THE NONCOMMUTATIVE SINGER–WERMER ... - CiteSeerX

J. London Math. Soc. 66 (2002) 710–720

Cf 2002

London Mathematical Society DOI: 10.1112/S0024610702003721

THE NONCOMMUTATIVE SINGER–WERMER CONJECTURE AND φ-DERIVATIONS ˇ M. BRESAR and A. R. VILLENA Abstract The question of when a φ-derivation on a Banach algebra has quasinilpotent values, and how this question is related to the noncommutative Singer–Wermer conjecture, is discussed.

1. Introduction In 1955, Singer and Wermer [10] proved that a continuous derivation on a commutative Banach algebra has its range in the (Jacobson) radical of the algebra. Simultaneously they conjectured that the hypothesis of continuity is superfluous. In 1969, Johnson [4] proved that this is true when the algebra is semisimple, but it took more than 30 years before this classical Singer–Wermer conjecture was finally confirmed for any commutative Banach algebra by Thomas [11]. There are many ways to extend the Singer–Wermer theorem to noncommutative algebras. One of them was noted by Sinclair [9]: every continuous derivation of a (possibly noncommutative) Banach algebra leaves primitive ideals of the algebra invariant. The question of whether the continuity assumption is redundant in this result is probably one of the most challenging unsolved problems in general Banach algebra theory. It is usually referred to as the noncommutative Singer–Wermer conjecture. It is known that for each derivation there can be only finitely many noninvariant primitive ideals, each of which is of finite codimension [12], but whether such derivations and ideals actually exist is still an open question. In this paper we approach this question via the so-called φ-derivations. These maps have been studied extensively in pure algebra. They were also implicitly used in a somewhat related recent paper [1] dealing with (Jordan) automorphisms of Banach algebras, but as far as we know they have not been treated systematically in Banach algebra theory. They are defined as follows. If φ is an automorphism of an algebra A, then a linear map ∆ : A −→ A is called a φ-derivation if it satisfies ∆(xy) = ∆(x)φ(y) + x∆(y)

for all x, y ∈ A

(we remark that some authors use a slightly different definition so that ∆(x)y + φ(x)∆(y) appears on the right hand side). Of course, 1A -derivations (where 1A is the identity on A) are just derivations. Further, for any automorphism φ, φ − 1A is a φ-derivation. Thus the concept of a φ-derivation may be viewed as an extension and unification of both the concepts of a derivation and an automorphism. A more general example of a φ-derivation is given by ∆ : x 7−→ cφ(x) − xc, where c is a fixed Received 23 May 2001; revised 1 March 2002. 2000 Mathematics Subject Classification 47B47, 46H40. The first author is supported by the Ministry of Science of Slovenia. The second author is supported by DGICYT grant PB98-1358.

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element. This type of map obviously extends the concept of an inner derivation. Our final example will be of particular importance for us. Given any derivation D of a unital algebra A and an invertible element b ∈ A, the map ∆ : x 7−→ D(x)b is a φb derivation, where φb : x 7−→ b−1 xb is an inner automorphism. Thus every derivation generates various φ-derivations with rather tractable (namely inner) automorphisms φ. This already suggests that general results on φ-derivations (especially in unital Banach algebras which have plenty of invertible elements) may give useful corollaries concerning derivations. On the other hand, when problems on derivations where the spectrum (implicitly) appears (such as the noncommutative Singer–Wermer conjecture) are studied, it may turn out to be useful to involve automorphisms, since they, unlike the derivations, behave extremely well with respect to the invertibility. We extend several existing results on derivations to φ-derivations. In particular, Theorem 3.1, Corollary 3.4, Theorem 4.2, Corollary 4.3 and Theorem 5.1 are well known in the case when ∆ is a derivation. We have taken the liberty of using (or, more accurately, appropriately modifying) various arguments from the papers on derivations in the references in this paper, notably from [12], without always specifically mentioning this. In the last section, we present several assertions on φderivations whose truthfulness would imply the truthfulness of the noncommutative Singer–Wermer conjecture. Some of them, and in particular in Theorem 5.2(ii), are known to be true for derivations [12], but unfortunately we have not been able to prove them for φ-derivations. As a final corollary to our investigations, we show that the noncommutative Singer–Wermer conjecture is equivalent to the conjecture that a derivation on a unital Banach algebra A may take invertible values only on such elements a ∈ A for which the two-sided ideal of A generated by a equals A. This is trivially true for inner derivations, and we show that it is also true for continuous derivations (Corollary 3.7). On the other hand, this conjecture obviously fails in some normed algebras which are not complete (such as, for instance, C∞ [a, b]). 2. φ-derivations In this section we gather together a few useful elementary observations on φderivations. We omit the proof of the first result since it is just a straightforward verification. Theorem 2.1. Let A be an algebra, let φ be an automorphism of A, and let ∆ be a φ-derivation on A. Set Λ0,n = φn ,

