AUTOMATIC CONTINUITY FOR BANACH ALGEBRAS WITH FINITE-DIMENSIONAL RADICAL

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1 AUTOMATIC CONTINUITY FOR BANACH ALGEBRAS WITH FINITE-DIMENSIONAL RADICAL HUNG LE PHAM Abstract. The paper [3] proved a necessary algebraic condition for a Banach algebra A with nite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sucient. We shall extend the above theorem. The conjecture will be conrmed in the case where A is separable and A/R is commutative, but will be shown to fail in general. Similar questions for derivations will be discussed. 1. Introduction This paper is concerned with the question of uniqueness of norm the question of when a Banach algebra has a unique complete norm, which means, by denition, that each complete algebra norm on the algebra is equivalent to the given norm. A celebrated theorem of B. E. Johnson ([6], [2, 5.1.6]) states that each semisimple Banach algebra has a unique complete norm. However, it has been known for a long time that even a commutative Banach algebra with 1-dimensional radical may fail to have this property; the rst such example is due to Feldman (see [2, 5.4.6]). There are other conditions that such an algebra needs to satisfy. The question of uniqueness of norm for algebras with nite-dimensional radical has been discussed in [12], in [11], and also implicitly in [4] (see also [2]). The most thorough treatment of this problem has been, however, in [3]. There, the authors propose necessary and sucient algebraic conditions for a Banach algebra with non-zero, nite-dimensional radical to have a unique complete norm. They proved the following ([3, Theorem 2.2], or [2, ]); for the notations see Ÿ2. Theorem. Let A be a unital Banach algebra with nite-dimensional radical. Suppose that there exists a minimal radical ideal K such that K λ K ρ has innite codimension in A. Then A does not have a unique complete norm. It was then conjectured in [3] that the converse of the above theorem holds. This conjecture was proved under various additional hypotheses, but, in its generality, was left unresolved. In Ÿ3, we shall give a new proof of the above theorem; moreover, we shall extend the result by proving that the algebra in consideration actually has a discontinuous automorphism. Then, in Ÿ4, the conjecture will be proved in the case where the algebra A is separable and the quotient A / (rad A) is commutative. This resolves the question, left open in [3], of uniqueness of norm for an interesting class of algebras. Finally, in Ÿ6, we shall construct a counter-example to show that the 2000 Mathematics Subject Classication. 46H40. Key words and phrases. Banach algebra, uniqueness of norm, automatic continuity, derivation, nite-dimensional. 1

2 2 HUNG LE PHAM conjecture fails, even in the case where the algebra is separable and the radical is 2-dimensional. It should be remarked that, for an algebra A with nite-dimensional radical, if A is separable with respect to a complete algebra norm, then A is separable with respect to each complete algebra norm. (See [3, Proposition 3.16].) There is a related question about the automatic continuity of derivations. It is proved in [8] that derivations on each semisimple Banach algebra are continuous. We shall consider two automatic continuity problems; one is for derivations on Banach algebras with nite-dimensional radical and the other is for derivations from Banach algebras into nite-dimensional Banach modules. In fact, our results on these problems are in parallel to those on the uniqueness of norm problem. Moreover, the results on constructing discontinuous automorphisms are all derived from similar results on constructing discontinuous derivations (see Ÿ3 and Ÿ6). There is another related topic, the Wedderburn decomposition of Banach algebras with nite-dimensional radical, which was developed in [7]. However, we shall not cover this topic here. In summary, we shall prove the following characterization theorems. Note that when A is non-unital we can always consider A # the unitization of A. Theorem (A). Let A be a separable Banach algebra with nite-dimensional radical R such that A/R is both unital and commutative. Then the following are equivalent: (a) each derivation on A is continuous; (b) each derivation D : A R with D 2 = 0 is continuous; (c) each automorphism of A is continuous; (d) each complete algebra norm on A is equivalent to the given norm; (e) for each minimal radical ideal K in A, the ideal K λ K ρ has nite codimension in A. Theorem (B). Let A be a separable, unital Banach algebra, and let X be a nitedimensional Banach A-bimodule such that A/X is commutative. Then the following are equivalent: (a) each derivation from A into X is continuous; (b) for each minimal sub-bimodule Y in X, the ideal Y λ Y ρ has nite codimension in A. 2. Preliminaries First, we recall some terminology explained in [3]; see also [2]. Let A be an algebra. For two subspaces E and F in A, we denote by EF the linear span of the set {ab : a E, b F }. A minimal radical ideal of A is a minimal, non-zero ideal of A which is contained in rad A, the radical of A. In each non-zero and nite-dimensional ideal contained in rad A, we can always nd a minimal radical ideal. When rad A = R is nite-dimensional, there exists n N such that R n = {0}, and, for each minimal radical ideal K, we have RK = KR = {0}. For a subspace I of A, dene the left and right annihilators of I by I λ = {a A : ai = {0}} and I ρ = {a A : Ia = {0}}, respectively. Dene the annihilator of I by I = I λ I ρ. Similarly, we can dene the left annihilator X λ of a left A-module X and right annihilator Y ρ of a right

3 BANACH ALGEBRAS WITH FINITE-DIMENSIONAL RADICAL 3 A-module Y ; X λ and Y ρ are ideals in A. For an A-bimodule Z, we also dene the annihilator Z to be Z λ Z ρ. For each m N, let M m denote the full matrix algebra of m m-matrices over C. For each m, n N, let M m,n denote the space of m n-matrices over C; the space M m,n is naturally considered as an M m -M n -bimodule. Let A and B be algebras. Recall that an A-B-bimodule X is minimal if X is non-zero and X has no other non-zero submodule, and is simple if, furthermore, AXB 0. For a nite-dimensional, simple A-B-bimodule X, there exist l, r N such that A/X λ = Ml and B/X ρ = Mr, and, when considered as an M l -M r - bimodule, X is isomorphic to M l,r ; see, for example, [2, page 65]. The module M l,r is the unique simple M l -M r -bimodule (up to isomorphism). Note also that each unital M l -M r -bimodule can be considered as a unital left M lr -module, and hence each unital M l -M r -bimodule is semisimple, so that it is an algebraic direct sum of its simple submodules, or equivalently, each of its submodules is an algebraic direct summand (in M l -mod-m r ). For the semisimplicity of unital left M n -modules see, for example, [9, pages 28 and 33]. The following are standard results from automatic continuity theory; for details, see [2]. Let T : E F be a linear operator from a Banach space E into another Banach space F. The separating space of T, denoted by S(T ), is dened as {v F : there exists a sequence (u n ) E with u n 0 and T (u n ) v}. The space S(T ) is a closed subspace of F, and, by the closed graph theorem, T is continuous if and only if S(T ) = {0}. Let S : F G be a bounded linear operator from F into a Banach space G. Then we have S(ST ) = S (S(T )) ; see [2, Theorem 5.2.2(ii)]. Now let θ : A B be a homomorphism from a Banach algebra A into a Banach algebra B. Then the separating space S(θ) is a closed ideal in θ(a). Dene the left and right continuity ideals of θ by I λ (θ) = {a A : b θ(ab), A B, is continuous}, and I ρ (θ) = {a A : b θ(ba), A B, is continuous}, respectively. By the previous paragraph, we have I λ (θ) = {a A : θ(a)s(θ) = {0}} and I ρ (θ) = {a A : S(θ)θ(a) = {0}}. As expected, I λ (θ) and I ρ (θ) are ideals in A; in general, they are not closed in A. Let D be a derivation from a Banach algebra A into a Banach A-bimodule X; by denition, D is a linear map from A into X satisfying D(ab) = a Db + (Da) b (a, b A). Then the separating space S(D) is a closed submodule of X. The left and right continuity ideals of D are dened to be I λ (D) = {a A : a S(D) = {0}} and I ρ (D) = {a A : S(D) a = {0}}, respectively. We see that I λ (D) = {a A : b D(ab), A X, is continuous}, and I ρ (D) = {a A : b D(ba), A X, is continuous}. In this case, the ideals I λ (D) and I ρ (D) are closed in A.

