Width bounds and Steinhaus property for unit groups of continuous rings
| Autor: | Bernard, Josefin, Schneider, Friedrich Martin |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 61 pages, no figures |
| Databáze: | arXiv |
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