Continuity of Lie Isomorphisms of Banach Algebras

We prove that if A and B are semisimple Banach algebras, then the separating subspace of every Lie isomorphism from A onto B is contained in the centre of B. Over the years, there has been considerable effort made and success in studying the structure of Lie isomorphisms of rings and Banach algebras [2–5, 7–15]. We are interested in investigating the continuity of Lie isomorphisms of Banach algebras. A Lie isomorphism from a Banach algebra A onto a Banach algebra B is a linear bijective mapping Φ from A onto B such that Φ([a1, a2]) = [Φ(a1),Φ(a2)] for all a1, a2 ∈ A. Here and subsequently, the bracket denotes the Lie product, [a, b] = ab − ba, on A and B. We measure the continuity of Φ by considering its separating subspace, which is defined as the subspace S(Φ) of those elements b ∈ B for which there exists a sequence {an} in A such that lim an = 0 and lim Φ(an) = b. S(Φ) is easily checked to be a Lie ideal of B, and the closed graph theorem shows that Φ is continuous if and only if S(Φ) = 0. The purpose of this paper is to prove the following result. Theorem. Let A and B be semisimple Banach algebras, and let Φ be a Lie isomorphism from A onto B. Then the separating subspace of Φ is contained in the centre of B. Accordingly, Φ is continuous if B has zero centre. For C∗-algebras, this result is contained in [17]. Its main ingredient is [17, Proposition 1.9], implying that the separating subspace of a Lie isomorphism from a Banach algebra A onto a Banach algebra B is contained in the weak radical of the Lie algebra of B. This, in turn, rests on an adaptation of Aupetit’s theorem that the separating subspace of a homomorphism from one Banach algebra onto another lies in the Jacobson radical. For certain von Neumann algebras and certain C∗-algebras, this result can alternatively be deduced from the results in [4] and [13], respectively. Our approach also relies on [17, Proposition 1.9], which has the subsequent consequence. From now on, for each element a in a Banach algebra A we denote by ad(a) the operator from A to itself defined by ad(a)(b) = [a, b]. Lemma 1. Let A and B be Banach algebras, let Φ be a Lie isomorphism from A onto B, and let b ∈ S(Φ). Then ad(b) is a quasinilpotent operator. Received 3 December 1997; revised 3 April 1998. 1991 Mathematics Subject Classification 17B40, 17B60, 46H40. Bull. London Math. Soc. 31 (1999) 6–10 continuity of lie isomorphisms of banach algebras 7 It is obvious that the weak radical of a Banach algebra A contains the centre of A. Unfortunately, we do not know whether the weak radical of the Lie algebra of a semisimple Banach algebra equals the centre of the algebra. Our method of proof therefore consists in showing that [B,S(Φ)] ⊂ P for every primitive ideal P of B. As is typical in automatic continuity theory, we have to distinguish between primitive ideals of finite and of infinite codimension. Our first result, covering the first case, can also be deduced from [17, Remark 3.4(ii)]; however, our argument is somewhat more direct. In the sequel, for a Banach algebra A, rad(A) and Z(A) stand for the radical and the centre of A, respectively. If P is a primitive ideal of A, then we shall denote by πP the quotient map from A onto A/P . To shorten notation, for each a ∈ A we write ȧ instead of πP (a). Lemma 2. Let A and B be complex Banach algebras, let Φ be a Lie isomorphism from A onto B, and let P be a primitive ideal of B of finite codimension. Then [B,S(Φ)] ⊂ P . Proof. It is well known that B/P is a simple Banach algebra. In fact, B/P is isomorphic to the matrix algebra Mn(C) for some n ∈ N. If dim(B/P ) = 1, then B/P is commutative and therefore [B,S(Φ)] ⊂ P . Assume that dim(B/P ) > 1. [7, Theorem 2 and Corollary 1] now show that either πP (S(Φ)) is contained in the centre of B/P and so [B,S(Φ)] ⊂ P , or πP (S(Φ)) = B/P . On the other hand, according to Lemma 1, the operator ad(b) from B to itself is quasinilpotent for every b ∈ S(Φ), and therefore the operator ad(ḃ) from B/P to itself is quasinilpotent