4 4 HUNG LE PHAM 3. Construction of discontinuous derivations and automorphisms In [3, Theorem 2.2], the authors construct directly, on a Banach algebra A, a complete algebra norm which is not equivalent to the given norm when A satises certain algebraic conditions. Theorem 3.3, below, extends that theorem by providing a discontinuous automorphism of A (as well as a discontinuous derivation on A). The method is to construct a discontinuous automorphism out of a certain derivation. First, we shall construct a discontinuous derivation from a Banach algebra A into a nite-dimensional Banach A-bimodule. Lemma 3.1. Let A be a Banach algebra, and let Y be a nite-dimensional, simple Banach A-bimodule. Suppose that A/(Y λ Y ρ ) is unital and has innite dimension. Let F be any nite-dimensional subspace of Y. Then there exists a discontinuous A-mod-A homomorphism T from Y onto Y such that T is zero on Y λ Y ρ + F. Proof. Since Y = Y λ Y ρ has nite codimension in A, we see that Y λ Y ρ is of innite codimension in Y. Then I = Y λ Y ρ + F + AF + F A + AF A is also an ideal of innite codimension in Y. Since A/(Y λ Y ρ ) is unital, the quotient space Y /I can be considered as a unital A/Y λ -A/Y ρ -bimodule. As discussed in Ÿ2, we have that both A/Y λ and A/Y ρ are isomorphic to some full matrix algebras, and that Y /I is an algebraic direct sum of simple A/Y λ -A/Y ρ -submodules. Since Y /I has innite dimension, there exist distinct such simple submodules, say E n (n N). For each n N, let T n be an A/Y λ -mod-a/y ρ homomorphism from Y /I onto Y such that T n is an isomorphism from E n onto Y and zero on the remaining submodules. Let π : Y Y /I be the quotient map. Then each T n π is an A-mod-A homomorphism onto Y, and T n π is zero on I. It is easily seen that Y = ker(t n π). n=1 The Baire category theorem applied to the Banach space Y then shows that ker(t n0 π) is not closed for some n 0 N. Hence, by setting T = T n0 π, we obtain the desired map T. Theorem 3.2. Let A be a Banach algebra, and let X be a nite-dimensional Banach A-bimodule such that A/(X λ X ρ ) is unital. Suppose that there exists a minimal sub-bimodule Y of X such that Y λ Y ρ has innite codimension in A. Then, for each nite-dimensional subspace F of Y, there exists a discontinuous derivation D : A X such that D(A) = Y and such that D(F ) = {0}. Proof. Let e A be such that e + X λ X ρ is the identity of A/(X λ X ρ ). Since Y is a minimal A-bimodule, either AY = {0} or AY = Y. Assume towards a contradiction that AY = {0}. Let a Y ρ. Since a ea X λ X ρ, we see that a AY ρ + X λ X ρ = Y λ Y ρ. So Y ρ Y λ Y ρ. This contradicts the assumption that Y λ Y ρ has innite codimension in A. Therefore AY = Y, and, similarly, we see that Y A = Y. Thus Y is indeed a simple A-bimodule. Consider the following two cases.

5 BANACH ALGEBRAS WITH FINITE-DIMENSIONAL RADICAL 5 Case 1: Y λ = Y ρ = Y. Then A/Y = M n for some n N. Let {p ij : i, j = 1,..., n} A be such that { p ij + Y : i, j = 1,..., n } corresponds to the standard basis of M n. Then each element a in A can be expressed uniquely as (3.1) a = α ij p ij + u, i,j n where α ij C for 1 i, j n and u Y. Let F 1 be the linear span of F {p ij p st δ js p it : 1 i, j, s, t n}. Then F 1 is a nite-dimensional subspace of Y. Let T : Y Y be a discontinuous A-mod-A homomorphism which is zero on (Y ) 2 + F 1, the existence of which is guaranteed by Lemma 3.1. Dene D : A Y by D(a) = T (u) for each a A written in the form specied in (3.1). It is routine to check that D is a derivation satisfying the requirement. Case 2: Y λ Y ρ. Since Y λ and Y ρ are distinct maximal ideals in A (by the simplicity of Y again), we have A = Y λ + Y ρ, and so Y ρ /Y = A/Yλ = Ml and Y λ /Y = A/Yρ = Mr, for some l, r N. Choose {p ij : i, j = 1,..., l} Y ρ and {q ij : i, j = 1,..., r} Y λ such that { p ij + Y : i, j = 1,..., l } and { q ij + Y : i, j = 1,..., r } correspond to the standard bases of M l and M r, respectively. Then we see that each element a in A can be expressed uniquely as (3.2) a = α ij p ij + β st q st + u, i,j l s,t r where α ij C for 1 i, j l, β st C for 1 s, t r, and u Y. Let F 1 be the linear span of F {p ij q st : 1 i, j l, and 1 s, t r} {p ij p st δ js p it : 1 i, j, s, t l} {q ij q st δ js q it : 1 i, j, s, t r}. Then F 1 is a nite-dimensional subspace of Y. By Lemma 3.1, we can nd a discontinuous A-mod-A homomorphism T : Y Y which is zero on Y λ Y ρ + F 1. Dene D : A Y by D(a) = T (u) for each a A written in the form specied in (3.2). Again, it can be checked that D satises our requirement. Theorem 3.3. Let A be a Banach algebra with nite-dimensional radical R such that A/(R + R λ R ρ ) is unital. Suppose that there exists a minimal radical ideal K such that K λ K ρ has innite codimension in A. Then: (i) there exists a discontinuous derivation D : A R with D 2 = 0; (ii) there exists a discontinuous automorphism on A. In particular, A has a complete algebra norm which is not equivalent to the given norm. Proof. (i) Since K is a minimal radical ideal, we have RK = KR = {0}. Thus K can be considered as a minimal A/R-bimodule. We see that A/R and K satisfy the hypothesis of Theorem 3.2, so there exists a discontinuous derivation D 0 : A/R

