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

This paper is devoted to local derivations on subalgebras on the algebra S(M, tau) of all tau-measurable operators affiliated with a von Neumann algebra M without abelian summands and with a faithful normal semi-finite trace tau. We prove that if is a solid *-subalgebra in S(M, tau) such that for all projection p a M with finite trace, then every local derivation on the algebra 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, tau) of all tau-compact operators affiliated with a semi-finite von Neumann algebra M and with a faithful normal semi-finite trace tau.