THE RANGE INCLUSION RESULTS FOR ALGEBRAIC NIL DERIVATIONS ON COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAS

The Pure and Applied Mathematics.
2013.
Nov,
20(4):
243-249

- Received : April 02, 2013
- Accepted : August 29, 2013
- Published : November 30, 2013

Download

PDF

e-PUB

PubReader

PPT

Export by style

Article

Metrics

Cited by

TagCloud

Let
A
be an algebra and
D
a derivation of
A
. Then
D
is called algebraic
nil
if for any
x
∈
A
there is a positive integer
n
=
n
(
x
) such that
D
^{n(x)}
(
P
(
x
)) = 0; for all
P
∈ ℂ (by convention
D
^{n(x)}
(α) = 0; for all α ∈ ℂ). In this paper, we show that any algebraic nil derivation (possibly unbounded) on a commutative complex algebra
A
maps into
N
(
A
) ; where
N
(
A
) denotes the set of all nilpotent elements of
A
. As an application, we deduce that any nilpotent derivation on a commutative complex algebra
A
maps into
N
(
A
).
Finally, we deduce two noncommutative versions of algebraic nil derivations inclusion range.
A
be a complex algebra. A linear map
D
from
A
to
A
is called a
derivation
if
D
(
xy
) =
D
(
x
)
y
+
xD
(
y
) holds for all
x, y
∈
A
. A derivation of
D
on
A
is called
nil
if for any
x
∈
A
there is a positive integer
n
=
n
(
x
) such that
D
^{n(x)}
= 0 (see
[6]
). Here, if the number
n
can be taken independently of
x
,
D
is called
nilpotent
. A derivation
D
of
A
is called
algebraic nil
if for any
x
∈
A
there is a positive integer
n
=
n
(
x
) such that
D
^{n(x)}
(
P
(
x
)) = 0, for all
P
∈ ℂ[
X
] (by convention
D
^{n(x)}
(α) = 0, for all α ∈ ℂ).
We will denote by
Q
(
A
) the set of all quasinilpotent elements in a Banach algebra
A
. In 1955, Singer and Wermer
[12]
proved that a continuous derivation on a commutative Banach algebra maps into the (Jacobson) radical, and they conjectured that this result holds even if the derivation is discontinuous. In 1988, Thomas
[13]
solved the long standing problem by showing that the conjecture is true.
In 1991, Kim and Jun
[10]
proved that if
D
is a derivation on a noncommutative Banach algebra
A
satisfying the condition [[
A, A
],
A
] = 0 then
D
(
A
) ⊂
Q
(
A
). In 1992, Vukman
[15]
proved that if
D
is a linear Jordan derivation on a noncommutative Banach algebra
A
such that the map
F
(
x
) = [[
Dx
,
x
],
x
] is commuting on
A
then
D
= 0. In 1992, Mathieu and Runde
[11]
proved that if
D
is a centralizing derivation on a Banach algebra A; then
D
(
A
) ⊂
rad
(
A
): In 1994, Bresar
[5]
showed that if
D
is a bounded derivation of a Banach algebra such that [
D
(
x
),
x
] ∈
Q
(
A
) for every
x
∈
A
; then
D
(
A
) ⊂
rad
(
A
) where rad(A) denotes the Jacobson radical of
A
.
To the best of our knowledge, there is no inclusion versions for derivations on arbitrary algebra, except the paper of Colville, Davis, and Keimel
[9]
in which they began studying positive derivations on f-rings (i.e.,
D
(
a
) ≥ 0, for all
a
≥ 0) and the papers of Boulabiar
[4]
, A. Toumi et al
[14]
and Ben Amor
[2]
, in which the authors studied exclusively positive and order bounded derivations on Archimedean almost
f
-algebras.
