Continuity of Lie mappings of the skew elements of Banach algebras with involution

Abstract

Abstract. 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 K A = {a ∈ A : a * = −a} which is a Lie subalgebraExamples 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 C p denote the usual class of those compact linear operators a on H for which a p = (where {µ n } 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], C p is a two-sided ideal of L(H) which becomes a complex Banach algebra for the norm · p . Since C p contains all the continuous linear operators with finitedimensional range, we deduce that C p is primitive. The involutions introduced in the preceding example leave invariant C p , 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.

Automatic Continuity in Associative and Nonassociative Context

“…Theorem 3.12. (Berenguer-Villena [8]). Let A and B be centrally closed prime Banach algebras with linear involution and assume that…”

“…Theorem 4.2 and part of Theorem 4.8 were extended to Lie derivations on Banach algebras in [7]. The remaining part of Theorem 4.8 was extended in [8] to a much more general context. Theorem 4.9.…”

“…Theorem 4.10. (Berenguer-Villena [8]). Let A be a centrally closed prime semisimple Banach algebra with linear involution, and let D be a derivation on K A .…”

Functional identities: a survey

“…For other applications of (G)FI's to Lie isomorphism and some related problems we refer to [BaM,BeBrCM3,BeC3,BeC6,BeC7,BV1,BV2,BV3,Bl,Br5,BrM1,C1,S,SB].…”

Epimorphisms Onto Derivation Algebras

“…The standard examples of Lie-Banach algebras are the closed Lie subalgebras of a Banach algebra A, such as A itself, the skew elements K A of an A in the case where A is endowed with a continuous linear involution, and the derivation algebra Der(A) of A. The continuity of Lie epimorphisms onto A was studied in [2] and the continuity of isomorphisms onto K A was studied in [3]. Throughout this paper, L stands for a Lie-Banach algebra, A stands for a complex Banach algebra with radical Rad(A), and Φ stands for an epimorphism from L onto Der(A).…”