Local Derivations on Subalgebras of τ-Measurable Operators with Respect to Semi-finite von Neumann Algebras

Abstract

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 ${\mathcal{A}}$ is a solid *-subalgebra in S(M, τ) such that ${p \in \mathcal{A}}$ for all projection p ∈ M with finite trace, then every local derivation on the algebra ${\mathcal{A}}$ 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 S ₀(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.