2025/11/19 by Daptari, Soumitra, Igarashi, Koki, Nakamura, Jumpei +1
Computer Science · Mathematics · #46B20 #46L08 #Advanced Algebra and Logic #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2511.15086
openalex publication_date 2025/11/19 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28
In this paper, we study three types of Birkhoff-James orthogonality in Hilbert C^*-modules, that is, the strong, quasi-strong, and original Birkhoff-James orthogonality. In general, the strong Birkhoff-James orthogonality is stronger than the quasi-strong Birkhoff-James orthogonality, and the quasi-strong Birkhoff-James orthogonality is stronger than the original Birkhoff-James orthogonality. Meanwhile, each reverse implication in this chain requires additional conditions. As the main results, we show that the strong and quasi-strong Birkhoff-James orthogonality are equivalent in a full Hilbert C^*-module if and only if the underlying C^*-algebra is commutative, and that the equivalence of the quasi-strong and original Birkhoff-James orthogonality in a full Hilbert C^*-module implies the primeness of the underlying C^*-algebra. Moreover, two examples, explaining the complexity of conditions for full Hilbert C^*-modules in which the quasi-strong and original Birkhoff-James orthogonality are equivalent, are given in the C^*-algebra settings.