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${}^{\ast }$-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${}^{\ast }$-triples preserving truncations in both directions. Let $A:E\to F$ be a surjective additive mapping between atomic JBW${}^{\ast }$-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 ${\tilde{C}}_{\sigma (k)}$ satisfying ${\inf }_k\{{\gamma }_k\}>0,$ and $$A(x)=({\gamma }_k{\Phi }_k\left({\pi }_k(x)\right){)}_{k\in {\Gamma }_1},for\; allx\in E,$$ where ${\pi }_k$ denotes the canonical projection of $E$ onto $C_k.$