The LFED Conjecture on images of locally finite derivations

Let $K$ be a field of characteristic $0$, and let $A$ be a $K$-algebra. A $K$-derivation of $A$ is a $K$-linear map satisfying the Leibniz rule; a $K$-$\mathcal{E}$-derivation is a $K$-linear map $\delta$ such that $\operatorname{id}-\delta$ is an algebra homomorphism. A linear operator is locally finite if every element of $A$ lies in a finite-dimensional invariant subspace. A subspace of $A$ is a Mathieu subspace if, whenever an element has all positive powers in the subspace, multiplying sufficiently high powers of that element by any fixed element of $A$ also gives elements of the subspace.

The LFED Conjecture. Images of locally finite $K$-derivations and $K$-$\mathcal{E}$-derivations of $K$-algebras are Mathieu subspaces.

This conjecture extends the known result that the image of any locally finite derivation of $K[x,y]$ is a Mathieu subspace, and is motivated by connections between Mathieu subspaces and the Jacobian conjecture. The claim is stated for general associative $K$-algebras, while the paper focuses on polynomial algebras; its resolution status is not established in the supplied text.