Complete description of derivations on $\tau$-compact operators for type I von Neumann algebras

TL;DR

This paper provides a comprehensive characterization of derivations on the algebra of τ-compact operators affiliated with type I von Neumann algebras, showing all derivations are spatial in the type I∞ case.

Contribution

It offers a complete description of derivations on S₀(M, τ) for type I von Neumann algebras, including the proof that all derivations are spatial for type I∞.

Findings

All derivations on S₀(M, τ) are spatial when M is of type I∞.

The paper characterizes derivations on τ-compact operator algebras for type I von Neumann algebras.

Provides a complete description of derivations on S₀(M, τ).

Abstract

Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau,$ let $S_0(M, \tau)$ be the algebra of all $\tau$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra $S_0(M, \tau).$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $S_0(M, \tau)$ is spatial.