This paper concerns the innerness of additive derivations of von Neumann algebras and it is motivated by a result of A. F. Ber et al. [Extr. Math. 21, No. 2, 107–147 (2006; Zbl 1129.46056)] stating that the algebra \(L^0(0,1)\) of all complex-valued measurable functions on the interval \((0,1)\) admits nontrivial derivations. The present authors claim that the existence of such pathological examples deeply depends on the commutativity of the underlying von Neumann algebra \(L^\infty(0,1)\).
Given a von Neumann algebra \(M\), in this paper they consider the algebra \(LS(M)\) of locally measurable operators and define its \(\ast\)-subalgebra \({\text{mix}}(M)\), called the central extension of \(M\). It is shown that \({\text{mix}}(M)=LS(M)\) if and only if \(M\) does not admit a type II direct summand. For properly infinite \(M\), it is proven that every additive derivation of \({\text{mix}}(M)\) is inner. Hence it follows that, if \(M\) is the direct sum of type \({\text{I}}_\infty\) and type III algebras, then every additive derivation of \(LS(M)\) is inner.