Derivations, local and 2-local derivations on some algebras of operators on Hilbert C*-modules

2017/05/26 by Jun He, He, Jun, Jiankui Li +3
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1705.09450

openalex publication_date 2017/05/26 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

For a commutative C*-algebra \mathcal A with unit e and a Hilbert~\mathcal A-module \mathcal M, denote by End_{\mathcal A}(\mathcal M) the algebra of all bounded \mathcal A-linear mappings on \mathcal M, and by End^*_{\mathcal A}(\mathcal M) the algebra of all adjointable mappings on \mathcal M. We prove that if \mathcal M is full, then each derivation on End_{\mathcal A}(\mathcal M) is \mathcal A-linear, continuous, and inner, and each 2-local derivation on End_{\mathcal A}(\mathcal M) or End^*_{\mathcal A}(\mathcal M) is a derivation. If there exist x₀ in \mathcal M and f₀ in \mathcal M^', such that f₀(x₀)=e, where \mathcal M^' denotes the set of all bounded \mathcal A-linear mappings from \mathcal M to \mathcal A, then each \mathcal A-linear local derivation on End_{\mathcal A}(\mathcal M) is a derivation.

Citations

• Derivations of nest algebras

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