6 6 HUNG LE PHAM K. Set D = D 0 π, where π : A A/R is the quotient map. Then D : A K is a discontinuous derivation with D(K) = {0}. (ii) Let D : A A be a discontinuous derivation such that D 2 = 0. Denote by id A the identity map on A. Set θ = id A + D. It is easy to see that θ is a discontinuous automorphism on A; its inverse map is ϕ = id A D. The map a θ(a), A R +, is an algebra norm on A, easily seen to be complete, and is not equivalent to the given norm. 4. Banach algebras having a unique complete norm topology In this section, we shall prove the mentioned conjecture for separable Banach algebras with nite-dimensional radical under some additional hypothesis. In particular, we shall prove the conjecture in the case where A is separable and A/(rad A) is commutative. This case was proved in [3] under the additional hypothesis that either rad A is central or (rad A) 2 = {0}. Part of our argument is an extension of the argument in [3]. Our approach is to utilize separating spaces as well as continuity ideals to prove the continuity of certain homomorphisms. Let A be an algebra. Recall that a composition series of an A-bimodule X is a chain X = X 0 X 1 X 2 X s 1 X s = {0} of submodules of X such that, for each 1 i s, the A-bimodule X i 1 /X i is minimal. We introduce the following concept. Denition 4.1. An admissible series for an A-bimodule X is a chain X = X 0 X 1 X 2 X s 1 X s = {0} of submodules of X such that the ideals (X i 1 /X i ) λ (X i 1 /X i ) ρ (1 i s) are each of nite codimension in A. Lemma 4.2. Let A be an algebra, and let X be an A-bimodule. Suppose that X has a composition series. Then the following are equivalent: (a) X has an admissible series; (b) X has an admissible series which is a composition series; (c) each composition series of X is an admissible series. When one of the above three conditions hold then each submodule of X has an admissible series. Proof. The implication (a) (b) follows by rening the given admissible series to obtain a composition series, which is easily seen to be admissible. The implication (b) (c) follows from the JordanHölder theorem [1, pp ]. The remaining implication (c) (a) is obvious. The nal assertion then follows from (c). Note that a nite-dimensional A-bimodule X always has a composition series. Lemma 4.3. Let A and B be commutative algebras, and let X be a nite-dimensional A-B-bimodule. Then X = n i=1 X i where X i are submodules of X (1 i n) such that, for each 1 i n, we have (i) either A/(X i ) λ is radical or both A/(X i ) λ is local and there exists a A such that a x = x (x X i ), and (ii) either B/(X i ) ρ is radical or both B/(X i ) ρ is local and there exists b B such that x b = x (x X i ).

7 BANACH ALGEBRAS WITH FINITE-DIMENSIONAL RADICAL 7 Proof. Since X is nite-dimensional, we can write X = n i=1 X i, where n N and each X i is an A-B-submodule of X with no non-trivial direct summand (in A-mod-B). Fix i with 1 i n. Then A i = A/(X i ) λ and B i = B/(X i ) ρ are nite-dimensional, commutative algebras which act faithfully on the left and right of X i, respectively. By the Wedderburn structure theorem (see, for example, [2, 1.5.8]), there exists an orthogonal set {p 1,..., p k } (which may be empty) of non-zero idempotents in A i such that k A i = Cp j rad A i. It then follows that X i = k j=1 j=1 ( p j X i 1 k j=1 p j ) X i, where p j X i (1 j k) and ( 1 k j=1 p j) Xi are easily seen to be A-B-bimodules, and each p j X i is non-zero. Therefore, by the assumption on X i, we must have either k = 0 or k = 1. Thus A i is either radical or local; in the later case we must have p 1 x = x (x X). The assertion (ii) is proved similarly. Lemma 4.4. Let A be an algebra, and let I 1, I 2, J 1, J 2 be nite codimensional ideals in A such that there exists m N with J m 1 I 1 and J m 2 I 2. Suppose that J 1 J 2 has nite codimension in A. Then I 1 I 2 has nite codimension in A. Proof. This is [3, Lemma 1.6]. Proposition 4.5. Let A be an algebra, and let X be a nite-dimensional A- bimodule, such that A/X is commutative. Then the following are equivalent: (a) for each minimal sub-bimodule Y of X, the ideal Y λ Y ρ has nite codimension in A; (b) there are sub-bimodules X 1,..., X n of X such that X = n i=1 X i, and such that (X i ) λ (X i ) ρ has nite codimension in A; (c) X has an admissible series. Proof. The implication (b) (c) is obvious, and the implication (c) (a) follows from Lemma 4.2. We shall prove the implication (a) (b). Suppose that (a) holds. Set B = A/X, so that B is a commutative algebra. Let X = n i=1 X i be the decomposition as in Lemma 4.3 (working in B-mod-B). Returning to A-mod-A, we see that the decomposition X = n i=1 X i still satises the conditions (i) and (ii) in Lemma 4.3. Fix 1 i n. Let Y be a minimal sub-bimodule of X i. First, if A/(X i ) λ is radical, then so is Y λ /(X i ) λ. Otherwise, if A/(X i ) λ is local, then, by Lemma 4.3, there exists a A such that a x = x (x X i ), and so a / Y λ. This implies that Y λ is a proper ideal in A, and therefore, by locality, Y λ /(X i ) λ is, again, radical. Thus, in both cases, Y λ /(X i ) λ is radical and nite-dimensional. It follows that (Y λ ) l (X i ) λ for some l N. Similarly, there exists r N such that (Y ρ ) r (X i ) ρ. By hypothesis, we have that Y λ Y ρ is of nite codimension in A. Hence, by Lemma 4.4, the ideal (X i ) λ (X i ) ρ also has nite codimension in A. This completes our proof.

