Maps preserving the truncation of triple products on Cartan factors
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910457180192768 |
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| author | Garcés, Jorge J. Li, Lei Peralta, Antonio M. Su, Shanshan |
| author_facet | Garcés, Jorge J. Li, Lei Peralta, Antonio M. Su, Shanshan |
| contents | Let $\{C_i\}_{i\in Γ_1},$ and $\{D_j\}_{j\in Γ_2},$ be two families of Cartan factors such that all of them have dimension at least $2$, and consider the atomic JBW$^*$-triples $A=\bigoplus\limits_{i\in Γ_1}^{\ell_{\infty}} C_i$ and $B=\bigoplus\limits_{j\in Γ_2}^{\ell_{\infty}} D_j$. Let $Δ:A \to B$ be a {\rm(}non-necessarily linear nor continuous{\rm)} bijection preserving the truncation of triple products in both directions, that is, $$\begin{aligned}
\boxed{a \mbox{ is a truncation of } \{b,c,b\}} \Leftrightarrow \boxed{Δ(a) \mbox{ is a truncation of } \{Δ(b),Δ(c),Δ(b)\}}
\end{aligned}$$ Assume additionally that the restriction of $Δ$ to each rank-one Cartan factor in $A$, if any, is a continuous mapping. Then we show that $Δ$ is an isometric real linear triple isomorphism. We also study some general properties of bijections preserving the truncation of triple products in both directions between general JB$^*$-triples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13489 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maps preserving the truncation of triple products on Cartan factors Garcés, Jorge J. Li, Lei Peralta, Antonio M. Su, Shanshan Operator Algebras Functional Analysis Let $\{C_i\}_{i\in Γ_1},$ and $\{D_j\}_{j\in Γ_2},$ be two families of Cartan factors such that all of them have dimension at least $2$, and consider the atomic JBW$^*$-triples $A=\bigoplus\limits_{i\in Γ_1}^{\ell_{\infty}} C_i$ and $B=\bigoplus\limits_{j\in Γ_2}^{\ell_{\infty}} D_j$. Let $Δ:A \to B$ be a {\rm(}non-necessarily linear nor continuous{\rm)} bijection preserving the truncation of triple products in both directions, that is, $$\begin{aligned} \boxed{a \mbox{ is a truncation of } \{b,c,b\}} \Leftrightarrow \boxed{Δ(a) \mbox{ is a truncation of } \{Δ(b),Δ(c),Δ(b)\}} \end{aligned}$$ Assume additionally that the restriction of $Δ$ to each rank-one Cartan factor in $A$, if any, is a continuous mapping. Then we show that $Δ$ is an isometric real linear triple isomorphism. We also study some general properties of bijections preserving the truncation of triple products in both directions between general JB$^*$-triples. |
| title | Maps preserving the truncation of triple products on Cartan factors |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2405.13489 |