Automatic continuity of $\sigma $-derivations on $C^*$-algebras

Abstract

Abstract. Let A be a C * -algebra acting on a Hilbert space H, let σ : A → B(H) be a linear mapping and let d : A → B(H) be a σ-derivation. Generalizing the celebrated theorem of Sakai, we prove that if σ is a continuous * -mapping, then d is automatically continuous. In addition, we show the converse is true in the sense that if d is a continuous * -σ-derivation, then there exists a continuous linear mapping Σ :The continuity of the so-called * -(σ, τ )-derivations is also discussed.

CONTINUITY OF (Α,β)-Derivatio OF OPERATOR ALGEBRAS

“…In particular, we show that if X is simple and α, β are surjective and continuous at zero, then each (α, β)-derivation on B(X) is continuous. In Section 4, using a similar argument to the proof in [13], we show that every (α, β)-derivation of a unital C * -algebra into its a Banach module is continuous if α, β are continuous at zero, which generalizes the main results in [10]. As corollaries, we also get the ultraweak continuity of (α, β)-derivations of von Neumann algebras when the ultraweak continuity and linearity on α and β are required.…”

“…On the continuity of (α, β)-derivations on C * -algebras, M. Mirzavazibri and S. Moslehian proved that each * -(α, β)-derivation from a C * -algebra A acting on a Hilbert space H into B(H) is continuous under the assumption that α and β are * -linear continuous mappings from A into B(H) ( [10,11,12]).…”

A generalization of amenability for topological semigroups and semigroup algebras

“…In the case where A = B, A is called ϕ-amenable. Several authors have studied ϕ-derivations, and ϕ-amenability of a Banach algebra A (see [7], [8], [15] and [16]).…”

A characterization of derivations on uniformly mean value Banach algebras

“…As is well known, the class of derivations is a very important class of linear mappings both in theory and applications and was studied intensively. Recently, a number of authors [1,4,8,9,14] have studied various generalized notions of derivations in the context of Banach algebras. Such mappings have been extensively studied in pure algebra; cf.…”