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Ultraweak Continuity of σ-derivations on von Neumann Algebras

  • Madjid Mirzavaziri
  • Mohammad Sal MoslehianEmail author
Article
  • 67 Downloads

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.

Keywords

Derivation ∗-homomorphism ∗-σ-derivation Inner σ-derivation von Neumann algebra Ultraweak topology Weak (operator) topology 

Mathematics Subject Classifications (2000)

Primary 46L57 Secondary 46L05 47B47 

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Pure MathematicsFerdowsi University of MashhadMashhadIran
  2. 2.Centre of Excellence in Analysis on Algebraic Structures (CEAAS)Ferdowsi University of MashhadMashhadIran

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