ON THE BEURLING ALGEBRAS A+α(D)—DERIVATIONS AND EXTENSIONS

Based on a description of the squares of cofinite primary ideals of A α + ( 𝔻 ) , we prove the following results: for α ≥ 1 , there exists a derivation from A α + ( 𝔻 ) into a finite-dimensional module such that this derivation is unbounded on every dense subalgebra; for m ∈ ℕ and α ∈ [ m , m + 1 ) , every finite-dimensional extension of A α + ( 𝔻 ) splits algebraically if and only if α ≥ m + 1 / 2 .

Clearly, A + α (D) is a Banach algebra with respect to this norm.These algebras have been considered in [13] where results on primary ideals were applied to operator theory.More recently, the algebras have appeared in the examination of finite-dimensional extensions of a whole range of commutative Banach algebras [4].The present paper deals with continuity problems of derivations from A + α (D) and with finite-dimensional extensions of this special type of Beurling algebras.Some of the results of the first paper will be the starting point for our investigation.
This paper is organized as follows: as a preparation, Section 2 describes the squares of cofinite primary ideals and exhibits an approximate identity for a special ideal in A + α (D).The results of Section 2 will be applied to the questions considered in Sections 3 and 4.These sections are investigating derivations and extensions, respectively, and are rather independent of each other.
Section 3, which is on derivations from A + α (D) into finite-dimensional Banach modules, follows the approach used by [2] for Banach algebras of differentiable functions on the unit interval.In our case, we are interested in derivations from A + α (D) which are discontinuous on the subalgebra of polynomials.For α > 1, we give an example of a derivation which is unbounded on every dense subalgebra.
Section 4 then turns to the problem of finite-dimensional extensions guided by the ideas of [4] which makes a comprehensive approach on extensions of Banach algebras in general.We solve a problem raised there: for m ∈ N and α ∈ [m, m + 1), every finite-dimensional extension of A + α (D) splits algebraically if and only if α ≥ m + 1/2.

Primary ideals of A +
α (D).Suppose that m ∈ N and that α ∈ [m, m + 1).Let f ∈ A + α (D).Then f is m-times continuously differentiable, and f (m) , the mth derivative of f , belongs to A + 0 (D).In fact, Therefore, we expect the algebras A + α (D) to resemble the Banach algebras of m-times continuously differentiable functions on the unit interval C m [0, 1] in more than one aspect.It will be of some use for us to turn this observation into a precise statement.
Of course, all results on the ideals I α,n hold for the corresponding primary ideals for any other point of T.
For an ideal I in a Banach algebra A, we define I 2 to be the linear span of the set I [2] := {a • b | a, b ∈ I}.We refer to I 2 as the square of I.
Lemma 2.2.Let m ∈ N. Suppose that α ∈ [m, m + 1).For g, h ∈ I α,0 , the function gh is (m + 1)-times differentiable at 1, and (gh) (m+1) (1) = m i=1 m + 1 i g (i) (1) • f (m+1−i) (1). (2.4) In order to describe the squares of these ideals, we use the same approach as used in [2] where the Banach algebras C m [0, 1] are investigated.For these algebras, the ideals are defined for n = 0,...,m.Let m ∈ N and let T represent the function which is given by [0, 1] [0, 1], t t.Then [2, Theorem 2.1] gives the following description: . We expect similar results to hold for A + α (D).Of course, we will require different arguments due to the different norm structure of A + α (D).The next result is [13,Lemma 2.1].We give a version which is a bit more precise.
As in [13,Lemma 2.2], it is easy to check that, for real numbers α, β > 0, where β is not an integer, (Z − 1) β ∈ A + α (D) if and only if β > α.In [13, Proposition 2.6], a sequence of polynomials (e n,m ) n∈N is defined by for every m ∈ Z + .It is shown that lim n→∞ (e n,m f ) = f for each f ∈ I α,m and a given α ∈ [m, m + 1).Note that, for n, m ∈ N, (2.7) Surprisingly, these polynomials will turn out to define an approximate identity for some other Banach algebra.The next lemma is our key observation.
