Derivations of noncommutative Arens algebras (original) (raw)
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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.
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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.
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 ).
Journal of Mathematical Analysis and Applications, 2013
This paper is devoted to derivations on the algebra S(M) of all measurable operators affiliated with a finite von Neumann algebra M. We prove that if M is a finite von Neumann algebra with a faithful normal semi-finite trace τ , equipped with the locally measure topology t, then every t-continuous derivation D : S(M) → S(M) is inner. A similar result is valid for derivation on the algebra S(M, τ) of τ-measurable operators equipped with the measure topology t τ .
Non-commutative Arens algebras and their derivations
Journal of Functional Analysis, 2007
Given a von Neumann algebra M with a faithful normal semi-finite trace τ , we consider the non-commutative Arens algebra L ω (M, τ ) = p 1 L p (M, τ ) and the related algebras L ω 2 (M, τ ) = p 2 L p (M, τ ) and M +L ω 2 (M, τ ) which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra M + L ω 2 (M, τ ) is inner and all derivations of the algebras L ω (M, τ ) and L ω 2 (M, τ ) are spatial and implemented by elements of M + L ω 2 (M, τ ). In particular we obtain that if the trace τ is finite then any derivation on the noncommutative Arens algebra L ω (M, τ ) is inner.
2009
Given a von Neumann algebra M we introduce the so-called central extension mix(M) of M . We show that mix(M) is a *-subalgebra in the algebra LS(M) of all locally measurable operators with respect to M, and this algebra coincides with LS(M) if and only if M does not admit type II direct summands. We prove that if M is a properly infinite von Neumann algebra then every additive derivation on the algebra mix(M) is inner. This implies that on the algebra LS(M), where M is a type I∞ or a type III von Neumann algebra, all additive derivations are inner derivations. MIRAMARE – TRIESTE August 2009 Senior Associate of ICTP. Corresponding author. sh ayupov@mail.ru karim2006@mail.ru
Additive derivations on algebras of measurable operators
2009
Given a von Neumann algebra MMM we introduce so called central extension mix(M)mix(M)mix(M) of MMM. We show that mix(M)mix(M)mix(M) is a *-subalgebra in the algebra LS(M)LS(M)LS(M) of all locally measurable operators with respect to M,M,M, and this algebra coincides with LS(M)LS(M)LS(M) if and only if MM M does not admit type II direct summands. We prove that if MMM is