8 8 HUNG LE PHAM Denition 4.6. Let A be an algebra with a nite-dimensional radical R. We say that A satises the A-property if the A-bimodule R R has an admissible series. The implication (a) (b) of the following proposition, in the case where R 2 = {0}, was implicit in the proof of [3, Theorem 3.13] (in fact, the argument there can be modied to work in the general case of (a) (b); our proof is dierent but still follows the idea of [3]). Proposition 4.7. Let A be an algebra with nite-dimensional radical R such that A/R is commutative. Then the following are equivalent: (a) for each minimal radical ideal K in A, the ideal K λ K ρ has nite codimension in A; (b) there are ideals R 1,..., R n in A such that R R = n i=1 R i, and such that (R i ) λ (R i ) ρ has nite codimension in A; (c) A satises the A-property. Proof. This is a corollary of Proposition 4.5, since each minimal radical ideal K is contained in R R. Lemma 4.8. Let A be a separable Banach algebra, and let E and F be closed linear subspaces of A with EF having nite codimension in A. Then EF is closed, and there exist a constant C > 0 and m N such that, for each a EF, there exist (x i ) m i=1 E and (y i) m i=1 F such that m m a = x i y i and x i y i C a. i=1 Proof. This is [3, Lemma 3.1] (see also [2, ] and [10]). i=1 The next result is an extension of Corollaries 3.7, 3.8, and 3.9 in [3]. However, we cannot use an induction scheme (like the one provided by [3, Theorem 3.6]) to prove the next theorem as was the case for those corollaries. Theorem 4.9. Let A be a separable Banach algebra with nite-dimensional radical. Suppose that A satises the A-property. Then each complete algebra norm on A is equivalent to the given norm. Proof. The radical of A is denoted by R. Consider another complete algebra norm on A. We need to prove that the identity map i : (A, ) (A, ) is continuous. By Johnson's automatic continuity theorem for epimorphisms, we have S(i) R. For each a R, since i is continuous on R a nite-dimensional ideal we have a I λ (i) I ρ (i). Thus, indeed S(i) R R. Assume towards a contradiction that S(i) {0}. Then there exists an ideal K which is maximal among the ideals properly contained in S(i). Since K is nitedimensional, K is closed in both topologies. Consider the map θ : (A, ) (A/K, ) induced by i. Then S(θ) = S(i)/K, and I λ (θ) = {a A : θ(a)s(θ) = {0}} = (S(i)/K) λ, and, similarly, I ρ (θ) = (S(i)/K) ρ. So I λ (θ) and I ρ (θ) are closed (in both topologies). Then, since the bilinear map T : (x, y) θ(xy), I λ (θ) I ρ (θ) A/K is separately continuous, it follows that T is bounded.