Lemma 2.4.Suppose that α ≥ 1, and let n ∈ Z + such that n ≤ α.Let g ∈ I α,n .Then there exists Proof.Since g(1) = 0 and α ≥ 1, there exists f ∈ A(D) with (Z − 1)f = g.Now suppose that (a n ) n∈Z + and (b n ) n∈Z + are the sequences of the Fourier coefficients for g and f , respectively.Then By an induction using Lemmas 2.3 and 2.4, we now obtain some useful estimates for the growth of functions in A + α (D).
Throughout the paper, we will make frequent use of the following corollary of Proposition 2.5.For completeness, we also include the above-mentioned variation of [13, Lemma 2.2] (cf. the remark after Lemma 2.3).
(i) Suppose that β is not an integer.Then (Z − 1) Next we apply the approach of [2, Theorem 2.1] to our situation.We are following the idea that, for the investigation of functions in I α,m , the common divisor (Z −1) m is redundant and therefore division by (Z − 1) m establishes a linear isomorphism.Naturally, the image is a Banach space with respect to the norm induced by this isomorphism.Since the image is also a subspace of A(D) and since I 2 α,m ⊆ I α,m , we assume that, with respect to some equivalent norm, this Banach space is in fact a Banach algebra. (2.9) Then the following hold: (i) with respect to an equivalent norm, B α is a Banach algebra; (ii) the Banach algebra B α has a sequential approximate identity; this approximate identity is bounded if and only if α = m; Proof.Note that all assertions hold for the case m = 0 (cf.[13] to verify (ii)).Now suppose that m ≥ 1.It follows by a simple induction from Lemma 2.4 that B α is a Banach space and that f Bα ≤ (m + 1) (Z − 1) m f α for f ∈ B α .Hence, (iii) holds.
In order to show that B α is a Banach algebra, we need a different characterization.Using Lemmas 2.1 and 2.4, it can be shown that (2.12) Thus, f h ∈ B α , that is, B α is an algebra.In fact, the multiplication is jointly continuous with respect to • Bα since for j = 0,...,m.Hence, (i) has been proved.
To show (ii), consider the sequence (e n,m ) n∈N described at the beginning of this section.We have already mentioned that, for g ∈ I α,m , we have lim n→∞ ge n,m −g = 0. Now let f ∈ B α .Then (Z − 1) m f ∈ I α,m and lim n→∞ (Z − 1) m (f e n,m − f ) α = 0.By (iii), lim n→∞ f e n,m − f Bα = 0. We have proved that (e n,m ) n∈N is an approximate identity for B α .
Suppose that α = m.We show by induction on m that ( e n,m Bm ) n∈N is bounded.This is equivalent to ( (Z −1) m e n,m m ) n∈N being bounded.The case m = 0 is obvious.Now let m ≥ 1.For n ∈ N, since this polynomial is of degree n + m.Hence, which is bounded in n by the induction hypothesis.
We now obtain a result analogous to [2, Theorem 2.1].
(2.17) By (i) and (ii), f g ∈ (Z − 1) m I α,m , and I 2 α,n = (Z − 1) n+1 I α,m follows.(iv) By the principle of uniform boundedness, the sequence (e n,m ) n is bounded as a sequence of multipliers on I α,m .Now consider the net (b α ) := (e n 1 •••e n m+1 ) n 1 ,...,n m+1 in I α,m , where the index set N m+1 is directed by the product order.Using the multiplier boundedness of (e n,m ) n , it is straightforward that (b λ ) is an approximate identity.
(v) and (vi) are now immediate.
Example 2.9.There is one question which remains to be checked in order to obtain a thorough comparison to the C m [0, 1] case; suppose that m ∈ N and α ∈ [m, m + 1): does I 2 α,0 = {f ∈ I α,1 | f (m+1) (1) exists} hold?For α > m, it is easy to see that the answer is in the negative since, for β ∈ (m, α), the fact that (Z −1) β+1 ∈ I α,1 provides a counterexample.
In order to decide the question for α = m, we may suppose for simplicity that α = 1.We know that I 1,1 ⊆ (Z − 1)I 0,0 .Define g ∈ A(D) by

.18)
We can now easily estimate the growth of the Fourier coefficients.In fact, for n ∈ N, n ≥ 2, and some constant C > 0. Hence, g Hence, the answer to the above question is again in the negative.
