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

Abstract

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 $B_n\to \mathrm{Aut}{C}_r^{\ast }({\mathbb{F}}_n)$ and $B_n\to \mathrm{Aut}{C}^{\ast }({\mathbb{F}}_n)$ into the automorphism groups of the corresponding group $C^{\ast }$-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 $C^{\ast }$-simple.

arXiv comment14 pages; some irrelevant paragraphs and remarks removed; to appear in J. Operator Theory