# Journal of Operator Theory

Volume 83, Issue 1, Winter 2020 pp. 55-72.

Dynamical systems and operator algebras associated to Artin's representation of braid groups

**Authors**:
Tron Omland

**Author institution:**Department of Mathematics, University of Oslo, NO-0316
Oslo, Norway \textit{and} Department of Computer Science, Oslo Metropolitan University, NO-0130 Oslo, Norway

**Summary: **Artin's representation is an injective homomorphism from the braid group $B_n$
on $n$ strands into $\mathrm{Aut}\,\mathbb{F}_n$, the automorphism group of the free
group $\mathbb{F}_n$ on $n$ generators. The representation induces maps
$B_n\to\mathrm{Aut}\, C^*_\mathrm r(\mathbb{F}_n)$ and $B_n\to\mathrm{Aut}\,
C^*(\mathbb{F}_n)$
into the automorphism groups of the corresponding group $C^*$-algebras of
$\mathbb{F}_n$. These maps also have natural restrictions to the pure braid group
$P_n$. In this paper, we consider twisted versions of the actions by cocycles
with values in the circle, and discuss the ideal structure of the associated
crossed products. Additionally, we make use of Artin's representation to show
that the braid groups $B_\infty$ and $P_\infty$ on infinitely many strands are
both $C^*$-simple.

**DOI: **http://dx.doi.org/10.7900/jot.2018jun12.2217

**Keywords: **Braid groups, $C^*$-dynamical systems, twisted group $C^*$-algebras, $C^*$-simplicity

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