9 BANACH ALGEBRAS WITH FINITE-DIMENSIONAL RADICAL 9 By the A-property of A and by Lemma 4.2, the ideal I λ (θ)i ρ (θ) is of nite codimension in A. Then, by Lemma 4.8, we see that θ is bounded on I λ (θ)i ρ (θ), which is, by the same lemma, a closed, nite-codimensional subspace of A. Thus θ is continuous on the whole of A, contradicting the fact that S(θ) = S(i)/K {0}. Hence, we must have S(i) = {0}, implying the continuity of i. The following theorem resolves a question that was left open in [3]. Theorem Let A be a separable Banach algebra with nite-dimensional radical R such that A/R is commutative. Suppose that K λ K ρ has nite codimension in A for each minimal radical ideal K in A. Then each complete algebra norm on A is equivalent to the given norm. Proof. This follows from Proposition 4.7, and Theorem 4.9. Remark. It can be seen that Theorem 4.10 and Proposition 4.7 still hold with the much weaker hypothesis that `A / R + R is commutative' instead of the hypothesis that `A/R is commutative' (with the same proofs). Note that the algebra A / R + R is only nite-dimensional. 5. The continuity of derivations Recall that we are interested in two classes of derivations: derivations on algebras with nite-dimensional radical, and derivations from algebras into nitedimensional bimodules. The proofs in this section are similar to those in Ÿ4. Theorem 5.1. Let D be a derivation from a separable Banach algebra A into a Banach A-bimodule X. Suppose that S(D) is nite-dimensional and has an admissible series as an A-bimodule. Then D is continuous. Proof. Assume towards a contradiction that S(D) {0}. Then there exists an A- bimodule Y which is maximal among the proper A-bimodules contained in S(D). Consider the map D 1 : A X/Y induced by D. Then D 1 is a derivation and S(D 1 ) = S(D)/Y. So I λ (D 1 ) = {a A : a S(D 1 ) = {0}} = (S(D)/Y ) λ, and, similarly, I ρ (D 1 ) = (S(D)/Y ) ρ. Continuing as in the proof of Theorem 4.9, we obtain a contradiction. Hence S(D) = {0}, implying the continuity of D. Corollary 5.2. Let A be a separable Banach algebra, and let X be a nite-dimensional Banach A-bimodule. Suppose that X has an admissible series as an A-bimodule. Then derivations from A into X are continuous. Proof. This follows from Proposition 4.2 and Theorem 5.1. Corollary 5.3. Let A be a separable Banach algebra, and let X be a nite-dimensional Banach A-bimodule such that A/X is commutative. Suppose that Y λ Y ρ has - nite codimension in A for each minimal sub-bimodule Y in X. Then all derivations from A into X are continuous. Proof. This follows from Proposition 4.5 and Corollary 5.2.

10 10 HUNG LE PHAM Remark. In [5], a Banach algebra A is constructed whose square A 2 has nite codimension but is not closed. Let X = C with trivial A-bimodule actions. Then A/X = 0. Thus A and X satises the hypothesis of Corollary 5.3 except for the separability of A. However, any discontinuous linear functional which is zero on A 2 is a discontinuous derivation from A into X. This shows that the separability condition in the hypothesis of the previous corollary and theorem is necessary. We shall see later that the commutativity condition is also necessary (Theorem 6.1). (Un)fortunately, the algebra A is neither commutative nor has nite-dimensional radical. So we cannot say anything about the separability hypothesis in other results. Now we consider derivations on Banach algebras. It is proved in [8] that, for a semisimple Banach algebra A, each derivation D on A is continuous, which means that S(D) = {0} = rad A. Although it is still open whether S(D) rad A holds for all Banach algebras A and all derivations D on A, we have the following. Lemma 5.4. Let A be a Banach algebra with radical R such that R 2 is closed and has nite codimension in R. Then, for each derivation D on A, we have S(D) R. Proof. We can assume that A is unital with the identity denoted by e. For each primitive ideal P in A, dene π P to be the natural projection from A onto A/P. Assume towards a contradiction that S(D) R. Then S(D) P 1 for some primitive ideal P 1 in A, which is equivalent to the discontinuity of π P1 D. By [8, Theorem 3.3], we have π P D is continuous for all except nitely many primitive ideals P i (1 i n), and each P i has nite codimension in A (hence is maximal). Now, let P 0 be the intersection of all the primitive ideals P dierent from P i (1 i n). Then ( ) P 0 /R n / n = P 0 + P i P i, i=1 which is nite-dimensional, and so R 2 is closed and has nite codimension in P 0. We have S(D) P 0, but S(D) P 1, so that P 0 P 1. Since P 1 is a maximal ideal in A, we see that P 0 + P 1 = A. In particular, we have e = p 0 + p 1, where