Λn,n = ∆n ,

Λk,n+1 = φΛk,n + ∆Λk−1,n

k = 1, . . . , n

for each n ∈ N. If a, b ∈ A and n ∈ N, then n X ∆n−j (a)Λj,n (b). ∆n (ab) = j=0

Accordingly, ∆n (ab) =

n   X n j=0

j

∆n−j (a)φn−j (∆j (b)),

provided that [∆, φ] = 0. Corollary 2.2. Let A be an algebra, let φ be an automorphism of A, and let ∆ be

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a φ-derivation on A such that [∆, φ] = 0. Suppose that a ∈ A is such that ∆2 (a) = 0. Then ∆n (aφ−1 (a) . . . φ−(n−1) (a)) = n!∆(a)n for each n ∈ N. Proof. To shorten notation, let cn stand for aφ−1 (a) . . . φ−(n−1) (a) and note that cn+1 = aφ−1 (cn ). If ∆n (cn ) = n!∆(a)n , then it is easily seen that ∆n+1 (cn ) = n!∆(∆(a)n ) = 0, and so, using Theorem 2.1, we get ∆n+1 (cn+1 ) = ∆n+1 (aφ−1 (cn ))  n+1  X n+1 ∆n+1−j (a)φn+1−j (φ−1 (∆j (cn ))) = j j=0

= (n + 1)∆(a)∆n (cn ) + aφ−1 (∆n+1 (cn )) = (n + 1)∆(a)n!∆(a)n .

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We remark that, appropriately modifying the proof of [1, Lemma], we can get another, more direct, proof of Corollary 2.2 which does not use Theorem 2.1. Theorem 2.3. Let A be an algebra, let φ be an automorphism of A, and let ∆ be a φ-derivation on A. Let I be a two-sided ideal of A, and suppose that both φ and [∆, φ] leave I invariant. Then ∆n (a1 a2 . . . an ) ∈ n!∆(a1 )∆(φ(a2 )) . . . ∆(φn−1 (an )) + I = n!∆(a1 )φ(∆(a2 )) . . . φn−1 (∆(an )) + I for all n ∈ N and a1 , . . . , an ∈ I. Proof.

We begin by observing that ∆(φk (a)) ∈ φk (∆(a)) + I

for all k ∈ N and a ∈ I. This is of course a straightforward consequence of the identity X φi [∆, φ]φj ∆φk = φk ∆ + [∆, φk ] = φk ∆ + i+j=k−1

together with the hypotheses of the theorem. Thus it suffices to prove that ∆n (a1 a2 . . . an ) ∈ n!∆(a1 )∆(φ(a2 )) . . . ∆(φn−1 (an )) + I

(2.1)

for all n ∈ N and a1 , . . . , an ∈ I. Assuming that (2.1) holds for n, we shall prove it for n + 1. Let a1 , . . . , an+1 ∈ I. We have ∆n+1 (a1 . . . an an+1 ) = ∆n (∆(a1 . . . an an+1 )) ! n+1 X a1 . . . ak−1 ∆(ak )φ(ak+1 ) . . . φ(an+1 ) = ∆n =

n X k=1

k=1

∆n (a1 . . . ak−1 (∆(ak )φ(ak+1 ))φ(ak+2 ) . . . φ(an+1 ))

