A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866911775729909760 |
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| author | Edeko, Nikolai Haase, Markus Kreidler, Henrik |
| author_facet | Edeko, Nikolai Haase, Markus Kreidler, Henrik |
| contents | In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky-Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, de Leeuw and Glicksberg). The proof rests heavily on the operator theory on Kaplansky-Hilbert modules, in particular the spectral theorem for Hilbert-Schmidt homomorphisms on such modules. As an application, a generalization of the celebrated Furstenberg-Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_04865 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem Edeko, Nikolai Haase, Markus Kreidler, Henrik Dynamical Systems Functional Analysis Operator Algebras Primary 37A15, 43A65, Secondary 46H25, 06E15 In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky-Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, de Leeuw and Glicksberg). The proof rests heavily on the operator theory on Kaplansky-Hilbert modules, in particular the spectral theorem for Hilbert-Schmidt homomorphisms on such modules. As an application, a generalization of the celebrated Furstenberg-Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established. |
| title | A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem |
| topic | Dynamical Systems Functional Analysis Operator Algebras Primary 37A15, 43A65, Secondary 46H25, 06E15 |
| url | https://arxiv.org/abs/2104.04865 |