© Hindawi Publishing Corp. DERIVATIONS ON BANACH ALGEBRAS

Let D be a derivation on a Banach algebra; by using the operator D2, we give necessary and sufficient conditions for the separating ideal of D to be nilpotent. We also introduce an ideal M(D) and apply it to find out more equivalent conditions for the continuity of D and for nilpotency of its separating ideal.


Introduction.
Let A be a Banach algebra.By a derivation on A, we mean a linear mapping D : A → A, which satisfies D(ab) = aD(b) + D(a)b for all a and b in A. The separating space of D is the set ( The set S(D) is a closed ideal of A which, by the closed-graph theorem, is zero if and only if D is continuous.
Definition 1.1.A closed ideal J of A is said to be a separating ideal if, for each sequence {a n } in A, there is a natural N such that The separating space of a derivation on A is a separating ideal [2, Chapter 5]; it also satisfies the same property for the left products.
The following assertions are of the most famous conjectures about derivations on Banach algebras: (C1) every derivation on a Banach algebra has a nilpotent separating ideal; (C2) every derivation on a semiprime Banach algebra is continuous; (C3) every derivation on a prime Banach algebra is continuous; (C4) every derivation on a Banach algebra leaves each primitive ideal invariant.
In the next section, we deal with (C1), and although, for a derivation D on a Banach algebra, the operators D n , n = 2, 3,..., are more complicated, by considering D 2 , we easily give some equivalent conditions for S(D) to be nilpotent.As a consequence, we reprove some of the results in [8].At the end of the next section, we introduce an ideal related to a derivation and apply it to obtain some equivalent conditions for continuity of D and for nilpotency of S(D).
We recall that S(D) is nilpotent if and only if 2. The results.From now on, A is a Banach algebra, and R and L denote the Jacobson radical and the nil radical of A, respectively, (see [6,Chapter 4] for definitions).Note that D is a derivation on A, and S(D) is the separating ideal of D.
and if all of B i 's coincide with each other, we denote this set by B n .
Consider a in S(D)∩J, then for each n ∈ N, a n ∈ (S(D)∩J) n , and since S(D) is a separating ideal, there exists N ∈ N such that

S(D)a n = S(D)a N (n ≥ N).
(2.1) Hence, by the Mittag-Leffler theorem [2, Theorem A.1.25]and the fact that S(D)a n ⊆ (S(D) ∩ J) n , we have ) , and by the hypothesis, D 2 (y n ) → 0 and D 2 (y n 2 ) → 0. On the other hand, The assertions of the following theorem were proved by Villena in [8], see also [9,Theorem 4.4].Using Theorem 2.1, we can reprove them in a different way.

Theorem 2.5. The derivation D is continuous if one of the following assertions hold:
(a) A is semiprime and dim(R ∩ ( In the sequel, we give other equivalent conditions for S(D) to be nilpotent, but first we introduce the set (2.4)

Obviously, M(D) is an ideal of A and (S(D) ∩ R) 2 ⊆ M(D).
The following theorems show that this ideal can help us to study the continuity of a derivation or nilpotency of its separating ideal.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning Proof.(a) By Corollary 2.4, S(D) is nilpotent, and since A is semiprime, D is continuous.(b) Without loss of generality, we may assume that A has an identity.By assumption, ∞ n=1 (aA∩R∩S(D)) n is finite dimensional; thus, D 2 is continuous on this space, and by Remark 2.2(ii), aA∩R ∩S(D) is a nil right ideal; therefore, a(S(D) ∩ R) is a nil right ideal, and by [6, Theorem 4.4.11],a(S(D) ∩ R) ⊆ L = {0}.Thus, AaA(S(D)∩ R) = {0}, where AaA is the ideal generated by a. Since a 2 ≠ 0 and A is prime, it follows that S(D)∩R = {0} and hence S(D) ⊆ L = {0}.(c) The same argument as in (b) shows that a(S(D) ∩ R) = {0}, and since A is an integral domain, S(D) ∩ R = {0} and D is continuous.
[2,e that in Theorem 2.1, we can replace J by a right ideal, see[2, Theorem 5.2.24].(ii)The argument of Theorem 2.1 shows that if J is not assumed to be closed and if D 2 is continuous on ∩ J) n , then S(D) ∩ J will be a nil ideal.If S(D) is nilpotent, then the result is obvious.Conversely, by Theorem 2.1, S(D) ∩ R is nilpotent, and by [1, Lemma 4.2], S(D) is nilpotent.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation