Operational 2-local automorphisms/derivations

Abstract: Let $\phi: A\to A$ be a (not necessarily linear, additive or continuous) map of a standard operator algebra. Suppose for any $a,b\in A$ there is an algebra automorphism $\theta_{a,b}$ of $ A$ such that \begin{align*} \phi(a)\phi(b) = \theta_{a,b}(ab). \end{align*} We show that either $\phi$ or $-\phi$ is a linear Jordan homomorphism. Similar results are obtained when any of the following conditions is satisfied: \begin{align*} \phi(a) + \phi(b) &= \theta_{a,b}(a+b), \ \phi(a)\phi(b)+\phi(b)\phi(a) &= \theta_{a,b}(ab+ba), \quad\text{or} \ \phi(a)\phi(b)\phi(a) &= \theta_{a,b}(aba). \end{align*} We also show that a map $\phi: M\to M$ of a semi-finite von Neumann algebra $ M$ is a linear derivation if for every $a,b\in M$ there is a linear derivation $D_{a,b}$ of $M$ such that $$ \phi(a)b + a\phi(b) = D_{a,b}(ab). $$