Ultraweak Continuity of σ-derivations on von Neumann Algebras
- 67 Downloads
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.
KeywordsDerivation ∗-homomorphism ∗-σ-derivation Inner σ-derivation von Neumann algebra Ultraweak topology Weak (operator) topology
Mathematics Subject Classifications (2000)Primary 46L57 Secondary 46L05 47B47
Unable to display preview. Download preview PDF.
- 1.Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. 1. C *- and W *-algebras, symmetry groups, decomposition of states, 2nd edn. Texts and Monographs in Physics. Springer, New York (1987)Google Scholar
- 3.Brešar, M., Villena, A.R.: The noncommutative Singer-Wermer conjecture and ϕ-derivations. J. Lond. Math. Soc. (2) 66(3), 710–720 (2002)Google Scholar
- 8.Mirzavaziri, M., Moslehian, M.S.: A Kadison-Sakai type theorem. Bull. Aust. Math. Soc. (in press)Google Scholar
- 10.Takesaki, M.: Theory of Operator Algebras. I, Reprint of the first (1979) edition. Encyclopaedia of Mathematical Sciences. 124. Operator Algebras and Non-commutative Geometry, 5. Springer, Berlin (2002)Google Scholar