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The free wreath product of a discrete group by a quantum automorphism group

Author: Lorenzo Pittau
Journal: Proc. Amer. Math. Soc. 144 (2016), 1985-2001
MSC (2010): Primary 46L65, 81R50
Published electronically: January 27, 2016
MathSciNet review: 3460161
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Abstract: Let $ \mathbb{G}$ be the quantum automorphism group of a finite dimensional C*-algebra $ (B,\psi )$ and $ \Gamma $ a discrete group. We want to compute the fusion rules of $ \widehat {\Gamma }\wr _* \mathbb{G}$. First of all, we will revise the representation theory of $ \mathbb{G}$ and, in particular, we will describe the spaces of intertwiners by using noncrossing partitions. It will allow us to find the fusion rules of the free wreath product in the general case of a state $ \psi $. We will also prove the simplicity of the reduced C*-algebra, when $ \psi $ is a trace, as well as the Haagerup property of $ L^\infty (\widehat {\Gamma }\wr _* \mathbb{G})$, when $ \Gamma $ is moreover finite.

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Lorenzo Pittau
Affiliation: Université de Cergy-Pontoise, 95000, Cergy-Pontoise, France — and — Univ. Paris Diderot, Sorbonne Paris Cité, IMJ-PRG, UMR 7586 CNRS, Sorbonne Universités, UPMC Univ Paris 06, F-75013, Paris, France

Received by editor(s): October 6, 2014
Received by editor(s) in revised form: May 14, 2015
Published electronically: January 27, 2016
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society