2024/12/18 by Lei Li, Siyu Liu, Li, Lei +3 · 1 citation
Computer Science · Mathematics · #Cellular Automata and Applications #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.2412.14394
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∈ E is a positive scalar multiple of a minimal tripotent in E if, and only if, its inner quadratic annihilator (that is, the set ^⊥q \a\ = \ b∈ 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→ 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: Γ1→ Γ2, a bounded family (γk)k∈ Γ1⊆ ℝ+, and a family (Φk)k∈ Γ1, where each Φk is a (complex) linear or a conjugate-linear (isometric) triple isomorphism from Ck onto \widetildeCσ(k) satisfying infk \γk \ >0, and A(x) = ( γk Φk (πk(x)) )k∈Γ1, \hbox for all x∈ E, where πk denotes the canonical projection of E onto Ck.