+∆n (a1 . . . an−1 (an ∆(an+1 ))),

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and so, by our assumption, ∆n+1 (a1 . . . an an+1 ) n X n!∆(a1 ) . . . ∆(φk−1 (∆(ak )φ(ak+1 ))) . . . ∆(φn−1 (φ(an+1 ))) ∈ k=1

+n!∆(a1 ) . . . ∆(φn−2 (φ(an−1 )))∆(φn−1 (an ∆(an+1 ))) + I.

(2.2)

On the other hand, we have ∆(φk−1 (∆(ak )φ(ak+1 ))) = ∆(φk−1 (∆(ak ))φk (ak+1 )) = ∆(φk−1 (∆(ak )))φk+1 (ak+1 ) + φk−1 (∆(ak ))∆(φk (ak+1 )) ∈ φk−1 (∆(ak ))∆(φk (ak+1 )) + I = ∆(φk−1 (ak ))∆(φk (ak+1 )) + I.

(2.3)

Similarly, we can see that ∆(φn−1 (an ∆(an+1 ))) ∈ ∆(φn−1 (an ))∆(φn (an+1 )) + I. From (2.2), (2.3) and (2.4), we obtain (2.1) for n + 1.

(2.4) q

Remark 2.4. Note that, in the case when φ is an inner automorphism, [∆, φ] leaves each ideal I invariant. Trivially, the same holds true for φ, and so the requirements of Theorem 2.3 are automatically satisfied. 3. Continuous φ-derivations Theorem 3.1. Let A be a Banach algebra, let φ be a continuous automorphism of A, and let ∆ be a continuous φ-derivation such that [∆, φ] = 0. Suppose that a ∈ A is such that ∆2 (a) = 0. Then ∆(a) is quasinilpotent. Proof. Write a1 = a and an+1 = aφ−1 (an ) for each n ∈ N. By induction, it is easily seen that kan k 6 kφ−1 kn−1 kakn . This fact, together with Corollary 2.2, gives k∆(a)n k1/n = k(n!)−1 ∆n (an )k1/n 6 (n!)−1/n k∆kkan k1/n 6 (n!)−1/n k∆kkφ−1 k1−1/n kak for each n ∈ N, and this latter term tends to 0 as n → ∞.

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Remark 3.2. (a) It should be noted that in the case when [∆, φ] 6= 0, Theorem 3.1 fails to hold true. Indeed, set A = M2 (C),       0 −1 0 1 1 0 a= , b= , e= , 1 0 1 0 0 0 and let ∆ be the φb -derivation on A defined by ∆(x) = [x, e]b for each x ∈ A. Then ∆(a) = 1 is not quasinilpotent; however, ∆2 (a) = 0. (b) One possible way of stating the Kleinecke–Shirokov theorem [6, 8] is the following. If D is a continuous derivation on a Banach algebra A and a ∈ A is such that D2 (a) = 0, then D(a) is quasinilpotent (see [7], for example). Of course this is Theorem 3.1 when φ = 1A .

