Spatiality of derivations on the algebra of τ-compact operators

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This paper is devoted to derivations on the algebra S₀(M, τ) of all τ-compact operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace τ. The main result asserts that every tτ-continuous derivation D:S₀(M, τ)→ S₀(M, τ) is spatial and implemented by a τ-measurable operator affiliated with M, where tτ denotes the measure topology on S₀(M, τ). We also show the automatic tτ-continuity of all derivations on S₀(M, τ) for properly infinite von Neumann algebras M. Thus in the properly infinite case the condition of tτ-continuity of the derivation is redundant for its spatiality.