# Notes on derivations of Murray--von Neumann algebras

## TL;DR

This paper investigates derivations on Murray-von Neumann algebras associated with type II$_1$ factors, extending classical results, introducing a new algebra of approximately differentiable operators, and exploring their derivations' properties.

## Contribution

It extends the understanding of derivations on Murray-von Neumann algebras, introduces the algebra AD(
{R}) as a noncommutative analogue of differentiable functions, and analyzes the extendability of derivations.

## Key findings

- Derivations from $S(
{M})$ into proper $
{M}$-bimodules are trivial.
- The algebra $AD(
{R})$ is a proper subalgebra of $S(
{R})$ and larger than von Neumann's continuous geometry.
- The classical approximate derivative extends to a non-spatial derivation from $AD(
{R})$ into $S(
{R})$.

## Abstract

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 $S(\mathcal{R}).$