Spatiality of derivations on the algebra of $\tau$-compact operators

Abstract

This paper is devoted to derivations on the algebra $S_0(M,\tau )$ of all $\tau$-compact operators affiliated with a von Neumann algebra $M$ and a faithful normal semi-finite trace $\tau .$ The main result asserts that every $t_{\tau }$-continuous derivation $D:S_0(M,\tau )\to S_0(M,\tau )$ is spatial and implemented by a $\tau$-measurable operator affiliated with $M$, where $t_{\tau }$ denotes the measure topology on $S_0(M,\tau )$. We also show the automatic $t_{\tau }$-continuity of all derivations on $S_0(M,\tau )$ for properly infinite von Neumann algebras $M$. Thus in the properly infinite case the condition of $t_{\tau }$-continuity of the derivation is redundant for its spatiality.

15 pages. arXiv admin note: substantial text overlap with arXiv:1306.0251

References in corpus (4)

• Derivations in algebras of operator-valued functions
• Additive derivations on algebras of measurable operators
• Description of Derivations on Measurable Operator Algebras of Type I
• Innerness of continuous derivations on algebras of measurable operators affiliated with finite von Neumann algebras

Cited by in corpus (3)

• Local And 2-Local Derivations On Algebras Of Measurable Operators
• Local derivations on measurable operators and commutativity
• Local derivations on subalgebras of $\tau$-measurable operators with respect to semi-finite von Neumann algebras