Ultraweak Continuity of σ-derivations on von Neumann Algebras

Abstract

Let σ be a surjective ultraweakly continuous ∗-linear mapping and d be a σ-derivation on a von Neumann algebra \(\mathfrak M\). We show that there are a surjective ultraweakly continuous ∗-homomorphism \(\Sigma:\mathfrak M\to\mathfrak M\) and a Σ-derivation \(D:\mathfrak M\to\mathfrak M\) such that D is ultraweakly continuous if and only if so is d. We use this fact to show that the σ-derivation d is automatically ultraweakly continuous. We also prove the converse in the sense that if σ is a linear mapping and d is an ultraweakly continuous ∗-σ-derivation on \(\mathfrak M\), then there is an ultraweakly continuous linear mapping \(\Sigma:\mathfrak M\to\mathfrak M\) such that d is a ∗-Σ-derivation.