2-Local derivations on matrix algebras and algebras of measurable operators (original) (raw)

Derivations, local and 2-local derivations on algebras of measurable operators

Contemporary Mathematics, 2016

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 2local 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.

Local Derivations on Algebras of Measurable Operators

Communications in Contemporary Mathematics, 2011

The paper is devoted to local derivations on the algebra S(M, τ ) of τmeasurable operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every local derivation on S(M, τ ) which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the condition of continuity of the local derivation is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra M for the existence of local derivations which are not derivations on algebras of measurable operators affiliated with M.

2-Local derivations on matrix algebras over commutative regular algebras

Linear Algebra and its Applications, 2013

The paper is devoted to 2-local derivations on matrix algebras over commutative regular algebras. We give necessary and sufficient conditions on a commutative regular algebra to admit 2-local derivations which are not derivations. We prove that every 2-local derivation on a matrix algebra over a commutative regular algebra is a derivation. We apply these results to 2-local derivations on algebras of measurable and locally measurable operators affiliated with type I von Neumann algebras.

2-Local derivations on von Neumann algebras

Positivity, 2014

The paper is devoted to the description of 2-local derivations on von Neumann algebras. Earlier it was proved that every 2-local derivation on a semi-finite von Neumann algebra is a derivation. In this paper, using the analogue of Gleason Theorem for signed measures, we extend this result to type III von Neumann algebras. This implies that on arbitrary von Neumann algebra each 2-local derivation is a derivation.

2-Local derivations on matrix algebras over semi-prime Banach algebras and onAW*-algebras

Journal of Physics: Conference Series, 2016

The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2 n (A), n ≥ 2, is a derivation. We apply this result to AW *-algebras and show that any 2-local derivation on an arbitrary AW *-algebra is a derivation.

Local derivations on measurable operators and commutativity

European Journal of Mathematics, 2016

We prove that a von Neumann algebra M is abelian if and only if the square of every derivation on the algebra S(M) of measurable operators affiliated with M is a local derivation. We also show that for general associative unital algebras this is not true.

Derivations on Algebras of Measurable Operators

2010

The present paper is a survey of recent results concerning derivations on various algebras of measurable operators affiliated with von Neumann algebras. A complete description of derivation is obtained in the case of type I von Neumann algebras. A special section is devoted to the Abelian case, namely to the existence of nontrivial derivations on algebras of measurable function. Local derivations on the above algebras are also considered.