Operational 2-local automorphisms/derivations

Abstract

Let $\ddot{I}•:A\to A$ 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 $\hat{I}{¸}_{a,b}$ of $A$ such that \begin{align*} ϕ(a)ϕ(b) = θ_{a,b}(ab). \end{align*} We show that either $\ddot{I}•$ or $-\ddot{I}•$ is a linear Jordan homomorphism. Similar results are obtained when any of the following conditions is satisfied: \begin{align*} ϕ(a) + ϕ(b) &= θ_{a,b}(a+b), \\ ϕ(a)ϕ(b)+ϕ(b)ϕ(a) &= θ_{a,b}(ab+ba), \quad\text{or} \\ ϕ(a)ϕ(b)ϕ(a) &= θ_{a,b}(aba). \end{align*} We also show that a map $\ddot{I}•:M\to M$ 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 $$\ddot{I}•(a)b+a\ddot{I}•(b)=D_{a,b}(ab).$$

10 pages; to appear in J. Nonlinear and Convex Analysis