for every ḃ ∈ πP (S(Φ)). If πP (S(Φ)) = B/P , then ad(ḃ) is quasinilpotent for all b ∈ B. Let ḃ be the matrix with 1 at the upper left corner and 0s elsewhere, and let ċ be the matrix with 1 at the upper right corner and 0s elsewhere. Since ad(ḃ)(ċ) = ċ, it follows that 1 is in the spectrum of ad(ḃ), which contradicts the quasinilpotence of this operator. To study the inclusion [B,S(Φ)] ⊂ P for a primitive ideal P of infinite codimension, we require the following result, on the structure of Lie isomorphisms of Banach algebras, whose proof is adapted from the proof of the remarkable Theorem 1 in [3]. Lemma 3. Let A and B be unital complex Banach algebras, let Φ be a Lie isomorphism from A onto B, and let P be a primitive ideal of B. Then one of the following assertions holds. (i) B/P is isomorphic to C. (ii) B/P is isomorphic to the matrix algebra M2(C). (iii) There exists a linear functional μ on A such that πPΦ(a ) − μ(a)πPΦ(a) ∈ Z(B/P ) for all a ∈ A. (iv) There exist a continuous homomorphism φ from A onto B/P , a linear mapping ξ from A to the centre of B/P , and α ∈ C such that πPΦ = αφ+ ξ. (v) There exist a continuous anti-homomorphism φ from A onto B/P , a linear mapping ξ from A to the centre of B/P , and α ∈ C such that πPΦ = αφ+ ξ. Proof. If B/P satisfies the standard polynomial identity S4, then either B/P is isomorphic to C or B/P is isomorphic to M2(C) (see [16, Theorem 7.1.14]). 8 m. i. berenguer and a. r. villena We now assume that B/P does not satisfy S4. It should be noted that B/P is a centrally closed algebra (see [11, Theorem 12]). For every b ∈ B, we have [(Φ−1(b))2,Φ−1(b)] = 0, and therefore 0 = Φ([(Φ−1(b))2,Φ−1(b)]) = [Φ((Φ−1(b))2),Φ(Φ−1(b))] = [Φ((Φ−1(b))2), b]. Consequently, the map q defined on B by q(b) = Φ((Φ−1(b))2) is a commuting trace of the bilinear map (b1, b2) 7→ Φ(Φ−1(b1)Φ−1(b2)) from B × B to B. We can proceed as in the proof of [3, Theorem 1], the only difference being in the application of [3, Lemmas 1 and 2] to B/P instead of B, in order to prove that there exist λ ∈ C, a linear functional η on B, and a mapping τ from B to the centre of B/P such that πPΦ((Φ −1(b))2) = λḃ + η(b)ḃ+ τ(b) for all b ∈ B. Taking μ = ηΦ and ν = τΦ, we can rewrite the preceding equality as πPΦ(a ) = λπP (Φ(a) ) + μ(a)πP (Φ(a)) + ν(a) for all a ∈ A. If λ = 0, then the above identity becomes πPΦ(a 2)− μ(a)πPΦ(a) = ν(a) ∈ Z(B/P ) for all a ∈ A, which gives assertion (iii) in the lemma. If λ 6= 0, then the mapping φ = λπPΦ + 12μ can be handled in the same way as in the proof of [3, Theorem 2] to prove that φ is a Jordan homomorphism from A to B/P . Since [1, A] = 0, we see that [πPΦ(1), B/P ] = [πPΦ(1), πPΦ(A)] = πPΦ([1, A]) = 0, which gives πPΦ(1) ∈ Z(B/P ) and so φ(1) ∈ Z(B/P ). On the other hand, if φ(1) = 0, then we would have φ(a) = 1 2 φ(1a+ a1) = φ(1)φ(a) = 0 for all a ∈ A, and therefore B/P = λπPΦ(A) ⊂ μ(A) ⊂ Z(B/P ), a contradiction. Thus φ(1) = ρ1̇ for some nonzero ρ ∈ C, and πPΦ(a) = λ −1φ(a)− λ−1 1 2 μ(a) = φ(λ−1a− λ−1 1 2 μ(a)ρ−1) for all a ∈ A. From this we deduce that φ is onto. By [6, Theorem H], φ is either a homomorphism or an anti-homomorphism from A onto B/P . On account of [1, Theorem 1], φ is continuous. Consequently, either assertion (iv) or assertion (v) is fulfilled with α = λ−1 and ξ = −αμ/2. Lemma 4. Let A and B be unital complex Banach algebras, and let Φ be a Lie isomorphism from A onto B. Then Φ([[a1, a2], [a1, a2]]) ∈ rad(B) for all a1 ∈ A and a2 ∈ Φ−1(S(Φ)). Accordingly, if B is semisimple, then [[a1, a2], [a1, a2]] = 0 for all a1 ∈ A and a2 ∈ Φ−1(S(Φ)). Proof. Let a1 ∈ A, a2 ∈ Φ−1(S(Φ)), and let P be a primitive ideal of B. The proof consists in proving that Φ([[a1, a2], [a1, a2]]) ∈ P . If either assertion (i) or assertion (ii) in Lemma 3 is satisfied, then Lemma 2 yields [B,S(Φ)] ⊂ P . If either continuity of lie isomorphisms of banach algebras 9 assertion (iv) or assertion (v) in Lemma 3 is satisfied, then it is easy to check that πP (S(Φ)) ⊂ Z(B/P ) and hence [B,S(Φ)] ⊂ P . In these four cases we have Φ([a1, a2]) = [Φ(a1),Φ(a2)] ∈ [B,S(Φ)] ⊂ P ,