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

Abstract

Let A be an algebra and D a derivation of A. Then D is called algebraic nil if for any $x{\in}A$ there is a positive integer n = n(x) such that $D^{n(x)}(P(x))=0$, for all $P{\in}\mathbb{C}[X]$ (by convention $D^{n(x)}({\alpha})=0$, for all ${\alpha}{\in}\mathbb{C}$). 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.

1. INTRODUCTION

Let 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 Dn(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 Dn(x) (P (x)) = 0, for all P ∈ ℂ[X] (by convention Dn(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.

 

2. THE MAIN RESULTS

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

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

Dn (x), Dn (x2), Dn (xn) ∈ N (A),

where N (A) denotes the set of all nilpotent elements of A. Then D (x) ∈ N (A).

Proof. Let x ∈ A with Dn (x), Dn (x2), Dn (xn) ∈ N (A). It follows that

Since Dn (x) ∈ N (A) ; we have

Moreover, letting such that n1 + n2 = n, this one has

By using the Leibnitz rule for Dk (xn1) and Dk (xn2) 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

Dn (x) = Dn (x2) = Dn (xn) = 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

(a1, b1) (a2, b2) = (a1a2 + a2a1, b1b2 + b2b1)

for all (a1, b1), (a2, b2) 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 Dn (x) = Dn (x2) = Dn (xn) = 0: Let a, b ∈ A. It follows that

Moreover, let n1; n2 ∈ℕ N such that n1 + n2 = n (x), then

By using the Leibnitz rule for aDk (xn1) b and aDk (xn2) b in Equality (5), by using Equality (4) and taking into account that Dn (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 Da.

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) D3 = 0; iii) D is induced by a nilpotent element.

Proof. i) ) ii) Let a ∈ A such that D = Da. Since any Archimedean d-algebra satisfies the condition (χ) ; then for all , we have

for all x ∈ A. Since Da is nilpotent, there exists n ∈ ℕ such that Therefore

for all x ∈ A. Consequently a2n+1 ∈ Z (A) = {0} : Hence a2n+1 = 0. By [3, Theorem 5.5], we deduce that a3 = 0: It follows that

ii) ⇒ iii) means that a3 = 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).