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

TL;DR

This paper explores the operator algebraic structures arising from Artin's braid group representations, investigates twisted actions via cocycles, and establishes the $C^*$-simplicity of infinite braid groups.

Contribution

It introduces twisted automorphism actions on group $C^*$-algebras, analyzes their ideal structures, and proves the $C^*$-simplicity of infinite braid groups using Artin's representation.

Findings

Analysis of ideal structures of crossed products

Introduction of cocycle-twisted automorphisms

Proof of $C^*$-simplicity for $B_ $ and $P_ $ with infinite strands

Abstract

Artin's representation is an injective homomorphism from the braid group $B_n$ on $n$ strands into $\operatorname{Aut}\mathbb{F}_n$, the automorphism group of the free group $\mathbb{F}_n$ on $n$ generators. The representation induces maps $B_n\to\operatorname{Aut}C^*_r(\mathbb{F}_n)$ and $B_n\to\operatorname{Aut}C^*(\mathbb{F}_n)$ 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.