Operational 2-local automorphisms/derivations

TL;DR

This paper characterizes certain nonlinear maps on operator algebras that behave like automorphisms or derivations under specific algebraic conditions, showing they are essentially linear Jordan homomorphisms or derivations.

Contribution

It extends the understanding of local automorphisms and derivations by establishing their linearity and structure under algebraic constraints in operator algebras.

Findings

Maps satisfying algebraic conditions are linear Jordan homomorphisms.

Similar results hold for maps satisfying additive or symmetrized conditions.

Maps on semi-finite von Neumann algebras are linear derivations under given conditions.

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 $$ \phi(a)b + a\phi(b) = D_{a,b}(ab). $$