Notes on derivations of Murray--von Neumann algebras

Let $\mathcal{M}$ be a type II$_1$ von Neumann factor and let $S(\mathcal{M})$ be the associated Murray-von Neumann algebra of all measurable operators affiliated to $\mathcal{M}.$ We extend a result of Kadison and Liu \cite{KL} by showing that any derivation from $S(\mathcal{M})$ into an $\mathcal{M}$-bimodule $\mathcal{B}\subsetneq S(\mathcal{M})$ is trivial. In the special case, when $\mathcal{M}$ is the hyperfinite type II$_1-$factor $\mathcal{R}$, we introduce the algebra $AD(\mathcal{R})$, a noncommutative analogue of the algebra of all almost everywhere approximately differentiable functions on $[0,1]$ and show that it is a proper subalgebra of $S(\mathcal{R})$. This algebra is strictly larger than the corresponding ring of continuous geometry introduced by von Neumann. Further, we establish that the classical approximate derivative on (classes of) Lebesgue measurable functions on $[0,1]$ admits an extension to a derivation from $AD(\mathcal{R})$ into $S(\mathcal{R})$, which fails to be spatial. Finally, we show that for a Cartan masa $\mathcal{A}$ in a hyperfinite II$_1-$factor $\mathcal{R}$ there exists a derivation $\delta$ from $\mathcal{A}$ into $S(\mathcal{A})$ which does not admit an extension up to a derivation from $\mathcal{R}$ to $S(\mathcal{R}).$