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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 0x1x2x3x1x2y3

0 0 −x1x2x3x1x2y3

0 0 0 1

0 0)x4y4

0 0

+x1y1

0 0 (0 1

0 0x2x3x4x2x4y3

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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