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(c) Let A be a Banach algebra, and let a, b, c ∈ A with b invertible. Considering the φb -derivation ∆ on A defined by ∆(a) = cφb (a) − ac for each a ∈ A, we see that Theorem 3.1 yields the following conclusion. If [b, c] = 0 and c2 ab2 +b2 ac2 = 2bcacb, then b−1 cab−ac is quasinilpotent. In the case when b = 1, this is the classical version of the Kleinecke–Shirokov theorem. (d) Considering the φ-derivation φ−1A , we get another corollary to Theorem 3.1. If φ is a continuous automorphism of a Banach algebra A and a ∈ A is such that 1 −1 2 (φ(a) + φ (a)) = a, then φ(a) − a is quasinilpotent. However, this is already known (see [1, 13] for more general results). Moreover, passing to the quotient algebra of A by its radical, it is easy to see that the assumption of continuity of φ is redundant (cf. [1]). (e) Let us point out an alternative proof of Theorem 3.1. We hope that its idea, that is, reducing the problem on φ-derivations of A to the problem on (ordinary) derivations on the Banach algebra of all continuous linear operators L(A) on A, may prove useful elsewhere. Assume that the conditions of Theorem 3.1 hold. An easy computation shows that L(A) also becomes a Banach algebra A when endowed with the product S ∗ T = Sφ−1 T and norm |S| = kφ−1 kkSk for all S, T ∈ A. Let D be defined on A by D(T ) = [∆, T ] for each T ∈ A. Since [∆, φ] = 0, we see that D is a derivation on A. For any b ∈ A, we denote by Lb ∈ A the operator Lb : x −→ bx. Note that D(La ) = L∆(a) φ and D2 (La ) = L∆2 (a) φ2 = 0. The Kleinecke–Shirokov theorem now shows that D(La ) = L∆(a) φ is quasinilpotent. It is a simple matter to show that ∆(a)n+1 = (L∆(a) φ)∗n (φ−1 (∆(a))) for each n ∈ N. Thus k∆(a)n+1 k = k(L∆(a) φ)∗n (φ−1 (∆(a)))k 6 k(L∆(a) φ)∗n kkφ−1 (∆(a))k = kφ−1 k−1 |(L∆(a) φ)∗n |kφ−1 (∆(a))k which shows that k∆(a)n+1 k1/n+1 6 kφ−1 k−1/n+1 kφ−1 (∆(a))k1/n+1 |(L∆(a) φ)∗n |1/n+1 for each n ∈ N and hence that lim k∆(a)n+1 k1/n+1 = 0. For an ideal I of an algebra A, we shall write πI for the quotient map from A onto A/I. Theorem 3.3. Let A be a unital Banach algebra, let φ be a continuous automorphism of A, and let ∆ be a continuous φ-derivation on A. Let I be a closed two-sided ideal of A, and suppose that both φ and [∆, φ] leave I invariant. Then πI (∆(a)) is quasinilpotent for each a ∈ I.

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Define an as in the proof of Theorem 3.1. Due to Theorem 2.3, we have πI (∆n (an )) = n!(πI (∆(a)))n ,

and so kπI (∆(a))n k1/n 6 (n!)−1/n k∆kkan k1/n 6 (n!)−1/n k∆kkφ−1 k1−1/n kak for each n ∈ N. Since the latter term tends to 0 as n → ∞, this yields the result. q Corollary 3.4. Let A be a unital Banach algebra, let φ be an inner automorphism of A, and let ∆ be a continuous φ-derivation on A. Then ∆ leaves each primitive ideal of A invariant. Proof. Let P be a primitive ideal of A. According to Theorem 3.3 together with Remark 2.4, πP (∆(P )) consists of quasinilpotent elements. On the other hand, it is easily seen that πP (∆(P )) is a two-sided ideal of A/P and thus πP (∆(P )) ⊂ Rad(A/P ) = 0. That is, ∆(P ) ⊂ P . q Corollary 3.5. Let A be a unital Banach algebra, let φ be a continuous automorphism of A, and let ∆ be a continuous φ-derivation on A. Let I be a two-sided ideal of A such that both φ and [∆, φ] leave I invariant. If ∆(a) = 1 for some a ∈ I, then I = A. Proof. Let I be the closure of I. Of course I is also invariant under φ and [∆, φ]. Theorem 3.3 shows that πI (1) = πI (∆(a)) is quasinilpotent. If I 6= A, then πI (1) is the identity of A/I, and so its spectral radius equals 1, a contradiction. Therefore I = A. However, if the closure of an ideal contains 1, then the same is true for the ideal. Hence I = A. q Corollary 3.6. Let A be a unital Banach algebra, let φ be an inner automorphism of A, and let ∆ be a continuous φ-derivation on A. Suppose that a ∈ A is such that ∆(a) = 1. Then the two-sided ideal of A generated by a equals A. Proof.

