Unbounded derivations of von Neumann algebras
Let ℳ be a semi-finite factor and let 𝓙(ℳ ) be the set of operators T in ℳ such that T = ETE for some finite projection E. We obtain a representation theorem for unitarily invariant norms on 𝓙(ℳ ) in terms of Ky Fan norms. As an application, we prove that the class of unitarily invariant norms on 𝓙(ℳ ) coincides with the class of symmetric gauge norms on a classical abelian algebra, which generalizes von Neumann's classical 1940 result on unitarily invariant norms on Mₙ(ℂ). As another application,...
Let 𝓐 be a Banach algebra and let ℳ be a W*-algebra. For a homomorphism Φ from 𝓐 into ℳ, we introduce and study ℳ -valued invariant Φ-means on the space of bounded linear maps from 𝓐 into ℳ. We establish several characterizations of existence of an ℳ -valued invariant Φ-mean on B(𝓐,ℳ). We also study the relation between existence of an ℳ -valued invariant Φ-mean on B(𝓐,ℳ) and amenability of 𝓐. Finally, for a character ϕ of 𝓐, we give some descriptions for ϕ-amenability of 𝓐 in terms of ℳ...