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Width bounds and Steinhaus property for unit groups of continuous rings

Josefin Bernard, Friedrich Martin Schneider

Abstract

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.

Width bounds and Steinhaus property for unit groups of continuous rings

Abstract

We prove an algebraic decomposition theorem for the unit group of an arbitrary non-discrete irreducible, continuous ring (in von Neumann's sense), which entails that every element of is both a product of commutators and a product of involutions. Combining this with further insights into the geometry of involutions, we deduce that has the so-called Steinhaus property with respect to the natural rank topology, thus every homomorphism from to a separable topological group is necessarily continuous. Due to earlier work, this has further dynamical ramifications: for instance, for every action of by homeomorphisms on a non-void metrizable compact space, every element of admits a fixed point in the latter. In particular, our results answer two questions by Carderi and Thom, even in generalized form.

Paper Structure

This paper contains 12 sections, 57 theorems, 319 equations.

Table of Contents

  1. Introduction
  2. Continuous geometries
  3. Regular rings
  4. Rank functions and continuous rings
  5. Subgroups induced by idempotents
  6. Geometry of involutions
  7. Dynamical independence of ideals
  8. Algebraic elements
  9. Approximation by matrix algebras
  10. Decomposition into locally special elements
  11. Steinhaus property
  12. Proofs of corollaries

Key Result

Theorem 1.1

Let $R$ be a non-discrete irreducible, continuous ring. Every element $a\in\mathop{\mathrm{GL}}\nolimits(R)$ admits a decomposition where

Theorems & Definitions (137)

  • Theorem 1.1: Theorem \ref{['theorem:decomposition']}
  • Corollary 1.2: Theorem \ref{['theorem:width']}
  • Theorem 1.3: Theorem \ref{['theorem:194-Steinhaus']}
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3: VonNeumannBook
  • ...and 127 more