Apply Corollary 3.5 and Remark 2.4.

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Corollary 3.7. Let A be a unital Banach algebra and let D be a continuous derivation on A. Suppose that a ∈ A is such that D(a) is invertible. Then the two-sided ideal of A generated by a equals A. Proof. Set b = D(a)−1 and define ∆ : A −→ A by ∆(x) = D(x)b. Then ∆ is a continuous φb -derivation and ∆(a) = 1. The preceding corollary now establishes the result. q 4. Automatic continuity of φ-derivations A key notion when studying the continuity of a linear map Φ from a Banach space X into a Banach space Y is that of the separating space S(Φ) of Φ, which is defined as follows: S(Φ) = {y ∈ Y : there exists (xn ) → 0 in X with (Φ(xn )) → y}.

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The separating space measures the closability of Φ, and the closed graph theorem shows that Φ is continuous if and only if S(Φ) = 0. A standard fact that we shall use throughout this section is that S(ΨΦ) = Ψ(S(Φ)), provided that Ψ is a continuous linear operator from Y into another Banach space Z. For a thorough discussion of the separating space and the basic principles in automatic continuity theory, we refer the reader to [2]. Lemma 4.1. Let A be a Banach algebra, let φ be an automorphism of A, and let ∆ be a φ-derivation on A. If P is a primitive ideal of A of infinite codimension, then πP ∆n is continuous for each n ∈ N. Proof. Suppose that the result is false. Let π be a continuous irreducible representation of A on a Banach space X such that P = ker π. It is easily checked that for every n ∈ N, the map πP ∆n is continuous if and only if the same is true for the map π∆n . We can define k to be the least natural number such that π∆k is discontinuous. We begin by proving that S(π∆k )π(A) ⊂ S(π∆k ). Let (an ) be a sequence in A with lim an = 0 and lim π∆k (an ) = T ∈ L(X), and let a ∈ A. According to Theorem 2.1, we have k X π∆k−j (an )π(Λj,k (φ−k (a))) π∆k (an φ−k (a)) = π∆k (an )π(a) + j=1

for each n ∈ N, and this latter term tends to T π(a) as n tends to infinity, because π∆i is continuous for each i = 0, . . . , k − 1. This clearly yields T π(a) ∈ S(π∆k ), as required. From the construction in [5, Theorem 2.2], which is obtained from [5, Lemma 2.1] by induction, it may be concluded that there exist sequences (an ) in A and (xn ) in X such that π(an . . . a1 )xn 6= 0 and π(an+1 an . . . a1 )xn = 0 for each n ∈ N. Let Rn : A −→ A and Sn : A −→ X be continuous linear operators given by Rn (a) = aφ−k (an ) and Sn (a) = π(a)xn for all a ∈ A and n ∈ N. Due to Theorem 2.1, for every a ∈ A, we have Sn ∆k R1 . . . Rm (a) = π(∆k (aφ−k (am . . . a1 )))xn = (π∆k )(a)π(am . . . a1 )xn +

k X

(π∆k−j )(a)π(Λj,k (φ−k (am . . . a1 )))xn ,

j=1

which equals k X

(π∆k−j )(a)π(Λj,k (φ−k (am . . . a1 )))xn

j=1

in the case when m > n. Since the operator a 7−→

k X j=1

(π∆k−j )(a)π(Λj,k (φ−k (am . . . a1 )))xn

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from A to X is continuous, the gliding hump argument [2] now shows that there exists an n ∈ N such that Sn ∆k R1 . . . Rn is continuous. Consequently, the operator a 7−→ (π∆k )(a)π(an . . . a1 )xn from A into X is continuous, and thus S(π∆k )π(an . . . a1 )xn = 0. From what has been proved at the beginning of this proof, it follows that S(π∆k )π(A)π(an . . . a1 )xn = 0, and hence that S(π∆k ) = 0. Thus π∆k is continuous, a contradiction.