In order to complete the picture, we give the following obvious results on primary ideals at points in D.

Derivations from A + α (D).
Let A be a Banach algebra, and E a Banach A-bimodule.
Derivations have been investigated for many years.In particular, there are numerous articles concerned with continuity questions for derivations.Whereas there are many results which automatically ensure continuity for a specific class of derivations [7, Section 10], some questions still remain open due to the few established methods for the construction of discontinuous derivations.One of these questions is [8, Question 2.3] which focuses on the disk algebra A(D) and on l 1 (Z + ) (which is A + 0 (D), of course).In a more general setting, we may put forward the following questions.
A Banach algebra A is a Banach algebra of power series if it can be embedded continuously into the algebra of formal power series C ] is given the topology of coordinatewise convergence (cf.[7, Section 5]).A Banach A-bimodule E is symmetric if the right and the left actions of A on E coincide.Then we may simply call E a Banach A-module.
(i) Let A be a Banach algebra of power series.Do there exist a Banach A-module E and a derivation D : A → E such that D is unbounded on every dense subalgebra?If an answer to this question does not seem to be achievable, we may weaken the question.
(ii) Let A be a Banach algebra of power series with C[Z] = A. Do there exist a Banach A-module E and a derivation D : A → E such that D is unbounded on the polynomials?Here we identify those elements of A which are mapped to C[X] by the embedding into C[[X]] as the polynomials of A, and we denote the subalgebra of these polynomials by C[Z] to obtain a formal distinction.
Although discontinuous derivations on A(D) (or arbitrary Banach algebras of power series) have been constructed [3,6,10], all these examples consist of derivations vanishing on the polynomials and thus do not even answer question (ii).There have been attempts to modify these constructions in order to obtain a positive answer for l 1 (Z + ) (see [12]).However, the problem is still open (for A(D) and l 1 (Z + )).
In this context, it is of some interest to consider other related Banach algebras of power series as there are the algebras A + α (D), subalgebras of A(D) and A + 0 (D), or weighted discrete convolution algebras l 1 (Z + ,ω), where ω is a radical weight.In the latter case, where the algebra contains A(D) and A + 0 (D), the author was able to find a positive solution for question (ii) (see [14]).
Surprisingly, it is not too difficult to give a positive answer to question (ii) for A + α (D) with α ≥ 1/2.Using the results of Section 2 on the ideal structure, we are even able to describe all derivations having finite-dimensional image.Again, we follow the approach of [2].However, it is not always possible to transfer their arguments in a straightforward way and we will make some observations differing from their results.In particular, we obtain an affirmative answer to the first question if α ≥ 1.
It is easy to see that, for a derivation D from a unital algebra A and for a polynomial p ∈ C[X], we have p (a) • D(a) = D(p(a)) for each a ∈ A. In particular, D(1) = 0. Note that this implies that, for a Banach algebra of power series in which the polynomials are dense, the set D(A) is a submodule of E for every derivation D : A → E.
First, we use arguments similar to [2, page 239] in order to show that a restriction of our investigations to a simple type of modules is justified.Let m ∈ Z + and α ∈ [m, m + 1).Suppose that E is a finite-dimensional Banach A + α (D)-module.Choosing a basis η 1 ,...,η n such that the matrix corresponding to the action of Z ∈ A + α (D) obtains its canonical Jordan form, we see that E can be decomposed into the direct sum of finite-dimensional submodules which correspond to the different Jordan blocks.If there exists only one summand of this type, E is called indecomposable.In this case, we see that the module multiplication by f ∈ A + α (D) corresponds to the matrix where λ ∈ σ (Z) = D.This also shows that the ideals I α,n (n = 0,...,m) are the only cofinite closed primary ideals for the character 1.Therefore, in the case where λ ∈ T, it follows that dim E ≤ m + 1.If λ ∈ D, then no restriction occurs.An indecomposable module of this type is referred to as a cyclic module at λ.This term implies that the module is of finite dimension.The basis η 1 ,...,η k is called the standard basis (which is unique if we demand that η 1 = 1).Note that, for λ ∈ D, we obtain a continuous linear map ρ : A → Ꮾ(E), Here we identify ξ and its coordinate vector, and Ꮾ(E) denotes the Banach algebra of bounded linear operators on E. This notation is consistent with our earlier definition of the mapping The same holds in the case where λ ∈ T, provided that dim E ≤ m.
