Innerness of continuous derivations on algebras of measurable operators affiliated with finite von Neumann algebras (original) (raw)
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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.
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Journal of Functional Analysis, 2009
Given a von Neumann algebra M denote by S(M) and LS(M) respectively the algebras of all measurable and locally measurable operators affiliated with M. For a faithful normal semi-finite trace τ on M let S(M, τ ) be the algebra of all τ -measurable operators from S(M). We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra M. In particular, we prove that if M is of type I ∞ then every derivation on LS(M) (resp. S(M) and S(M, τ )) is inner.
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This paper is devoted to local derivations on subalgebras on the algebra S(M, τ) of all τ-measurable operators affiliated with a von Neumann algebra M without abelian summands and with a faithful normal semi-finite trace τ. We prove that if A is a solid *-subalgebra in S(M, τ) such that p ∈ A for all projection p ∈ M with finite trace, then every local derivation on the algebra A is a derivation. This result is new even in the case of standard subalgebras on the algebra B(H) of all bounded linear operators on a Hilbert space H. We also apply our main theorem to the algebra S0(M, τ) of all τ-compact operators affiliated with a semi-finite von Neumann algebra M and with a faithful normal semi-finite trace τ.
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Given a type I von Neumann algebra MMM with a faithful normal semi-finite trace tau,\tau,tau, let L(M,tau)L(M, \tau)L(M,tau) be the algebra of all tau\tautau-measurable operators affiliated with M.M.M. We give a complete description of all derivations on the algebra L(M,tau).L(M, \tau).L(M,tau). In particular, we prove that if MMM is of type I$_\infty$ then every derivation on L(M,tau)L(M, \tau)L(M,tau) is inner.
Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra
Siberian Advances in Mathematics, 2008
Let M be a type I von Neumann algebra with the center Z, a faithful normal semi-finite trace τ. Let L(M, τ ) be the algebra of all τ -measurable operators affiliated with M and let S 0 (M, τ ) be the subalgebra in L(M, τ ) consisting of all operators x such that given any ε > 0 there is a projection p ∈ P(M ) with τ (p ⊥ ) < ∞, xp ∈ M and xp < ε. We prove that any Z-linear derivation of S 0 (M, τ ) is spatial and generated by an element from L(M, τ ).
Innerness of derivations on subalgebras of measurable operators
Lobachevskii Journal of Mathematics, 2008
Given a von Neumann algebra M with a faithful normal semifinite trace τ , let L(M, τ) be the algebra of all τ -measurable operators affiliated with M . We prove that if A is a locally convex reflexive complete metrizable solid * -subalgebra in L(M, τ), that can be embedded into a locally bounded weak Fr´echet M -bimodule, then any derivation on A is inner.
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.
Spatiality of Derivations on the Algebra of τ-Compact Operators
Integral Equations and Operator Theory, 2013
This paper is devoted to derivations on the algebra S 0 (M, τ ) of all τ -compact operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace τ. The main result asserts that every t τ -continuous derivation D : S 0 (M, τ ) → S 0 (M, τ ) is spatial and implemented by a τ -measurable operator affiliated with M , where t τ denotes the measure topology on S 0 (M, τ ). We also show the automatic t τ -continuity of all derivations on S 0 (M, τ ) for properly infinite von Neumann algebras M . Thus in the properly infinite case the condition of t τ -continuity of the derivation is redundant for its spatiality.
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.