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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