Local Derivations on Algebras of Measurable Operators

Abstract

The paper is devoted to local derivations on the algebra S(M, τ) of τ -measurable 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 continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra M for the algebra S(M, τ) to admit local derivations which are not derivations.Lemma 3.1. Given any element a ∈ A there exists an idempotent e ∈ ∇ such that (i) ea is integral with respect to K c (∇), moreover in this case ea ∈ K c (∇); (ii) if e = 1, then e ⊥ a is weakly transcendental with respect to K c (∇).Proof. Denote by ∇ int the set of all idempotents e ∈ ∇ such that ea is integral with respect to K c (∇). By [5, Proposition 3.8] each integral element with respect to K c (∇) in fact belongs to K c (∇). Therefore ∇ int = {e ∈ ∇ : ea ∈ K c (∇)}. We set e = sup ∇ int . Since ∇ is a complete Boolean algebra of countable type (see [5, Proposition 2.7]), there exists a countable family of mutually disjoint elements {e k } k≥1 in ∇ with sup k≥1 e k = e, such that given any e ∈ ∇ int there exists k ≥ 1, for which e k ≤ e . It is clear that from e k ≤ e and e ∈ ∇ int it follows that e k ∈ ∇ int , and thus e k a ∈ K c (∇). Therefore ea = k≥1 e k a ∈ K c (∇). Further since s(a) ⊥ a = 0 ∈ K c (∇) we have that s(a) ⊥ ≤ e, i.e. e ⊥ ≤ s(a) and hence s(e ⊥ a) = e ⊥ . Now let us show that if e = 1 then e ⊥ a is weakly transcendental with respect to K c (∇). Suppose the opposite, i.e. there exists a non zero idempotent q ≤ e ⊥ = s(e ⊥ a) such that qa is integral with respect to K c (∇). This means that q ∈ ∇ int , i.e. q ≤ e. This is a contradiction with 0 = q ≤ e ⊥ . Therefore e ⊥ a is weakly transcendental with respect to K c (∇). The proof is complete.

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

“…In this section we study local derivations on the algebra S(M, τ ) of τ -measurable operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace τ. The results presented here are due to Albeverio, Ayupov, Kudaybergenov and Nurjanov (see [7], [18]).…”

Local and 2-local derivations on noncommutative Arens algebras

“…He also showed that every local derivation from a C * -algebra A into any Banach A-bimodule is continuous. In [4], [6] local derivations have been investigated on the algebra S(M ) of all measurable operators with respect a von Neumann algebra M . In particular, we have proved that for type I von Neumann algebras without abelian direct summands every local derivation on S(M ) is a derivation.…”

Local derivations on measurable operators and commutativity

“…In [2], we have studied local derivations on the algebra S(M, τ ) of all τ -measurable operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace τ . One of our main results in [2] (Theorem 2.1) presents an unbounded version of Kadison's Theorem A from [12], and it asserts that every local derivation on S(M, τ ) which is continuous in the measure topology automatically becomes a derivation.…”

“…One of our main results in [2] (Theorem 2.1) presents an unbounded version of Kadison's Theorem A from [12], and it asserts that every local derivation on S(M, τ ) which is continuous in the measure topology automatically becomes a derivation. In particular, for type I von Neumann algebras M all such local derivations on S(M, τ ) are inner derivations.…”