Derivations in the Banach ideals of τ-compact operators

2012/04/18 by A. F. Ber, Fedor Sukochev, Ber, A. F. +2
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA) #math.OA

paper · pdf · doi:10.48550/arxiv.1204.4052

arxiv created 2012/04/18 · openalex publication_date 2012/04/18 · arxiv updated 2012/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a von Neumann algebra equipped with a faithful normal semi-finite trace τ and let S₀(τ) be the algebra of all τ-compact operators affiliated with M. Let E(τ)⊆ S₀(τ) be a symmetric operator space (on M) and let E be a symmetrically-normed Banach ideal of τ-compact operators in M. We study (i) derivations δ on M with the range in E(τ) and (ii) derivations on the Banach algebra E. In the first case our main results assert that such derivations are continuous (with respect to the norm topologies) and also inner (under some mild assumptions on E(τ)). In the second case we show that any such derivation is necessarily inner when M is a type I factor. As an interesting application of our results for the case (i) we deduce that any derivation from M into an L_p-space, L_p(M,τ), (1<p<∞) associated with M is inner.

Related

• Derivations on symmetric quasi-Banach ideals of compact operators
• Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra
• Complete description of derivations on τ-compact operators for type I von Neumann algebras
• Structure of derivations on various algebras of measurable operators for type I von Neumann algebras
• Description of Derivations on Measurable Operator Algebras of Type I