Local derivations on subalgebras of $\tau$-measurable operators with respect to semi-finite von Neumann algebras

Abstract

This paper is devoted to local derivations on subalgebras on the algebra $S(M,\tau )$ 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 $\mathcal{A}$ is a solid $\ast$-subalgebra in $S(M,\tau )$ such that $p\in \mathcal{A}$ for all projection $p\in M$ with finite trace, then every local derivation on the algebra $\mathcal{A}$ is a derivation. This result is new even in the case 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 $\tau .$

9 pages, Meditterian J. Math. (in press)

References in corpus (3)

• 2-Local derivations on matrix algebras over commutative regular algebras
• Innerness of continuous derivations on algebras of measurable operators affiliated with finite von Neumann algebras
• Spatiality of derivations on the algebra of $\tau$-compact operators