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 Bn on n strands into AutFn, the automorphism group of the free group Fn on n generators. The representation induces maps Bn→AutC∗r(Fn) and Bn→AutC∗(Fn) into the automorphism groups of the corresponding group C∗-algebras of Fn. These maps also have natural restrictions to the pure braid group Pn. 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∞ and P∞ on infinitely many strands are both C∗-simple.