Local And 2-Local Derivations On Algebras Of Measurable Operators

Abstract

The present paper presents a survey of some recent results devoted to derivations, local derivations and 2-local derivations on various algebras of measurable operators affiliated with von Neumann algebras. We give a complete description of derivation on these algebras, except the case where the von Neumann algebra is of type II${}_1$. In the latter case the result is obtained under an extra condition of measure continuity of derivations. Local and 2-local derivations on the above algebras are also considered. We give sufficient conditions on a von Neumann algebra $M$, under which every local or 2-local derivation on the algebra of measurable operators affiliated with $M$ is automatically becomes a derivation. We also give examples of commutative algebras of measurable operators admitting local and 2-local derivations which are not derivations.

accepted to Contem. Math. AMS, 21 pages. arXiv admin note: text overlap with arXiv:0901.2983, arXiv:0808.0149

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• Derivations in algebras of operator-valued functions
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Cited by in corpus (2)

• Derivations and 2-local derivations on matrix algebras over commutative algebras
• 2-Local derivations on matrix algebras and algebras of measurable operators