗∗ of tri-derivation D:A×A×A−→ Xis a
tri-derivation if and only if
1. π∗∗r∗
2(D∗∗∗∗(A∗∗ , A∗∗ , A∗∗ ), X ∗)⊆A∗,
2. D∗∗∗∗r∗(π∗∗∗∗
1(X∗, A∗∗), A∗∗ , A∗∗ )⊆A∗,
3. D∗∗∗∗∗(π∗∗∗∗
1(X∗, A∗∗), A, A)⊆A∗,
4. D∗∗∗∗∗∗(A∗∗ , π∗∗∗∗
1(X∗, A∗∗), A)⊆A∗,
5. D∗∗∗∗∗∗∗(A∗∗ , A∗∗ , π∗∗∗∗
1(X∗, A∗∗)) ⊆A∗.
For case 7, fourth adjoint D∗∗∗∗ of tri-derivation D:A×A×A−→ Xis a
tri-derivation if and only if
1. π∗∗∗∗
2(X∗, D∗∗∗∗ (A, A∗∗ , A∗∗ )) ⊆A∗,
2. π∗∗r∗
2(D∗∗∗∗(A, A, A∗∗ ), X ∗)⊆A∗,
3. D∗∗∗∗∗(π∗∗∗∗
1(X∗, A∗∗), A, A)⊆A∗,
4. D∗∗∗∗∗∗(A∗∗ , π∗∗∗∗
1(X∗, A∗∗), A)⊆A∗,

Regularity and the Fourth Adjoint 65
5. D∗∗∗∗∗∗∗(A∗∗ , A∗∗ , π∗∗∗∗
1(X∗, A∗∗)) ⊆A∗.
For case 8, fourth adjoint D∗∗∗∗ of tri-derivation D:A×A×A−→ Xis a
tri-derivation if and only if
1. π∗∗∗∗
2(X∗, D∗∗∗∗ (A, A∗∗ , A∗∗ )) ⊆A∗,
2. π∗∗r∗
2(D∗∗∗∗(A∗∗ , A∗∗ , A∗∗ ), X ∗)⊆A∗,
3. D∗∗∗∗r∗(π∗∗∗∗
1(X∗, A∗∗), A∗∗ , A∗∗ )⊆A∗,
4. D∗∗∗∗∗(π∗∗∗∗
1(X∗, A∗∗), A, A)⊆A∗,
5. D∗∗∗∗∗∗(A∗∗ , π∗∗∗∗
1(X∗, A∗∗), A)⊆A∗,
6. D∗∗∗∗∗∗∗(A∗∗ , A∗∗ , π∗∗∗∗
1(X∗, A∗∗)) ⊆A∗.
Remark 2 For adjoint Dr∗∗∗∗rof tri-derivation D:A×A×A−→ Xwe have
the same argument.
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