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

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

2007-03-12

arxiv.org/abs/math/0703331v1

Mathematics

Operator Algebras

14 pages

Scientific paper

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 $\|xp\|<\epsilon.$ We prove that any $Z$-linear derivation of $S_0(M, \tau)$ is spatial and generated by an element from $L(M, \tau).$

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