p 0 P 0 and p 1 P 1. Then e = e 2 = p + p 2 1, where p = p p 0 p 1 + p 1 p 0 = p 0 + p 1 p 0 P 0. Thus, for each a A, we have π P1 D(a) = π P1 D(ea) = π P1 D (pa). This implies that π P1 D is discontinuous on P 0. However, it easily seen that π P1 D is zero on R 2, a closed subspace of P 0 of nite codimension, and so π P1 D is continuous on P 0, a contradiction. Corollary 5.5. Let A be a separable Banach algebra with nite-dimensional radical. Suppose that A satises the A-property. Then derivations on A are continuous. Proof. The radical of A is denoted by R. Let D : A A be a derivation. By Lemma 5.4, we have S(D) R. As in the proof of Theorem 4.9, we see that S(D) R R. Hence, the result follows from Theorem 5.1. Corollary 5.6. Let A be a separable Banach algebra with nite-dimensional radical R such that A / R + R is commutative. Suppose that K λ K ρ has nite codimension in A for each minimal radical ideal K in A. Then each derivation on A is continuous. Λ i=1

11 BANACH ALGEBRAS WITH FINITE-DIMENSIONAL RADICAL 11 Remark. The results in this section hold without modication for a more general class of operators, the intertwining operators. Let A be a Banach algebra. A linear operator T from a Banach A-bimodule X into a Banach A-bimodule Y is said to be intertwining over A if, for each a A, the maps x T (a x) a T (x) and x T (x a) T (x) a, both from X to Y, are continuous; see, for instance, [2, 2.7.1]. Thus each derivation from A into a Banach A-bimodule is an intertwining operator. 6. The general case of dimension at least two In this section, we present a counter-example to the main conjecture in [3] mentioned earlier. For a separable Banach algebra A with 1-dimensional radical R, the only minimal radical ideal is R itself. We see that the conjecture holds in this case; A has a unique complete norm if (and only if) R λ R ρ has nite codimension in A, by Theorem 4.9 (and 3.3) or by [3, Corollary 3.9]. However, for greater dimensions, even for dimension 2, the problem becomes more complicated. In [3, Example 5.5], a separable, unital Banach algebra with 2-dimensional radical was successfully constructed which satises the hypothesis of the conjecture, but not the hypothesis of any result in [3] (nor does it satisfy the hypothesis of our results); the construction is rather involved. Yet it was proved directly that this algebra has a unique complete norm. Our example will be a separable, unital Banach algebra with 2-dimensional radical (showing that the conjecture fails for each dimension greater than 1). Following the approach of our previous construction, we shall rst construct similar examples for the problem of derivations from Banach algebras into 2-dimensional Banach modules, and then for the problem of derivations on Banach algebras with 2-dimensional radical. Theorem 6.1. There exist a separable, semisimple, unital Banach algebra A and a 2-dimensional, unital Banach A-bimodule X such that both the following hold: (a) for each minimal sub-bimodule Y in X, the ideal Y λ Y ρ has nite codimension in A; (b) there exist a discontinuous derivation D from A into X. The following provides a counter-example to the conjecture. Corollary 6.2. There exists a separable, unital Banach algebra A with 2-dimensional radical such that all the following hold: (a) for each minimal radical ideal K, the ideal K λ K ρ has nite codimension in A; (b) there exists a discontinuous derivation on A; (c) there exists a discontinuous automorphism on A. In particular, there exists a complete algebra norm on A which is not equivalent to the given norm. Proof. Let A, X and D : A X be as in Theorem 6.1. Set A = A X. Then, with l 1 -norm and with product given by (a x)(b y) = ab (a y + x b) (a, b A, x, y M),

12 12 HUNG LE PHAM A is a separable, unital Banach algebra with rad A = X. We extend D by linearity to the whole of A by mapping X to 0. Then we obtain a discontinuous derivation on A. Dene θ : a x a (Da + x), A A. Then it is easily seen that θ is a discontinuous automorphism on A. The map a x θ(a x), A R +, gives a complete algebra norm, say, on A, and is not equivalent to the given norm. Finally, condition (a) follows from Theorem 6.1 (a). Proof of Theorem 6.1. Denote by M u 2 the algebra of upper triangular 2 2-matrices (over C). Let B be a separable, commutative, semisimple Banach algebra such that B 2 has innite codimension in B (many such examples exist, for example, l 1 with pointwise product). Denote by M 2 (B) the Banach algebra of 2 2-matrices with coecients in B. Then M 2 (B) is also semisimple and separable. Set A generic element of A has the form ( α β 0 γ A = M u 2 M 2 (B). ) ( s u t v