Related papers: On the Baum--Connes conjecture for discrete quantu…

In this paper, we use the KK-theory of Kasparov to prove exactness of sequences relating the K-theory of a real C^*-algebra and of its complexification (generalizing results of Boersema). We use this to relate the real version of the…

We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group $\mathbb{G}$ and certain left invariant C*-subalgebras of $C_0(\mathbb{G})$. We also prove that every compact…

We develop a generalization of quantitative $K$-theory, which we call controlled $K$-theory. It is powerful enough to study the $K$-theory of crossed product of $C^*$-algebras by action of \'etale groupoids and discrete quantum groups. In…

We develop a fundamental theory of compact quantum group equivariant finite extensions of C*-algebras. In particular we focus on the case of quantum homogeneous spaces and give a Tannaka-Krein type result for equivariant correspondences. As…

For a large class of C*-algebras $A$, we calculate the $K$-theory of reduced crossed products $A^{\otimes G}\rtimes_rG$ of Bernoulli shifts by groups satisfying the Baum--Connes conjecture. In particular, we give explicit formulas for…

The Gromov-Lawson-Rosenberg conjecture for a group G states that a compact spin manifold with fundamental group G admits a metric of positive scalar curvature if and only if a certain topological obstruction vanishes. It is known to be true…

We introduce a K\"unneth 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 K\"unneth formula by…

We compute the group homology, the topological K-theory of the reduced C^*-algebra, the algebraic K-theory and the algebraic L-theory of the group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by Z/4.…

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed…

Let $H$ be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra $A$. It was recently conjectured that there does not exist an equivariant $*$-homomorphism from $A$ (type-I case) or $H$ (type-II case) to…

We show that complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups, satisfy a categorical version of the Baum-Connes conjecture with trivial coefficients. This approach, based on…

Given a (not necessarily discrete) proper metric space $M$ with bounded geometry, we define a groupoid $G(M)$. We show that the coarse Baum--Connes conjecture with coefficients, which states that the assembly map with coefficients for G(M)…

Let G be a locally compact group and rho a non-unitary finite dimensional representation of G. We consider tensor products of rho by some unitary representations of G in order to define two Banach algebras analogous to the group…

Let G be a finite group. We systematically exploit general homological methods in order to reduce the computation of G-equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor assigning to a…

We associate a non-commutative $C^*$-algebra with any locally finite simplicial complex. We determine the $K$-theory of these algebras and show that they can be used to obtain a conceptual explanation for the Baum-Connes conjecture.

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…

A torsion class $\mathcal{T}$ of the module category $\operatorname{\mathsf{mod}} A$ of a finite dimensional algebra $A$ over a field $K$ is said to be compact if there exists a module $M \in \operatorname{\mathsf{mod}} A$ such that…

We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{i\pi/\ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor…

In this paper we study the $ K $-theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are $ K $-amenable and…

Given an action of a discrete quantum group (in the sense of Van Daele, Kustermans and Effros-Ruan) ${\cal A}$ on a $C^*$-algebra ${\cal C}$, satisfying some regularity assumptions resembling the proper $\Gamma$-compact action for a…