Characterization of minimal tripotents via annihilators and its application to the study of additive preservers of truncations

Abstract

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 $\sigma: \Gamma_1\to \Gamma_2$, a bounded family $(\gamma_k)_{k\in \Gamma_1}\subseteq \mathbb{R}^+$, and a family $(\Phi_k)_{k\in \Gamma_1},$ where each $\Phi_k$ is a (complex) linear or a conjugate-linear (isometric) triple isomorphism from $C_k$ onto $\widetilde{C}_{\sigma(k)}$ satisfying $\inf_{k} \{\gamma_k \} >0,$ and $$A(x) = \Big( \gamma_{k} \Phi_k \left(\pi_k(x)\right) \Big)_{k\in\Gamma_1},\ \hbox{ for all } x\in E,$$ where $\pi_k$ denotes the canonical projection of $E$ onto $C_k.$