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

2016/09/15 by Omland, Tron
#20F36 #22D25 #46L05 (Primary) #46L55 (Secondary) #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1609.04737

Abstract

Artin's representation is an injective homomorphism from the braid group B_n on n strands into Aut\mathbbF_n, the automorphism group of the free group \mathbbF_n on n generators. The representation induces maps B_n\toAutC^*_r(\mathbbF_n) and B_n\toAutC^*(\mathbbF_n) into the automorphism groups of the corresponding group C^*-algebras of \mathbbF_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_∞ and P_∞ on infinitely many strands are both C^*-simple.

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