Derivations on the Algebra of $\tau$-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 $\tau .$ Let $L(M,\tau )$ be the algebra of all $\tau$-measurable operators affiliated with $M$ and let $S_0(M,\tau )$ be the subalgebra in $L(M,\tau )$ consisting of all operators $x$ such that given any $\epsilon >0$ there is a projection $p\in \mathcal{P}(M)$ with $\tau (p^{\perp })<\infty ,xp\in M$ and $\Vert xp\Vert <\epsilon .$ We prove that any $Z$-linear derivation of $S_0(M,\tau )$ is spatial and generated by an element from $L(M,\tau ).$

14 pages