Maps preserving the truncation of triple products on Cartan factors

2024/05/22 by Garcés, Jorge J., Li, Lei, Peralta, Antonio M. +1
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2405.13489

Abstract

Let \C_i\_{i∈ Γ1}, and \D_j\_{j∈ Γ2}, be two families of Cartan factors such that all of them have dimension at least 2, and consider the atomic JBW^*-triples A=\bigoplus_{i∈ Γ1}^ℓ_∞ C_i and B=\bigoplus_{j∈ Γ2}^ℓ_∞ D_j. Let Δ:A → B be a \rm(non-necessarily linear nor continuous\rm) bijection preserving the truncation of triple products in both directions, that is, \beginaligned \boxeda is a truncation of \b,c,b\ ⇔ \boxedΔ(a) is a truncation of \Δ(b),Δ(c),Δ(b)\ \endaligned Assume additionally that the restriction of Δ to each rank-one Cartan factor in A, if any, is a continuous mapping. Then we show that Δ 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.

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