Identifying JBW^*-algebras through their spheres of positive elements

2025/05/06 by Antonio M. Peralta, Pedro Saavedra, Peralta, Antonio M. +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2505.03287

openalex publication_date 2025/05/06 · openalex created_date 2025/10/16 · openalex updated_date 2026/08/01

Abstract

Let \mathfrakA and \mathfrakB be JBW^*-algebras with projection lattices P (\mathfrakA) and P (\mathfrakB), and let Θ: P (\mathfrakA)→ P(\mathfrakB) be an order isomorphism. We prove that if \mathfrakA does not contain any type I₂ direct summand and Θ preserves points at distance 1, then Θ extends to a Jordan ^*-isomorphism from \mathfrakA onto \mathfrakB. We also establish that if \mathfrakA and \mathfrakB are two atomic JBW^*-algebras of type I₂ and Θ: P (\mathfrakA)→ P(\mathfrakB) preserves points at distance (√(2))/(2), then \mathfrakA is Jordan ^*-isomorphic to \mathfrakB. Furthermore, if \mathfrakA and \mathfrakB are two general JBW^*-algebras such that the type I₂ part of \mathfrakA is atomic and Θ is an isometry, we prove the existence of an extension of Θ to a Jordan ^*-isomorphism from \mathfrakA onto \mathfrakB. We provide a positive answer to Tingley's problem for positive spheres showing that if \mathfrakA and \mathfrakB are JBW^*-algebras such that the type I₂ part of \mathfrakA is atomic, then every surjective isometry from the set, S_\mathfrakA⁺, of positive norm-one elements of \mathfrakA onto the positive norm-one elements of \mathfrakB extends to a Jordan ^*-isomorphism from \mathfrakA onto \mathfrakB. We prove a metric characterization of projections in JBW^*-algebras as follows: if a is a norm-one positive element in a JBW^*-algebra \mathfrakA, then a is a projection if, and only if, it satisfies the double sphere property, that is, \c ∈ S_\mathfrakA⁺ : ‖c - b‖ = 1 for all b ∈ S_\mathfrakA⁺ with ‖b - a‖ = 1\ = \a\.

Citations

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Related

• Tingley's problem through the facial structure of an atomic JBW*-triple
• On the extension of isometries between the unit spheres of a JBW^*-triple and a Banach space
• Order isomorphisms between cones of JB-algebras
• Can one identify two unital JB^*-algebras by the metric spaces\n determined by their sets of unitaries?
• Exploring new solutions to Tingley's problem for function algebras