Publications

• arXiv version, HAL version.

• Abstract. We introduce a Künneth class in the quantum equivariant setting inspired by the pioneer work by J. Chabert, H. Oyono-Oyono and S. Echterhoff, which allows to relate the quantum Baum-Connes property with the Künneth formula by generalising some key results of Chabert-Oyono-Oyono-Echterhoff to discrete quantum groups. Finally, we make the observation that the C*-algebra defining a compact quantum group with dual satisfying the strong quantum Baum-Connes property belongs to the Künneth class. This allows to obtain some K-theory computations for quantum direct products based on earlier work by Voigt and Vergnioux-Voigt.

• arXiv version, HAL version.

• Abstract. We prove that torsion-freeness in the sense of Meyer-Nest is preserved under divisible discrete quantum subgroups. As a consequence, we obtain some stability results of the torsion-freeness property for relevant constructions of quantum groups (quantum (semi-)direct products, compact bicrossed products and quantum free products). We improve some stability results concerning the Baum-Connes conjecture appearing already in a previous work of the author. For instance, we show that the (resp. strong) Baum-Connes conjecture is preserved by discrete quantum subgroups (without any torsion-freeness or divisibility assumption).

• arXiv version, HAL version.

• Abstract. We study the theory of projective representations for a compact quantum group G, i.e. actions of G on B(H) for some Hilbert space H. We show that any such projective representation is inner, and is hence induced by an Ω-twisted representation for some unitary measurable 2-cocycle Ω on G. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators K(H), if and only if the associated 2-cocycle is regular, and that this condition is automatically satisfied if G is of Kac type. This allows in particular to characterise the torsion of projective type of G^ in terms of the projective representation theory of G. For a given regular unitary 2-cocycle Ω, we then study Ω-twisted actions on C*-algebras. We define deformed crossed products with respect to Ω,  obtaining a twisted version of the Baaj-Skandalis duality and a quantum version of the Packer-Raeburn's trick. As an application, we provide a twisted version of the Green-Julg isomorphism and obtain the quantum Baum-Connes assembly map for permutation torsion-free discrete quantum groups.

• arXiv version, HAL version.

• Published version.

• Abstract. The well known "associativity property" of the crossed product by a semi-direct product of discrete groups is generalized into the context of discrete quantum groups. This decomposition allows to define an appropriate triangulated functor relating the Baum-Connes property for the quantum semi-direct product to the Baum-Connes property for the discrete quantum groups involved in the construction. The corresponding stability result for the Baum-Connes property generalizes a result of J. Chabert for a quantum semi-direct product under torsion-freeness assumption. The K-amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena. The analogous strategy can be applied for the dual of a quantum direct product. In this case, we obtain, in addition, a connection with the Künneth formula, which is the quantum counterpart to a result of J. Chabert, S. Echterhoff and H. Oyono-Oyono. Again the K-amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena.

• arXiv version, HAL version.

• Published version.

• Abstract. We classify torsion actions of free wreath products of arbitrary compact quantum groups by SN+ and use this to prove that if G is a torsion-free compact quantum group satisfying the strong Baum-Connes property, then G≀*SN+ also satisfies the strong Baum-Connes property. We then compute the K-theory of free wreath products of classical and quantum free groups by SOq(3).

• Abstract. Quantum games are mental experiments designed to demonstrate the effects of quantum mechanics on information. In this article, we look at one such game in particular, in which two players, Alice and Bob, work out a joint strategy to win the game, then get separated and questioned by a game master (referee). Their answers determine their joint win. The aim of physicists in inventing these "games" is to show that the strategies possible in the world of classical physics can be beaten by exploiting quantum effects!

We are going to try to understand how this is possible in the case of the CHSH game (invented by Clauser, Horne, Shimony and Hett). Quantum physics will lead us to do a bit of complex geometry, i.e. to place ourselves in spaces whose coordinates can take on complex values. The aim of this article is to show how complex numbers, invented by the algebraists of the 16th century, can be used by today's physicists.

We will show that a classical deterministic strategy can win this game 75% of the time, while a quantum strategy can improve this score and win more than 85% of the time!