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Theorem 4.2. Let A be a Banach algebra, let φ be an automorphism of A, and let ∆ be a φ-derivation on A. Then the following assertions hold: (i) For every n ∈ N, πP ∆n is continuous for all primitive ideals P of A except possibly finitely many exceptional primitive ideals. (ii) If φ is an inner automorphism, then πP ∆n is continuous for each n ∈ N and for each primitive ideal P of A except possibly finitely many exceptional primitive ideals. Moreover, every exceptional ideal is of finite codimension. Proof. Suppose that the first assertion in the theorem is false. Let k be the least natural number for which this assertion fails to be true. Due to Lemma 4.1, there exists a sequence (Pn ) of pairwise different exceptional ideals of A which must necessarily be of finite codimension such that πPn ∆k is discontinuous and πPn ∆j is continuous for all n ∈ N and j < k. By [5, Lemma 3.2], there exists a sequence (an ) in A such that πPn (am ) = 0 whenever m > n and πPn (am ) is invertible if m 6 n. Let Rn : A −→ A and Sn : A −→ A/Pn be continuous linear operators given by Rn (a) = aφ−k (an ) and Sn (a) = πPn (∆k−1 (a)) for all a ∈ A and n ∈ N. Due to Theorem 2.1, for every a ∈ A, we have Sn ∆R1 . . . Rm (a) = πPn (∆k (aφ−k (am . . . a1 ))) = πPn (∆k (a)(am . . . a1 ))   k X ∆k−j (a)Λj,kn (φ−k (am . . . a1 )) , +πPn  j=1

which equals k X

πPn (∆k−j (a))πPn (Λj,k (φ−k (am . . . a1 )))

j=1

in the case when m > n. Since the operator a 7−→

k X j=1

πPn (∆k−j (a))πPn (Λj,k (φ−k (am . . . a1 )))

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from A to A/Pn is continuous, the gliding hump argument [2] now shows that there exists an n ∈ N such that Sn ∆R1 . . . Rn is continuous. Consequently, the operator a 7−→ πPn (∆k (a))πPn (an . . . a1 ) from A into A/Pn is continuous. Since πPn (an . . . a1 ) is invertible, it may be concluded that the operator πPn ∆k is continuous, which contradicts the definition of k. We now assume that φ is an inner automorphism and that the second assertion in the theorem is false. Then there exists a sequence (Pn ) of pairwise different exceptional ideals of A which must necessarily be of finite codimension. For every n ∈ N, let kn be the least natural number such that πPn ∆kn is discontinuous. Let (an ) be as before, and let Rn : A −→ A and Sn : A −→ A/Pn be defined by Rn (a) = aan and Sn (a) = πPn (∆kn −1 (a)) for all a ∈ A and n ∈ N. Due to Theorem 2.1, for every a ∈ A, we have Sn ∆R1 . . . Rm (a) = πPn (∆kn (a(am . . . a1 ))) = πPn (∆kn (a)φkn (am . . . a1 ))   kn X +πPn  ∆kn −j (a)Λj,kn (am . . . a1 ) , j=1

which equals kn X

πPn (∆kn −j (a))πPn (Λj,kn (am . . . a1 ))

j=1

in the case when m > n, since am ∈ Pn and so πPn (φkn (am )) = 0. On the other hand, the operator kn X πPn (∆k−j (a))πPn (Λj,k (am . . . a1 )) a 7−→ j=1

