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

Abstract

Let $M$ be a type I von Neumann algebra with the center $Z,$ a faithful normal semi-finite trace $τ.$ Let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M$ and let $S_0(M, τ)$ be the subalgebra in $L(M, τ)$ consisting of all operators $x$ such that given any $ε>0$ there is a projection $p\in\mathcal{P}(M)$ with $τ(p^{\perp})<\infty, xp\in M$ and $\|xp\|<ε.$ We prove that any $Z$-linear derivation of $S_0(M, τ)$ is spatial and generated by an element from $L(M, τ).$