Next, suppose that E = ⊕ n i=1 E i , where E 1 ,...,E n are indecomposable submodules of E. Then there exist pairwise orthogonal projections P 1 ,...,P n onto E 1 ,...,E n , respectively, such that each projection commutes with the module action.Now let D : A + α (D) → E be a derivation.Then D i := P i D is a derivation into E i for each i = 1,...,n, and Now consider the case where E is infinite dimensional and the image of D is of finite dimension.Then D(A) is closed and hence, as mentioned above, a submodule.Thus, when considering the continuity of derivations with finite-dimensional image (as a map from A + α (D) or from C[Z]), we may always suppose that the module E is finitedimensional and indecomposable, or, equivalently, that E is cyclic at some λ ∈ D. In this situation, a derivation D : In order to describe all cyclic derivations for A + α (D), we need the notion of a singular derivation.For a Banach algebra of power series A, a derivation D is called singular if D vanishes on the polynomials.Thus, a derivation which is bounded on the polynomials can be written as the sum of a continuous and a singular derivation.Such a derivation is called decomposable.Our main interest is to find derivations for which this decomposition is not possible.
First, note that, for λ ∈ D, every derivation into a cyclic Banach A + α (D)-module at λ is continuous.This follows from Proposition 2.10 and the fact that the elements of A + α (D) are infinitely differentiable at λ.In this situation, every derivation is given by f f •ξ for some ξ ∈ E.
The last observation implies that, when looking for derivations unbounded on C[Z], we have to consider cyclic modules at points of T. At the beginning of this section, we have seen that their dimension is necessarily less than m + 1.Hence, we are dealing merely with point derivations if m = 0. We have to consider this case separately.Recall that there is a one-to-one correspondence between point derivations at 1 and those linear functionals on I α,0 which vanish on I 2 α,0 (cf.[9, Proposition 1.8.8]).(iii) We will construct the required derivation at 1.By Proposition 2.5, (Z −1) ∈ (I α,0 ) 2 .Now define a linear functional D on A + α (D) such that That D is a point derivation at 1 can be easily verified since, for In particular, D(p) = p (1) for every p ∈ C[Z].Hence, D is unbounded on the polynomials.
Note that, for α = 0, every point derivation at 1 is zero (and hence continuous) since I 2 α,0 = I α,0 .Note further that the last proposition does not provide a positive answer to our initial question (i) in the case 1/2 ≤ α < 1: every point derivation at 1 vanishes on span{1,I 2 α,0 } which is a dense subalgebra.The first implication of the following result can be proved in exactly the same way as [2, Theorem 5.2].The second implication is immediate if we recall the definition of the map f f • ξ above.
→ E be a continuous derivation.Then the height of D is at most m, and, for some ξ ∈ E, we have by Corollary 2.8.Therefore, we may define a linear functional µ on A + α (D) such that, for l = 0,...,2m + 1, We are now constructing a derivation on A + α (D) which is unbounded on the polynomials.The result should be compared with [2,Theorem 5.4].Recall that the dual Banach space of A + α (D) can be identified naturally and isometrically with the space
Let p ∈ C[Z].We may find sequences (f n ) n∈N and (g n ) n∈N in Ꮾ with lim n→∞ f n = (Z − 1)p, lim n→∞ g n = (Z − 1) m , and (1)  n ( 1)g (m)  n (1) (3.11) by Lemma 2.2.Hence f (1)  n ( 1)g (m)  n ( 1) 1). (3.12) In other words, γ coincides with µ on For n ∈ Z + , define ξ m+n ∈ C m+1 by ξ m+n = (b m+n ,...,b n ), and define further M ∈ M m+1 (C) by Then The characteristic polynomial of M is (Z − 1) m+1 again.On the other hand, the minimal polynomial of M is of degree m + 1 since {M k e n+1 } m k=1 is a linearly independent set.Here e n+1 is the (n + 1)th canonical basis vector.Therefore, the Jordan form N of M is given by and, for n ≥ m, we have If we define the norm on M m+1 (C) to be the maximum of the moduli of the matrix coefficients, then The theorem is somewhat surprising when we take into account that the derivation maps into a finite-dimensional module and that ker D is a cofinite subalgebra.However, for m ∈ N, we are able to extend this result to the algebra C m [0, 1].In fact, using the embedding the proof of the theorem can be easily modified to obtain the following.
