On relationship among three types of Birkhoff-James orthogonality
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| Format: | Preprint |
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2025
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| _version_ | 1866908665012813824 |
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| author | Daptari, Soumitra Igarashi, Koki Nakamura, Jumpei Tanaka, Ryotaro |
| author_facet | Daptari, Soumitra Igarashi, Koki Nakamura, Jumpei Tanaka, Ryotaro |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_15086 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On relationship among three types of Birkhoff-James orthogonality Daptari, Soumitra Igarashi, Koki Nakamura, Jumpei Tanaka, Ryotaro Functional Analysis 46L08, 46B20 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. |
| title | On relationship among three types of Birkhoff-James orthogonality |
| topic | Functional Analysis 46L08, 46B20 |
| url | https://arxiv.org/abs/2511.15086 |