Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries

De Rijdt, An; Vander Vennet, Nikolas

Annales de l'Institut Fourier, Tome 60 (2010) no. 1, pp. 169-216.

La notion de l’équivalence monoïdale pour les groupes quantiques compacts a été introduite récemment par Bichon, De Rijdt et Vaes. Dans cet article, nous montrons  : étant donné deux groupes quantiques compacts à équivalence monoïdale, alors il existe une correspondance bijective entre leurs actions. Cette correspondance s’avère être très utile pour obtenir la relation entre les frontières de Poisson et Martin des deux groupes quantiques compacts à équivalence monoïdale. Finalement, nous appliquons ces résultats au calcul des frontières de Poisson des duals associés aux groupes quantiques d’automorphismes.

The notion of monoidal equivalence for compact quantum groups was recently introduced by Bichon, De Rijdt and Vaes. In this paper we prove that there is a natural bijective correspondence between actions of monoidally equivalent quantum groups on unital $C^{*}$-algebras or on von Neumann algebras. This correspondence turns out to be very useful to obtain the behavior of Poisson and Martin boundaries under monoidal equivalence of quantum groups. Finally, we apply these results to identify the Poisson boundary for the duals of quantum automorphism groups.

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