Corollary 3.5.Let m ∈ N.There exists a derivation from C m [0, 1] which is unbounded on every dense subalgebra and maps into a finite-dimensional module.This establishes a simpler example of a derivation of this special type than that given by [2, Proposition 6.2].
We would like to conclude that every derivation into an (m + 1)-dimensional, cyclic module at 1 can be decomposed into the sum of a continuous derivation, a singular derivation, and the scalar multiple of a fixed derivation which is unbounded on the polynomials.Obviously, we cannot use the same argument as [2] since A + α (D) is not a regular Banach function algebra.Nevertheless, we have the following lemma.Lemma 3.6.Let m ≥ 1, α ∈ [m, m + 1), and let E be a cyclic, (m + 1)-dimensional A + α (D)-module at 1. Suppose that D is a cyclic derivation of height m + 1 at 1 such that, with respect to the standard basis η 1 ,...,η m+1 , for linear functionals µ 1 ,...,µ m+1 on A + α (D).Suppose further that µ m+1 vanishes on the polynomials.Then D is decomposable.
Proof.Let j be the maximum integer with for linear functionals µ 1 ,...,µ m+1 .It follows that µ m+1 is a point derivation.By Proposition 3.2, µ m+1 is decomposable, and there exist κ ∈ C and a singular point derivation λ on A + α (D) such that We doubt that the last corollary remains true in the case where α > m.In this situation, there exists a singular derivation of height m + 1 into a cyclic module at 1.It might happen that this singular derivation coincides with (−D µ ) on a dense subalgebra Ꮽ.However, Ꮽ has to satisfy additional properties, that is, 4. Finite-dimensional extensions.This section now turns to finite-dimensional extensions of Beurling algebras.As it is shown here, the splitting problem for (finitedimensional) extensions is closely connected to the structure of (cofinite) ideals.Thus, our results are mainly consequences of Section 2.
An extension Σ(Ꮽ,I) of a Banach algebra A is a short exact sequence of Banach algebras The Banach algebra I is usually considered as an ideal of Ꮽ.The extension is called radical (nilpotent, finite-dimensional) if I is radical (nilpotent, finite-dimensional).The extension is commutative if Ꮽ is commutative, and singular if I 2 = {0}.In the latter case, we can regard I as a Banach A-bimodule, and there is a corresponding concept of a singular extension of a Banach algebra A by a Banach A-bimodule E.
An extension is admissible if the sequence splits as a sequence of Banach spaces, that is, there exists a continuous linear map Φ : A → Ꮽ with p • Φ = Id A .Thus, every finitedimensional extension is admissible.An extension splits algebraically if the sequence splits as a sequence of complex algebras, that is, if there exists a homomorphism ρ : A → Ꮽ such that p •ρ = Id A .It splits strongly if it splits algebraically and if the splitting homomorphism ρ can be chosen to be continuous, or, equivalently, if the sequence splits as a sequence of Banach algebras.For a detailed discussion of extensions of Banach algebras in a more general context, see [4].
As usual, the principal tool for the investigation of a singular extension Σ(Ꮽ,E) of A by a Banach A-bimodule E is the continuous Hochschild cohomology groups Ᏼ n (A, E), where n ∈ N.For a definition, see [11].All admissible, singular extensions of A by E split strongly if and only if Ᏼ 2 (A, E) = {0}.Ꮾ n (A, E) denotes the Banach space of continuous n-linear maps from A into E.For the connecting maps of the Hochschild-Kamowitz complex, we write δ n : Ꮾ n (A, E) → Ꮾ n+1 (A, E).The Hochschild cohomology groups are given by Ᏼ n (A, E) = ker δ n / im δ n−1 .Further, we set ᐆ n (A, E) = ker δ n and This equation is called the cocycle identity.µ is a (continuous) 2-coboundary if there exists a continuous linear map λ : For A commutative and E symmetric, this is equivalent to the commutativity of the corresponding extension.