It is well-known that the notion of nil derivations is a generalization of the notion of nilpotent derivations. The latter, because of its close relation with automorphisms and the existence of a Jordan decomposition into semisimple and nilpotent parts for a large family of derivations (it is a generalization of that of algebraic derivations), has received considerable attention (see
[6
,
7
,
8]
). In this paper we shall be concerned principally with the range of algebraic nil derivations
D
on commutative algebra, on noncommutative archimedean d-algebra and on noncommutative algebra
A
satisfying the following condition; [[
A
,
A
],
A
] = 0.
Proposition 2.1.
Let A be a commutative complex algebra, n be a positive integer, D be a derivation on A and x ∈ A such that
D^{n}
(
x
),
D^{n}
(
x
^{2}
),
D^{n}
(
x^{n}
) ∈
N
(
A
),
where N
(
A
)
denotes the set of all nilpotent elements of A
.
Then
D
(
x
) ∈
N
(
A
).
Proof
. Let
x
∈ A with
D^{n}
(
x
),
D^{n}
(
x
^{2}
),
D^{n}
(
x^{n}
) ∈
N
(
A
). It follows that
Since
D^{n}
(
x
) ∈
N
(
A
) ; we have
Moreover, letting
such that
n_{1}
+
n_{2}
=
n
, this one has
By using the Leibnitz rule for
D^{k}
(
x^{n1}
) and
D^{k}
(
x^{n2}
) in Equality (3) and by using the relation (2), we deduce that
(
D
(
x
))
^{n}
∈
N
(
A
)
and then
D
(
x
) ∈
N
(
A
).□
From the above result, we deduce the following:
Proposition 2.2.
Let A be a commutative complex algebra, n be a positive integer, D be a derivation on A and x ∈ A such that
D^{n}
(
x
) =
D^{n}
(
x
^{2}
) =
D^{n}
(
x^{n}
) = 0.
Then (
D
(
x
))
^{n}
= 0.
The below theorem is an immediate consequence of Proposition 2.2.
Theorem 2.3.
Let A be a commutative complex algebra and let D be an algebraic nil derivation on A. Then D
(
A
)
is contained in N
(
A
).
Since any nilpotent derivation is algebraic nil, we have the following:
Corollary 2.4.
Let A be a commutative complex algebra and let D be a nilpotent derivation on A. Then D
(
A
)
is contained in N
(
A
).
In what follows, we shall deal with the range of algebraic nil derivation on non-commutative algebras. In order to hit this mark, we will need the following lemma.
Lemma 2.5
([10, Lemma 3.1]).
Let A be a complex algebra satisfying the condition
[[
A, A
],
A
] = 0.
Let A
⊕
A be the vector space direct sum. Define a multiplication in A
⊕
A by setting
(
a_{1}, b_{1}
) (
a_{2}, b_{2}
) = (
a_{1}a_{2}
+
a_{2}a_{1}, b_{1}b_{2}
+
b_{2}b_{1}
)
for all
(
a_{1}, b_{1}
), (
a_{2}, b_{2}
)
in A
⊕
A
.
Then A
⊕
A is a commutative algebra
.
Using the previous lemma, we deduce the following result. Its proof is inspired from [10, Theorem 3.2].
Theorem 2.6.
Let A be a complex algebra satisfying the condition
[[
A, A
],
A
] = 0
and let D be an algebraic nil derivation on A. Then D
(A)
is contained in N
(
A
).
Proof
. By the previous lemma,
A
⊕
A
is a commutative algebra. Now we define the linear mapping
:
A
⊕
A
→
A
⊕
A
by
(
a, b
) = (
D
(
a
),
D
(
b
)).
Since
D
is an algebraic nil derivation on
A
, it is not hard to prove that
is an algebraic nil derivation on
A
⊕
A
. By Theorem 1, we have
(
A
⊕
A
) ⊂
N
(
A
⊕
A
) =
N
(
A
) ⊕
N
(
A
). Therefore
D
(
A
) ⊂
N
(
A
).□
Corollary 2.7.
Let A be a complex algebra satisfying the condition
[[
A
,
A
],
A
] = 0
and let D be a nilpotent derivation on A. Then D
