Width bounds and Steinhaus property for unit groups of continuous rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914106865352704 |
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| author | Bernard, Josefin Schneider, Friedrich Martin |
| author_facet | Bernard, Josefin Schneider, Friedrich Martin |
| contents | We prove an algebraic decomposition theorem for the unit group $\mathrm{GL}(R)$ of an arbitrary non-discrete irreducible, continuous ring $R$ (in von Neumann's sense), which entails that every element of $\mathrm{GL}(R)$ is both a product of $7$ commutators and a product of $16$ involutions. Combining this with further insights into the geometry of involutions, we deduce that $\mathrm{GL}(R)$ has the so-called Steinhaus property with respect to the natural rank topology, thus every homomorphism from $\mathrm{GL}(R)$ to a separable topological group is necessarily continuous. Due to earlier work, this has further dynamical ramifications: for instance, for every action of $\mathrm{GL}(R)$ by homeomorphisms on a non-void metrizable compact space, every element of $\mathrm{GL}(R)$ admits a fixed point in the latter. In particular, our results answer two questions by Carderi and Thom, even in generalized form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Width bounds and Steinhaus property for unit groups of continuous rings Bernard, Josefin Schneider, Friedrich Martin Group Theory Rings and Algebras 22A05, 37B02, 06C20, 16E50 We prove an algebraic decomposition theorem for the unit group $\mathrm{GL}(R)$ of an arbitrary non-discrete irreducible, continuous ring $R$ (in von Neumann's sense), which entails that every element of $\mathrm{GL}(R)$ is both a product of $7$ commutators and a product of $16$ involutions. Combining this with further insights into the geometry of involutions, we deduce that $\mathrm{GL}(R)$ has the so-called Steinhaus property with respect to the natural rank topology, thus every homomorphism from $\mathrm{GL}(R)$ to a separable topological group is necessarily continuous. Due to earlier work, this has further dynamical ramifications: for instance, for every action of $\mathrm{GL}(R)$ by homeomorphisms on a non-void metrizable compact space, every element of $\mathrm{GL}(R)$ admits a fixed point in the latter. In particular, our results answer two questions by Carderi and Thom, even in generalized form. |
| title | Width bounds and Steinhaus property for unit groups of continuous rings |
| topic | Group Theory Rings and Algebras 22A05, 37B02, 06C20, 16E50 |
| url | https://arxiv.org/abs/2412.17480 |