from A to A/Pn is continuous, and thus the gliding hump argument [2] again shows that there exists an n ∈ N such that Sn ∆R1 . . . Rn is continuous. Therefore, the operator a 7−→ πPn (∆kn (a))πPn (φkn (an . . . a1 )) from A into A/Pn is continuous. Since πPn (an . . . a1 ) is invertible, it follows that so is πPn (φkn (an . . . a1 )), and therefore that the operator πPn ∆kn is continuous, which contradicts the definition of kn . q Corollary 4.3. Let A be a semisimple Banach algebra, let φ be an automorphism of A, and let ∆ be a φ-derivation on A. Then ∆ is continuous. Proof. We begin by pointing out that φ is continuous [3]. Let I be the intersection of all the primitive ideals P of A such that πP ∆ is continuous. Of course S(∆) ⊂ I. The preceding theorem shows that πP ∆ is continuous for all primitive ideals P of A except for possibly finitely many exceptional ideals P1 , . . . , Pn . Since A/P1 ⊕ . . . ⊕ A/Pn is finite-dimensional and the map a 7−→ (a + P1 , . . . , a + Pn ) from I into A/P1 ⊕ . . . ⊕ A/Pn is injective, it follows that dim I < ∞. Consequently, ∆ is continuous when restricted to I. On the other hand, I is a two-sided ideal of a semisimple algebra, and therefore it is also a semisimple algebra. Since I is

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finite-dimensional, the Wedderburn theorem shows that I must have an identity e. If (an ) is a sequence in A with lim an = 0 and lim ∆(an ) = a for some a ∈ I, then 0 = lim ∆(ean ) = lim(∆(e)φ(an ) + e∆(an )) = ea = a, which shows the continuity of ∆.

q

5. Noncommutative Singer–Wermer conjecture Theorem 5.1. Let A be a unital Banach algebra, let φ be an inner automorphism of A, and let ∆ be a φ-derivation on A. Then the following assertions hold: (i) ∆ leaves each primitive ideal of A invariant except for possibly finitely many exceptional primitive ideals which are of finite codimension. (ii) If [∆, φ] = 0 and a ∈ A is such that ∆2 (a) = 0, then πP (∆(a)) is quasi-nilpotent for all primitive ideals P of A except for possibly finitely many exceptional primitive ideals of finite codimension. Accordingly, the spectrum of ∆(a) is finite. Proof. Assume that P is a primitive ideal of A such that πP ∆n is continuous for each n ∈ N. By [14, Lemma 3.1], which was inspired by [12, Lemma 1.1], there exists a constant C such that kπP ∆n k 6 C n for each n ∈ N. Taking into account Remark 2.4, we can proceed analogously to the proof of Theorem 3.3, with k∆k replaced by C, to obtain the fact that πP (∆(P )) consists of quasinilpotent elements. As in the proof of Corollary 3.4 this gives ∆(P ) ⊂ P . Finally, Theorem 4.2 establishes the first assertion. We now proceed to prove the second assertion in the theorem. For every invariant primitive ideal P , φ drops to an inner automorphism φP of A/P and ∆ drops to a φP derivation ∆P on A/P defined by φP (πP (a)) = πP (φ(a)) and ∆P (πP (a)) = πP (∆(a)) for each a ∈ A. It is clear that [∆P , φP ] = 0 and that ∆P 2 (πP (a)) = 0. Corollary 4.3 shows that ∆P is continuous, and Theorem 3.1 now shows that ∆P (πP (a)) = πP (∆(a)) is quasinilpotent. The proof is completed by applying the first assertion together with the well known fact that the spectrum of an element b ∈ A is the union of the q spectra of πP (b), where P runs over all primitive ideals of A. The question of whether or not there are no exceptional primitive ideals in each of the assertions of the preceding theorem is closely related to the noncommutative Singer–Wermer conjecture. Theorem 5.2. Consider the following assertions: (i) For every inner automorphism φ and every φ-derivation ∆ of a unital Banach algebra A, ∆(P ) ⊂ P for each primitive ideal P of A. (ii) For every inner automorphism φ and every φ-derivation ∆ of a unital Banach algebra A, ∆(a) is quasinilpotent whenever a ∈ Rad(A) is such that ∆2 (a) = 0. (iii) For every inner automorphism φ and every φ-derivation ∆ of a unital Banach algebra A, ∆(a) 6= 1 for every a ∈ Rad(A). (iv) Every derivation on a Banach algebra A leaves each primitive ideal of A invariant. (v) Every derivation on a unital Banach algebra A takes invertible values only on such elements a ∈ A for which the two-sided ideal of A generated by a equals A. Then i ⇒ ii ⇒ iii ⇒ iv ⇔ v.