where α, β, γ C and s, t, u, v B. Then A is an algebra with pointwise addition and with product being the matrix multiplication, so that M u 2 and M 2 (B) can be naturally identied with subalgebras of A. Further, ), M u 2 M 2 (B) M 2 (B) and M 2 (B) M u 2 M 2 (B). Giving A the l 1 -norm, we obtain a separable, unital Banach algebra (with the identity given by the identity of M u 2), and M 2 (B) is a closed ideal of A. We see that A has the following two obvious characters: ( ) ( ) s u s u ϕ : ( α β 0 γ ) t v α and ψ : ( α β 0 γ We claim that A is semisimple. Indeed, let ( ) ( ) α β s u a = rad A. 0 γ t v ) Then, since ϕ(a) = ψ(a) = 0, we must have α = γ = 0. We next see that ( ) ( ) ( ) ( ) 1 0 s a = and a =, 0 0 t t v t v γ. so the two elements on the right are in M 2 (B) rad A = rad M 2 (B), and so s = t = v = 0. Now, let w B. Multiplying a on the left by ( ) 0 0, w 0 we obtain ( βw + wu )

13 BANACH ALGEBRAS WITH FINITE-DIMENSIONAL RADICAL 13 which, again, must be in the radical of M 2 (B), and so βw + wu = 0 (w B). Since B cannot have a right identity, we must have β = 0, and then, since B is semisimple, we see that u = 0. Thus, we have proved that rad A = {0}. Next, we claim that Mϕ 2 has nite codimension in A, where M ϕ is the kernel of ϕ. Indeed, for each s, t, u, v B, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) s u u = + +, t v 0 1 t v s 0 so that M 2 (B) Mϕ. 2 In fact, we can check that Mϕ 2 = M ϕ. Consider X = C 2 as a unital Banach A-bimodule by dening the module multiplications as M 2 (B) X = X M 2 (B) = {0} and the multiplications by M u 2 as ( ) ( ) ( ) ( ) ( ) ( ) α β ζ αζ + βη ζ α β αζ = and =, 0 γ η γη η 0 γ αη for α, β, γ, ζ, η C. Then X has exactly one minimal A-mod-A submodule, namely {( ) } ζ Y = : ζ C. 0 We see that Y λ = Y ρ = M ϕ, so that Y λ Y ρ = Mϕ 2 has codimension 1 in A. Now, since B 2 has innite codimension in B, there exists a discontinuous linear functional λ on B such that λ is zero on B 2. Dene a discontinuous linear map D : A X by ( α β D : 0 γ It can be checked that D is a derivation. ) ( s u t v ) 7. Conclusion ( λ(s) λ(t) It remains open to determine conditions which are both necessary and sucient for a (separable) Banach algebra with nite-dimensional radical to have a unique complete norm. We have seen that the necessary condition (in the unital case) that for each minimal radical ideal K, the ideal K λ K ρ has nite codimension is not sucient. Our construction is, however, not systematic enough to provide a clue on nding and proving additional necessary conditions. Acknowledgements The author wishes to thank his supervisor, Professor H. Garth Dales, for much advice and support. This work was undertaken while the author was studying for a PhD at the University of Leeds, supported nancially by the ORS, the School of Mathematics, and the University of Leeds. References [1] E. A. Behrens, Ring theory, Translated from the German by Clive Reis, Pure and Applied Mathematics, Vol. 44 (Academic Press, New York, 1972). [2] H. G. Dales, Banach algebras and automatic continuity, volume 24 of London Mathematical Society Monographs. New Series (The Clarendon Press Oxford University Press, Oxford, 2000). [3] H. G. Dales and R. J. Loy, `Uniqueness of the norm topology for Banach algebras with nite-dimensional radical', Proc. London Math. Soc. (3) 74 (1997), [4] H. G. Dales and J. P. McClure, `Continuity of homomorphisms into certain commutative Banach algebras', Proc. London Math. Soc. (3) 26 (1973), ).

14 14 HUNG LE PHAM [5] P. G. Dixon, `Nonseparable Banach algebras whose squares are pathological', J. Functional Analysis 26 (1977), [6] B. E. Johnson, `The uniqueness of the (complete) norm topology', Bull. Amer. Math. Soc. 73 (1967), [7], `The Wedderburn decomposition of Banach algebras with nite dimensional radical', Amer. J. Math. 90 (1968), [8] B. E. Johnson and A. M. Sinclair, `Continuity of derivations and a problem of Kaplansky', Amer. J. Math. 90 (1968), [9] T. Y. Lam, A rst course in noncommutative rings, volume 131 of Graduate Texts in Mathematics (Springer-Verlag, New York, 1991). [10] R. J. Loy, `Multilinear mappings and Banach algebras', J. London Math. Soc. (2) 14 (1976), [11], `The uniqueness of norm problem in Banach algebras with nite-dimensional radical', in: Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), Lecture Notes in Math. 975 (Springer, Berlin, 1983) pp [12] V. M. Zinde, `The property of uniqueness of the norm for commutative Banach algebras with nite-dimensional radical', Vestnik Moskov. Univ. Ser. I Mat. Meh. 25 (1970), 38. Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England address: hung@maths.leeds.ac.uk