Operational 2-local automorphisms/derivations
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911955110854656 |
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| author | Wang, Liguang Wong, Ngai-Ching |
| author_facet | Wang, Liguang Wong, Ngai-Ching |
| contents | 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 $$ ϕ(a)b + aϕ(b) = D_{a,b}(ab). $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10150 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Operational 2-local automorphisms/derivations Wang, Liguang Wong, Ngai-Ching Operator Algebras Functional Analysis 46L10, 46L50 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 $$ ϕ(a)b + aϕ(b) = D_{a,b}(ab). $$ |
| title | Operational 2-local automorphisms/derivations |
| topic | Operator Algebras Functional Analysis 46L10, 46L50 |
| url | https://arxiv.org/abs/2407.10150 |