Continuity of Lie Mappings of the Skew Elements of Banach Algebras with Involution

Let A and B be centrally closed prime complex Banach algebras with linear involution. If A is semisimple, then any Lie derivation of the skew elements of A is continuous and any Lie isomorphism from the skew elements of B onto the skew elements of A is continuous. The Lie product [a, b] = ab−ba induces on any Banach algebra A a Lie structure of great interest for their intimate connections with the geometry of manifolds modeled on Banach spaces. In case A has a linear involution ∗, then the skew elements are the linear subspace KA = {a ∈ A : a∗ = −a} which is a Lie subalgebra of A. A Lie derivation of KA is a linear mapping d from KA to itself which satisfies d([a, b]) = [d(a), b] + [a, d(b)] for all a, b ∈ KA. If B is another Banach algebra with linear involution, then a Lie isomorphism from KB onto KA is a linear bijection φ from KB onto KA satisfying φ([a, b]) = [φ(a), φ(b)] for all a, b ∈ KB. Examples 1. Let H be a complex Hilbert space. Let L(H) denote the primitive C∗-algebra of all continuous linear operators on H , and for each a ∈ L(H), let a• denote the usual adjoint operator of a. 1. If J is a conjugation of H , then it is easy to check that the mapping ∗ from L(H) to itself defined by a∗ = Ja•J is a linear involution on L(H). If J is an anticonjugation of H , then the mapping a∗ = −Ja•J is a linear involution on L(H). The skew elements relative to the preceding involutions are classical complex Banach-Lie algebras of bounded operators (see [3]). 2. Let us denote by C∞ the set of all compact linear operators on H and let ‖·‖∞ be the usual operator norm. For 1 ≤ p < ∞, let Cp denote the usual class of those compact linear operators a on H for which ‖a‖p = ( ∑∞ n=1 μ p n) < ∞, where {μn} is the sequence of eigenvalues of the operator (a•a)1/2 arranged in decreasing order and repeated according to multiplicity. According to [2, Lemmas XI.9, XI.10, and XI.14], Cp is a two-sided ideal of L(H) which becomes a complex Banach algebra for the norm ‖·‖p. Since Cp contains all the continuous linear operators with finitedimensional range, we deduce that Cp is primitive. The involutions introduced in the preceding example leave invariant Cp, and their skew elements are classical complex Banach-Lie algebras of compact operators (see [3]). It was proved in [3] that Lie derivations and Lie •-automorphisms of all the preceding Banach-Lie algebras are continuous. Received by the editors February 7, 1997. 1991 Mathematics Subject Classification. Primary 46H40, 17B40. c ©1998 American Mathematical Society