Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

Abstract

Let M be a type I von Neumann algebra with the center Z, and let LS(M ) be the algebra of all locally measurable operators affiliated with M. We prove that every Z-linear derivation on LS(M ) is inner. In particular all Z-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements from LS(M ).

Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

“…In the present paper we give a complete description of all derivations on the algebra LS(M) of all locally measurable operators affiliated with a type I von Neumann algebra M, and also on its subalgebras S(M)-of measurable operators and S(M, τ ) of τ -measurable operators, where τ is a faithful normal semi-finite trace on M. We prove that the above mentioned construction of derivations D δ from [2] gives the general form of pathological derivations on these algebras and these exist only in the type I fin case, while for type I ∞ von Neumann algebras M all derivations on LS(M), S(M) and S(M, τ ) are inner. Moreover we prove that an arbitrary derivation D on each of these algebras can be uniquely decomposed into the sum D = D a + D δ where the derivation D a is inner (for LS(M), S(M) and S(M, τ )) while the derivation D δ is constructed in the above mentioned manner from a nontrivial derivation δ on the center of the corresponding algebra.…”

“…The present paper continues the series of papers of the authors [1,2] devoted to the study and a description of derivations on the algebra LS(M) of locally measurable operators with respect to a von Neumann algebra M and on various subalgebras of LS(M).…”

Continuous derivations on algebras of locally measurable operators are inner

“…For a von Neumann algebra of type I and III this problem is solved in [1,2]. In the present paper it is proven that for every t(M)-continuous derivation δ acting on an EW * -algebra A with the bounded part A b = M, there exists a ∈ A, such that δ(x) = ax − xa = [a, x] for all x ∈ A, i.e.…”

Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra

“…Moreover[2, Example 4.6] gives an example of a non Z-linear (and hence non spatial) derivation on S 0 (M, τ ) = L(M, τ ) for an appropriate von Neumann algebra M with a faithful normal finite trace τ.On the other hand if the lattice of projections in a von Neumann algebra M is atomic then any derivation on S 0 (M, τ ) is automatically Z-linear (cf [2,. Example 4.6]).…”