Characterization of minimal tripotents via annihilators and its application to the study of additive preservers of truncations
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2024
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| _version_ | 1866929639858896896 |
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| author | Li, Lei Liu, Siyu Peralta, Antonio M. |
| author_facet | Li, Lei Liu, Siyu Peralta, Antonio M. |
| contents | The contributions in this note begin with a new characterization of (positive) scalar multiples of minimal tripotents in a general JB$^*$-triple $E$, proving that a non-zero element $a\in E$ is a positive scalar multiple of a minimal tripotent in $E$ if, and only if, its inner quadratic annihilator (that is, the set $^{\perp_{q}}\!\{a\} = \{ b\in E: \{a,b,a\} =0\}$) is maximal among all inner quadratic annihilators of single elements in $E$. We subsequently apply this characterization to the study of surjective additive maps between atomic JBW$^*$-triples preserving truncations in both directions. Let $A: E\to F$ be a surjective additive mapping between atomic JBW$^*$-triples, where $E$ contains no one-dimensional Cartan factors as direct summands. We show that $A$ preserves truncations in both directions if, and only if, there exists a bijection $σ: Γ_1\to Γ_2$, a bounded family $(γ_k)_{k\in Γ_1}\subseteq \mathbb{R}^+$, and a family $(Φ_k)_{k\in Γ_1},$ where each $Φ_k$ is a (complex) linear or a conjugate-linear (isometric) triple isomorphism from $C_k$ onto $\widetilde{C}_{σ(k)}$ satisfying $\inf_{k} \{γ_k \} >0,$ and $$A(x) = \Big( γ_{k} Φ_k \left(π_k(x)\right) \Big)_{k\inΓ_1},\ \hbox{ for all } x\in E,$$ where $π_k$ denotes the canonical projection of $E$ onto $C_k.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_14394 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization of minimal tripotents via annihilators and its application to the study of additive preservers of truncations Li, Lei Liu, Siyu Peralta, Antonio M. Operator Algebras Functional Analysis The contributions in this note begin with a new characterization of (positive) scalar multiples of minimal tripotents in a general JB$^*$-triple $E$, proving that a non-zero element $a\in E$ is a positive scalar multiple of a minimal tripotent in $E$ if, and only if, its inner quadratic annihilator (that is, the set $^{\perp_{q}}\!\{a\} = \{ b\in E: \{a,b,a\} =0\}$) is maximal among all inner quadratic annihilators of single elements in $E$. We subsequently apply this characterization to the study of surjective additive maps between atomic JBW$^*$-triples preserving truncations in both directions. Let $A: E\to F$ be a surjective additive mapping between atomic JBW$^*$-triples, where $E$ contains no one-dimensional Cartan factors as direct summands. We show that $A$ preserves truncations in both directions if, and only if, there exists a bijection $σ: Γ_1\to Γ_2$, a bounded family $(γ_k)_{k\in Γ_1}\subseteq \mathbb{R}^+$, and a family $(Φ_k)_{k\in Γ_1},$ where each $Φ_k$ is a (complex) linear or a conjugate-linear (isometric) triple isomorphism from $C_k$ onto $\widetilde{C}_{σ(k)}$ satisfying $\inf_{k} \{γ_k \} >0,$ and $$A(x) = \Big( γ_{k} Φ_k \left(π_k(x)\right) \Big)_{k\inΓ_1},\ \hbox{ for all } x\in E,$$ where $π_k$ denotes the canonical projection of $E$ onto $C_k.$ |
| title | Characterization of minimal tripotents via annihilators and its application to the study of additive preservers of truncations |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2412.14394 |