Operational 2-local automorphisms/derivations

Liguang Wang; Ngai-Ching Wong

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arXiv · 2407.10150

Operational 2-local automorphisms/derivations

Liguang Wang·Ngai-Ching Wong

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Abstract

Let $ϕ: A\to A$ 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 $θ_{a,b}$ of $ A$ such that \begin{align*} ϕ(a)ϕ(b) = θ_{a,b}(ab). \end{align*} We show that either $ϕ$ or $-ϕ$ 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 $ϕ: M\to M$ 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 $$ ϕ(a)b + aϕ(b) = D_{a,b}(ab). $$

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BibTeX RIS

Liguang Wang, Ngai-Ching Wong. 2024-07-14. Operational 2-local automorphisms/derivations. https://arxiv.org/abs/2407.10150

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