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∈ 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: 1 2, a bounded family (γk)k∈ 1⊂eq R+, and a family (k)k∈ 1, where each k is a (complex) linear or a conjugate-linear (isometric) triple isomorphism from Ck onto Cσ(k) satisfying ∈fk \γk \ >0, and A(x) = ( γk k (πk(x)) )k∈1,\ for all x∈ E, where πk denotes the canonical projection of E onto Ck.
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