Partially defined σ-derivations on semisimple Banach algebras

Abstract

Abstract. 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.

Automatic Continuity of Some Types of Double Derivations on Semisimple Banach Algebras

“…Automatic continuity of derivations are studied by many researcher, we mention some of them of our present work see [1], [2], [5], [6] and [7].…”

Characterizations of Centralizers and Derivations on Some Algebras

“…Remark In [9], the authors prove that for a prime semisimple Banach algebra A with nontrival idempotents and a linear mapping ∆ from A into itself, the condition ∆(A)B + A∆(B) = 0 for each A and B in A with AB = 0 implies that ∆ is bounded. By Lemma 4.1, we have that for a von Neumann algebra A, the result holds still even if A is not prime.…”

Skew Derivations in Banach Algebras

“…The skew derivations appear in q-Weyl algebras, enveloping algebras of solvable Lie superalgebras and coordinate rings of quantum matrices [17]. See [1,6,8,9,12,14,18,[21][22][23][24]28] for some recent results concerning skew derivations in Banach algebras. Brešar and Villena [9] proved that if δ is a continuous σ-derivation of A satisfying δ 2 (a) = 0 for some a ∈ A, where σ is a continuous automorphism of A such that δσ = σδ, then δ(a) is quasinilpotent.…”