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Dynamical systems and operator algebras associated to Artin’s representation of braid groups. (English) Zbl 1463.46082

The author presents a study on Artin’s representations, letting the braid group \(B_n\) of \(n\) strands act on the various group \(C^*\)-algebras of the free group \({\mathbb{F}}_n\) on \(n\) generators, and also allowing the permutations to be twisted by a scalar-valued cocycle. To do this, \(C^*\)-dynamical systems are defined and the corresponding crossed products \(C^*\)-algebras are investigated. Notions on braid groups \(B_n\) and \(P_n\) and Artin’s representation are presented and preliminaries on group \(C^*\)-algebras and semidirect products are recalled. After that, the main results on twisted versions of Artin’s representation are given. Applying techniques developed in [E. Bédos and T. Omland, J. Noncommut. Geom. 12, No. 3, 947–996 (2018; Zbl 1412.46069)], complete proofs of the main theorems are given and more results on twisted group \(C^*\)-algebras and the second cohomology group are obtained. Using Artin’s representation on the relative Kleppner condition, it is proved that the braid groups \(B_\infty\) and \(P_\infty\) on infinitely many strands are both \(C^*\)-simple.

MSC:

46L05 General theory of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
20F36 Braid groups; Artin groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations

Citations:

Zbl 1412.46069