Search

We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C*-algebra admits at most one KMS-state for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups., Comment: 45 pages, v2 corrects a few minor points. A final version of the paper will appear in the Analysis & PDE

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced., La notion de propriété d'approximation de Haagerup, introduite à l'origine pour les algèbres de von Neumann ayant une trace finie, normale, et fidèle, est généralisée pour les algèbres de von Neumann arbitraires. Nous discutons deux caractérisations équivalentes : une du point de vue de la représentation standard et une autre du point de vue des applications linéaires approximantes liées à un poids fidèle, normal, semifini. Quelques propriétés de permanence, en particulier celles concernant les produits croisés, sont établies et certains exemples sont introduits.