(
A
)
is contained in N
(
A
).
Next, we will be interested with the range of derivations on noncommutative algebra
A
satisfying the following condition;
(χ)
a
[
A, A
]
b
= 0
for all
a,b
∈
A
.
Theorem 2.8.
Let A be a complex algebra satisfying the condition (χ) and let D be an algebraic nil derivation on A. Then D(A) is contained in N
(
A
).
Proof
. Let
x
∈
A
. Then there exists
such that
D^{n}
(
x
) =
D^{n}
(
x
^{2}
) =
D^{n}
(
x^{n}
) = 0: Let
a, b
∈
A
. It follows that
Moreover, let
n
_{1}
;
n
_{2}
∈ℕ N such that
n
_{1}
+
n
_{2}
=
n
(
x
), then
By using the Leibnitz rule for
aD^{k}
(
x^{n1}
)
b
and
aD^{k}
(
x^{n2}
)
b
in Equality (5), by using Equality (4) and taking into account that
D^{n}
(
x
) = 0, we deduce that
a
(
D
(
x
))
^{n} b
= 0
for all
a, b
∈
A
. Consequently (
D
(
x
))
^{n+2}
= 0. Therefore
D
(
A
) ⊂
N
(
A
).□
Corollary 2.9.
Let A be a complex algebra satisfying the condition
(χ)
and let D be a nilpotent derivation on A, then D
(
A
)
is contained in N
(
A
).
In the following lines, we recall definitions and some basic facts about latticeordered algebras. For more information about this field, one can refer to
[1,3]
. A (real) algebra
A
which is simultaneously a vector lattice such that the partial ordering and the multiplication in
A
are compatible, that is
a, b
∈
A
^{+}
implies
ab
∈
A
^{+}
is called
lattice-ordered algebra
( briefly ℓ-algebra). The ℓ-algebra
A
is said to be a
d
-algebra whenever
a
Λ
b
= 0 in
A
implies
ac
Λ
bc
=
ca
Λ
cb
= 0, for all 0 ≤
c
∈
A
. In general,
d
-algebras are not commutative, see
[3]
.
Since any Archimedean
d
-algebra satisfies the condition (χ) ; see [3, Corollary 5.7], we deduce the following result:
Corollary 2.10.
Let A be an Archimedean d-algebra and let D be an algebraic nil derivation on A.Then D
(
A
)
is contained in N
(
A
) :
Definition 2.11.
Let A be an algebra. For a fixed a ∈ A, define D : A → A by D(
x
) = [
x, a
] =
xa
−
ax
, for all
x
∈
A
. Then
D
is called
inner derivation
of
A
associated with
a
and is generally denoted by
D_{a}
.
Theorem 2.12.
Let A be an Archimedean d-algebra with the condition Z
(
A
) = {0},
where Z
(
A
)
denotes the center of A and let D be an inner derivation on A
.
Then the following assertions are equivalent
:
i) D is nilpotent; ii) D ^{3} = 0; iii) D is induced by a nilpotent element .
Proof
. i) ) ii) Let
a
∈
A
such that
D
=
D_{a}
. Since any Archimedean
d
-algebra satisfies the condition (χ) ; then for all
, we have
for all
x
∈
A
. Since
D_{a}
is nilpotent, there exists
n
∈ ℕ such that
Therefore
for all
x
∈
A
. Consequently
a^{2n+1}
∈
Z
(
A
) = {0} : Hence
a^{2n+1}
= 0. By [3, Theorem 5.5], we deduce that
a
^{3}
= 0: It follows that
ii) ⇒ iii)
means that
a
^{3}
= 0. Therefore
a
∈
N
(
A
).
iii) ⇒ i) This path is obvious.□
Remark 2.13.
It is obvious that algebraic nil derivations are nil derivations. The simple-minded attempt to extend Theorem 1,2 and 3 to nil derivations obviously fails. This is illustrated in the following example.
Example 2.14.
Let
A
= ℂ[
X
]
and D
:
A
→
A defined by
It is not hard to prove that D is a nil derivation but not an algebraic nil derivation, whereas
D
(
A
) =
A
≠
N
(
A
).