In [4], a related class of groups, H 2 (A, E), is defined.For this definition, N 2 (A, E) is taken to be the set of all continuous cocycles which are coboundaries in the algebraic sense, that is, the set of all µ ∈ ᐆ 2 (A, E) such that there exists a (not necessarily continuous) linear map λ : A → E satisfying (4.3).Now we set H 2 (A, E) := ᐆ 2 (A, E)/ N 2 (A, E).All singular admissible extensions of A by E split algebraically if and only if H 2 (A, E) = {0}.
An important observation (for the case n = 2, but it is obvious that the proof holds for each n ∈ N) is made in the remark after [4, Proposition 2.2]: let A be a unital Banach algebra and let M be a maximal ideal in A. Let E be a unital A-module.
Recall that, for finite-dimensional extensions, the problem of strong splitting can be reduced to singular, one-dimensional extensions.However, for the investigation of possible algebraic splittings, one has to consider all finite-dimensional singular extensions by a certain type of modules [4, pages 63-64].
Extensions of the algebras A + α (D) have been considered before in [4].For the case α = 0, we have A + 0 (D) = l 1 (Z + ), and every finite-dimensional extension splits strongly since every maximal ideal has a bounded approximate identity [4,Proposition 4.4].For α > 0, we have the following result on strong splittings which is [4, Proposition 5.9(i)].Proposition 4.1.Let α > 0. Then there exists a one-dimensional extension of A + α (D) which does not split strongly.
Thus our objective is to establish algebraic splitting of extensions of A + α (D).Proposition 5.9 in [4] also shows that each one-dimensional extension of A + α (D) splits algebraically (α ≥ 0), and that there exists a two-dimensional extension which does not split algebraically provided that 1 ≤ α < 3/2.The case α ≥ 3/2 remains unsolved.
In this section, we prove that, for m ∈ N and α ∈ [m, m+1), every finite-dimensional extension splits algebraically if and only if α ≥ m + 1/2.
Note that there is a simple solution for the case where α ∈ (0, 1).Then each maximal ideal of A + α (D) either has an approximate identity or is a principal ideal.Thus, every finite-dimensional extension splits algebraically by [4,Theorem 4.13].
To cover the case α ≥ 1, we begin with a reduction to the case of singular, commutative extensions.The result may be proved in a way similar to [4,Theorem 5.5].We think that one should be more careful showing this reduction.However, this does not require any new arguments but simple (albeit tedious) matrix manipulations.Therefore, we omit the proof.
The proof would also contain arguments showing that µ ∈ ᐆ 2 (A + α (D), E) is symmetric provided that E is a symmetric module, that is, every extension by a symmetric module is commutative.
Then at least one of the following assertions is true: (i) Σ splits algebraically; (ii) Σ is singular and commutative.
Decomposing symmetric modules as shown in Section 3, it suffices to consider cyclic modules at an arbitrary λ ∈ D.
Using exactly the same arguments as in the proof of [4,Theorem 5.6], it follows that µ is a continuous 2-cocycle which is not algebraically cobound.Thus, H 2 (I α,0 ,E) ≠ {0} and therefore It is obvious from [4,Theorem 5.6] that the proof of Lemma 4.3 depends on the fact that, by the hypothesis, (Z − 1) 2m+1 ∈ I 2 α,m .By Proposition 2.5, this does not hold for α ≥ m + 1/2.We will show that this observation forces every cocycle to cobound algebraically for α ≥ m + 1/2.
The proof of our following main result is simplified considerably for the case m = 1.
Inspecting the proof of the previous result carefully, the hypothesis that α ≥ m+1/2 is needed only once: we have to ensure that (Z − 1) 2m+1 ∈ I 2 α,m in order to define λ m+1 consistently.Thus, our approach can be modified to obtain two interesting observations.
