On relationship among three types of Birkhoff-James orthogonality

Soumitra Daptari; Koki Igarashi; Jumpei Nakamura; Ryotaro Tanaka

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arXiv · 2511.15086

On relationship among three types of Birkhoff-James orthogonality

Soumitra Daptari·Koki Igarashi·Jumpei Nakamura·Ryotaro Tanaka

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Abstract

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.

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Soumitra Daptari, Koki Igarashi, Jumpei Nakamura, Ryotaro Tanaka. 2025-11-19. On relationship among three types of Birkhoff-James orthogonality. https://arxiv.org/abs/2511.15086

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