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the noncommutative singer–wermer conjecture

Proof. First, it is clear that (i) implies (ii), and that (ii) implies (iii). Suppose that (iv) is not true. By a well known (though unpublished) result by Thomas, there exist a radical Banach algebra R and a derivation D on the unitisation R 1 of R such that b = D(a) does not lie in R for some a ∈ R. Then b is invertible, and the map ∆ defined on R 1 by ∆(x) = D(x)b−1 for each x ∈ R 1 is a φb−1 -derivation on R 1 . Since ∆(a) = 1, the assertion (iii) fails to hold true. On the other hand, if assertion (v) were true, it would follow that the two-sided ideal Ia of R 1 generated by a equalled R 1 . However, Ia ⊂ R, a contradiction. We now assume that assertion (iv) holds true. Let D be a derivation on a Banach algebra A, and let a ∈ A be such that D(a) is invertible. Since we have assumed that (iv) is true, we have D(R) ⊂ R, where we write R = Rad(A) for brevity. Therefore D drops to a derivation DR on A/R defined by DR (a + R) = D(a) + R for each a ∈ A. Since A/R is semisimple, it follows that DR is continuous [5]. Since DR (a + R) is invertible, Corollary 3.7 now yields Ia + R = A. In particular, there exists a b ∈ Ia such that 1 − b ∈ R, and this shows that b is invertible and so Ia = A. q Remark 5.3. The second assertion of Theorem 5.2 is currently known to be true for derivations [12]. In fact, in that paper it is proved that if D is a derivation on a Banach algebra A and a ∈ A is such that D2 (a) = 0, then D(a) is quasinilpotent. References 1. M. Bresˇar, A. Fosˇner and M. Fosˇner, ‘A Kleinecke–Shirokov type condition with Jordan automorphisms’, Studia Math. 147 (2001) 237–242. 2. H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs (Oxford University Press, 2001). 3. B. E. Johnson, ‘The uniqueness of the (complete) norm topology’, Bull. Amer. Math. Soc. 73 (1967) 537–539. 4. B. E. Johnson, ‘Continuity of derivations on commutative Banach algebras’, Amer. J. Math. 91 (1969) 1–10. 5. B. E. Johnson and A. M. Sinclair, ‘Continuity of derivations and a problem of Kaplansky’, Amer. J. Math. 90 (1968) 1067–1073. 6. D. C. Kleinecke, ‘On operator commutators’, Proc. Amer. Math. Soc. 8 (1957) 535–536. 7. M. Mathieu and G. J. Murphy, ‘Derivations mapping into the radical’, Arch. Math. (Basel) 57 (1991) 469–474. 8. F. V. Shirokov, ‘Proof of a conjecture of Kaplansky’, Uspekhi Mat. Nauk 11 (1956) 167–168 (Russian). 9. A. M. Sinclair, ‘Continuous derivations on Banach algebras’, Proc. Amer. Math. Soc. 29 (1969) 166–170. 10. I. M. Singer and J. Wermer, ‘Derivations on commutative normed algebras’, Math. Ann. 129 (1955) 260–264. 11. M. P. Thomas, ‘The image of a derivation is contained in the radical’, Ann. of Math. 128 (1988) 435–460. 12. M. P. Thomas, ‘Primitive ideals and derivations on non-commutative Banach algebras’, Pacific J. Math. 159 (1993) 139–152. 13. Yu. V. Turovskii, ‘Homomorphisms and derivations of rings and algebras’, Spectral Theory of Operators and its Applications 7 (ELM, Baku, 1986) 201–207 (Russian). 14. A. R. Villena, ‘Lie derivations on Banach algebras’, J. Algebra 226 (2000) 390–409.

Department of Mathematics University of Maribor PF, Koroˇska 160 Slovenia [email protected]

Departamento de An´ alisis Matem´ atico Facultad de Ciencias Universidad de Granada 18071 Granada Spain [email protected]