Maps preserving the truncation of triple products on Cartan factors

TL;DR

This paper characterizes maps that preserve the structure of triple product truncations in Cartan factors and shows such maps are essentially isometric linear isomorphisms under certain conditions.

Contribution

It proves that bijections preserving triple product truncations on atomic JBW*-triples are isometric real linear triple isomorphisms, extending understanding of structure-preserving maps.

Findings

Preserving truncation of triple products implies isometric linearity.

Maps are characterized as triple isomorphisms under certain conditions.

Results extend to general JB*-triples with additional properties.

Abstract

Let $\{C_i\}_{i\in \Gamma_1},$ and $\{D_j\}_{j\in \Gamma_2},$ be two families of Cartan factors such that all of them have dimension at least $2$, and consider the atomic JBW$^*$-triples $A=\bigoplus\limits_{i\in \Gamma_1}^{\ell_{\infty}} C_i$ and $B=\bigoplus\limits_{j\in \Gamma_2}^{\ell_{\infty}} D_j$. Let $\Delta :A \to B$ be a {\rm(}non-necessarily linear nor continuous{\rm)} bijection preserving the truncation of triple products in both directions, that is, $$\begin{aligned} \boxed{a \mbox{ is a truncation of } \{b,c,b\}} \Leftrightarrow \boxed{\Delta(a) \mbox{ is a truncation of } \{\Delta(b),\Delta(c),\Delta(b)\}} \end{aligned}$$ Assume additionally that the restriction of $\Delta$ to each rank-one Cartan factor in $A$, if any, is a continuous mapping. Then we show that $\Delta$ 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.