Innerness of derivations on subalgebras of measurable operators (original) (raw)
Related papers
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 .
Mediterranean Journal of Mathematics, 2014
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.
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.
Structure of derivations on various algebras of measurable operators for type I von Neumann algebras
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.
Description of Derivations on Measurable Operator Algebras of Type I
2007
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 ).
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
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.
2-Local derivations on matrix algebras and algebras of measurable operators
2017
Let \(\mathcal{A}\) be a unital Banach algebra such that any Jordan derivation from \(\mathcal{A}\) into any \(\mathcal{A}\)-bimodule \(\mathcal{M}\) is a derivation. We prove that any 2-local derivation from the algebra Mn(mathcalA)M_n(\mathcal{A})Mn(mathcalA) into Mn(mathcalM)M_n(\mathcal{M})Mn(mathcalM) (ngeq3)(n\geq 3)(ngeq3) is a derivation. We apply this result to show that any 2-local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation.