Universal covering groups of unitary groups of von Neumann algebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929631840436224 |
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| author | Sarkowicz, Pawel |
| author_facet | Sarkowicz, Pawel |
| contents | We give a short and simple proof, utilizing the pre-determinant of P. de la Harpe and G. Skandalis, that the universal covering group of the unitary group of a II$_1$ von Neumann algebra $\mathcal{M}$, when equipped with the norm topology, splits algebraically as the direct product of the self-adjoint part of its center and the unitary group $U(\mathcal{M})$. Thus, when $\mathcal{M}$ is a II$_1$ factor, the universal covering group of $U(\mathcal{M})$ is algebraically isomorphic to the direct product $\mathbb{R} \times U(\mathcal{M})$. In particular, the question of P. de la Harpe and D. McDuff of whether the universal cover of $U(\mathcal{M})$ is a perfect group is answered in the negative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universal covering groups of unitary groups of von Neumann algebras Sarkowicz, Pawel Operator Algebras 46L10 46L05 (Primary), 46L80 (Secondary) We give a short and simple proof, utilizing the pre-determinant of P. de la Harpe and G. Skandalis, that the universal covering group of the unitary group of a II$_1$ von Neumann algebra $\mathcal{M}$, when equipped with the norm topology, splits algebraically as the direct product of the self-adjoint part of its center and the unitary group $U(\mathcal{M})$. Thus, when $\mathcal{M}$ is a II$_1$ factor, the universal covering group of $U(\mathcal{M})$ is algebraically isomorphic to the direct product $\mathbb{R} \times U(\mathcal{M})$. In particular, the question of P. de la Harpe and D. McDuff of whether the universal cover of $U(\mathcal{M})$ is a perfect group is answered in the negative. |
| title | Universal covering groups of unitary groups of von Neumann algebras |
| topic | Operator Algebras 46L10 46L05 (Primary), 46L80 (Secondary) |
| url | https://arxiv.org/abs/2408.13710 |