1. INTRODUCTION

Let
2. THE MAIN RESULTS

To prove our first theorem, we shall need the following algebraic result.
PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

Acknowledgements

The referees have reviewed the paper very carefully. The author expresses his deep thanks for the comments.

Aliprantis C.D.
,
Burkinshaw O.
1985
Positive Operators
Academic Press
Orlando

Ben Amor F.
2010
On orthosymmetric bilinear maps
Positivity
14
(1)
123 -
134
** DOI : 10.1007/s11117-009-0009-4**

Bernau S.J.
,
Huijsmans C.B.
1990
Almost f-algebras and d-algebras
Math. Proc. Cam. Phil. Soc.
107
287 -
308
** DOI : 10.1017/S0305004100068560**

Boulabiar K.
2002
Positive Derivations on Almost f-rings
Order
19
385 -
395
** DOI : 10.1023/A:1022869819129**

Bresar M.
1994
Derivations of noncommutative Banach algebras II
Arch. Math.
63
56 -
59
** DOI : 10.1007/BF01196299**

Chung L.O.
1985
Nil derivations
J. Algebra
95
20 -
30
** DOI : 10.1016/0021-8693(85)90089-4**

Chung L.O.
,
Luh J.
1984
Nilpotency of derivations on an ideal
Proc. Amer. Math. Soc.
90
(2)
211 -
214
** DOI : 10.1090/S0002-9939-1984-0727235-3**

Chung L.O.
,
Kobayashi Y.
1985
Nil derivations and chain conditions in prime rings
Proc. Amer. Math. Soc.
94
(2)
201 -
205
** DOI : 10.1090/S0002-9939-1985-0784162-4**

Colville P.
,
Davis G.
,
Keimel K.
1977
Positive derivations on f-rings
J. Austral. Math. Soc.
23
371 -
375
** DOI : 10.1017/S1446788700019017**

Kim B.D.
,
Jun K.-W.
1991
The range of derivations on noncommutative Banach algebras
Bull. Korean Math. Soc.
28
65 -
68

Mathieu M.
,
Runde V.
1992
Derivations mapping into the radical II
Bull. London Math. Soc.
24
485 -
487
** DOI : 10.1112/blms/24.5.485**

Singer I.M.
,
Wermer: J.
1955
Derivations on commutative normed algebras
Math. Ann.
129
260 -
264
** DOI : 10.1007/BF01362370**

Thomas M.
1988
The image of a derivations contained in the radical
Ann. of Math.
128
435 -
460
** DOI : 10.2307/1971432**

Toumi A.
,
Toumi M.A.
2010
Order bounded derivations on Archimedean almost f-algebras
Positivity
14
(2)
239 -
245
** DOI : 10.1007/s11117-009-0013-8**

Vukman J.
1992
On derivations in prime rings and Banach algebras
Proc. Amer. Math. Soc.
116
(4)
877 -
884
** DOI : 10.1090/S0002-9939-1992-1072093-8**

Citing 'THE RANGE INCLUSION RESULTS FOR ALGEBRAIC NIL DERIVATIONS ON COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAS
'

@article{ SHGHCX_2013_v20n4_243}
,title={THE RANGE INCLUSION RESULTS FOR ALGEBRAIC NIL DERIVATIONS ON COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAS}
,volume={4}
, url={http://dx.doi.org/10.7468/jksmeb.2013.20.4.243}, DOI={10.7468/jksmeb.2013.20.4.243}
, number= {4}
, journal={The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={TOUMI, MOHAMED ALI}
, year={2013}
, month={Nov}