Direct integral of locally Hilbert spaces over a locally measure space
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918115990831104 |
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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 by generalizing the classical notion of a measure space to that of a locally measure space. We establish that the direct integral of a family of locally Hilbert spaces over a locally measure space forms a locally Hilbert space. We then define two important subclasses of locally bounded operators on such direct integrals, namely decomposable locally bounded operators and diagonalizable locally bounded operators. We show that each of these subclasses forms a locally von Neumann algebra, and in particular, that the locally von Neumann algebra of diagonalizable operators is abelian. Finally, we prove that the locally von Neumann algebra of diagonalizable operators coincides with the commutant of the locally von Neumann algebra of decomposable operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Direct integral of locally Hilbert spaces over a locally measure space Kulkarni, Chaitanya J. Pamula, Santhosh Kumar Functional Analysis Operator Algebras 46A13, 46M40, 47L10, 46A03, 46C05 In this work, we introduce the concept of the direct integral of locally Hilbert spaces by generalizing the classical notion of a measure space to that of a locally measure space. We establish that the direct integral of a family of locally Hilbert spaces over a locally measure space forms a locally Hilbert space. We then define two important subclasses of locally bounded operators on such direct integrals, namely decomposable locally bounded operators and diagonalizable locally bounded operators. We show that each of these subclasses forms a locally von Neumann algebra, and in particular, that the locally von Neumann algebra of diagonalizable operators is abelian. Finally, we prove that the locally von Neumann algebra of diagonalizable operators coincides with the commutant of the locally von Neumann algebra of decomposable operators. |
| title | Direct integral of locally Hilbert spaces over a locally measure space |
| topic | Functional Analysis Operator Algebras 46A13, 46M40, 47L10, 46A03, 46C05 |
| url | https://arxiv.org/abs/2508.03779 |