Partially defined σ-derivations on semisimple Banach algebras
Let A be a semisimple Banach algebra with a linear automorphism σ and let δ : I → A be a σ-derivation, where I is an ideal of A. Then Φ(δ)(I ∩ σ(I)) = 0, where Φ(δ) is the separating space of δ. As a consequence, if I is an essential ideal then the σ-derivation δ is closable. In a prime C∗-algebra, we show that every σ-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Banach algebra with nontrivial idempotents is continuous if it satisfies the σ-derivation expansion formula on zero products. 1. Results. Throughout the paper, A is always a unital Banach algebra over the complex field C and σ is a linear endomorphism of A. Let 1A denote the identity automorphism of A. By a σ-derivation of A we mean a linear map δ : A → A such that δ(xy) = σ(x)δ(y) + δ(x)y for all x, y ∈ A. Clearly, the map σ − 1A is a σ-derivation and 1A-derivations are just ordinary derivations. Thus the concept of σ-derivations can be regarded as a generalization of both derivations and endomorphisms. Let I be a nonzero ideal of A. A linear map δ : I → A is called a σ-derivation defined on I if δ(xy) = σ(x)δ(y)+δ(x)y for all x, y ∈ I. An ideal I of A is called essential if I has nontrivial intersection with any nonzero ideal of A. For a semisimple algebra A, this is equivalent to saying that aI = 0 where a ∈ A implies a = 0. A σ-derivation δ : I → A is called essentially defined on an ideal I if I is an essential ideal of A. Kaplansky conjectured that every derivation on a C∗-algebra is continuous [16] and that every derivation on a semisimple Banach algebra is continuous [17]. Sakai confirmed Kaplansky’s conjecture for C∗-algebras in [22]. The second conjecture was confirmed by Johnson and Sinclair in [15]. 2000 Mathematics Subject Classification: 46H40, 47B47, 46H15.