Direct integral of locally Hilbert spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918115374268416 |
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| author | Kulkarni, Chaitanya J. Pamula, Santhosh Kumar |
| author_facet | Kulkarni, Chaitanya J. Pamula, Santhosh Kumar |
| contents | In this work, we introduce the concept of the direct integral of locally Hilbert spaces. This notion is formulated such that the direct integral of locally Hilbert spaces forms a locally Hilbert space. We then define two classes of locally bounded operators on the direct integral of locally Hilbert spaces namely the class of decomposable locally bounded operators and the class of diagonalizable locally bounded operators. We prove that the set of all decomposable locally bounded operators forms a locally von Neumann algebra, while the set of diagonalizable locally bounded operators forms an abelian von Neumann algebra, and we show that they are commutant of each other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Direct integral of locally Hilbert spaces Kulkarni, Chaitanya J. Pamula, Santhosh Kumar Functional Analysis Operator Algebras 46A03, 46L05, 46A13, 47L40, 46A40 In this work, we introduce the concept of the direct integral of locally Hilbert spaces. This notion is formulated such that the direct integral of locally Hilbert spaces forms a locally Hilbert space. We then define two classes of locally bounded operators on the direct integral of locally Hilbert spaces namely the class of decomposable locally bounded operators and the class of diagonalizable locally bounded operators. We prove that the set of all decomposable locally bounded operators forms a locally von Neumann algebra, while the set of diagonalizable locally bounded operators forms an abelian von Neumann algebra, and we show that they are commutant of each other. |
| title | Direct integral of locally Hilbert spaces |
| topic | Functional Analysis Operator Algebras 46A03, 46L05, 46A13, 47L40, 46A40 |
| url | https://arxiv.org/abs/2508.03407 |