Proof.(i) We may suppose that α < m+1/2.Now suppose that E is a cyclic A + α (D)module, that µ ∈ ᐆ 2 (A + α (D), E), and that k := dim E ≤ m.As in the earlier remark, we may consider E as a submodule of an (m + 1)-dimensional, cyclic module F , and µ = (µ 1 ,...,µ k , 0,...,0) with respect to the standard basis of F .If we now define λ m+1 as we did in the proof, we might obtain an inconsistency since (Z − 1) 2m+1 ∈ I 2 α,m .In fact, we have Now there no longer occurs any obstruction for the definition of a functional λ m+1 and we may proceed as before.The claim follows.
This section can be summarized as follows.(i) Suppose that α < m+1/2.Then every finite-dimensional extension of A + α (D) with dimension at most m splits algebraically, and there exists an (m+1)-dimensional extension which does not split algebraically.(ii) Suppose that α ≥ m+1/2.Then every finite-dimensional extension of A + α (D) splits algebraically.
It is remarkable that, for certain α's, α ≥ 1, all finite-dimensional extensions split, whereas, considering the so closely related algebras C m [0, 1], this does not hold for any m ∈ N.

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

1 .
Introduction.Let α be a positive real number.By D, we denote the open unit disk.The Beurling algebra A + α (D) is a subalgebra of the classical disk algebra A(D).For f ∈ A(D) with power series expansion f (z) = ∞ n=0 a n z n (z ∈ D), the function f belongs to A + α (D) if and only if ∞ n=0 |a n |(n + 1) α < ∞.In this case, we define f α := ∞ n=0 |a n |(n + 1) α .

. 6 )
(ii) Suppose that α = m.Then there exists a singular derivation of height n into E if and only if n ≤ m. (iii) Suppose that α > m.Then there exists a singular derivation of height n into E.Proof.Again, the result may be shown almost completely as the corresponding result in [2, Theorem 5.3], taking into account that this theorem actually deals with height k + 1.Note that (ii) follows since I 2 α,m = (Z − 1) m I α,m for α = m.The only implication we still have to prove is the following: for α > m, there exists a singular derivation of height n = m + 1 into E.In fact,

. 10 )
is a derivation which is unbounded on C[Z].Furthermore, D is discontinuous on every dense subalgebra.Proof.By Lemma 2.2, µ is well defined since (Z − 1) ∈ I 2 α,0 and 1 ∈ I α,0 .It is easily checked that D is indeed a derivation.Clearly D(C[Z]) = E. Since dim E = m + 1, it follows that D is unbounded on the polynomials by Proposition 3.2.

Theorem 3 . 7 .
.20) D 1 can be extended to a continuous derivation into E. Then D = D 1 + D 2 , where D 2 is a singular derivation.Let m ∈ N and let α ∈ [m, m + 1).Suppose that E is a cyclic, (m + 1)dimensional A + α (D)-module at 1 and µ is a generalized derivative of order m + 1 at 1. Let D : A + α (D) → E be a derivation.Then there exist κ ∈ C, a continuous derivation D c , and a singular derivation D s such that D = D c + D s + κD µ .Proof.If D is of height less than m + 1, then the claim holds for κ = 0 by Proposition 3.2.Thus we may suppose that D is of height m + 1.With respect to the standard basis,

Corollary 3 . 8 .
.22) Now set D = D − κF.Then the derivation D maps the polynomials into (Z − 1)E.By Lemma 3.6, D is decomposable, and the claim follows.Let m ∈ N. Suppose that E is a finite-dimensional A + m (D)-module.Then a derivation D : A + m (D) → E is unbounded on the polynomials if and only if D is unbounded on every dense subalgebra.
Suppose that α > 0. Then the ideal I 2 α,n is of infinite codimension for 0 ≤ n ≤ m.
and let D : A + α (D) → E be a derivation.Then D is continuous and hence decomposable.As a next step, we describe singular derivations from A + Let D : A + α (D) → E be a derivation.Then D is singular if and only if a functional µ on A + α (D) with µ| C[Z] = 0 and µ((Z − 1) n A + α (D) into cyclic modules at 1. Proposition 3.3.Let m, n ∈ N with n ≤ m+1, and α ∈ [m, m+1).Let E be a cyclic, n-dimensional Banach A + α (D)-module at 1. Suppose that η 1 ,...,η n form the standard basis.(i)α (D)) = {0} exists such that

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