Stefaan Vaes

Locally compact quantum groups

These lecture notes are intended as an introduction to the theory of locally compact quantum groups that are studied in the framework of operator algebras, i.e. C*-algebras and von Neumann algebras. The presentation revolves around the definition of a locally compact quantum group as given in [KuV00a] and [KuV03].

Extensions of locally compact quantum groups and the bicrossed product construction

Abstract In the framework of locally compact quantum groups, we study cocycle actions. We develop the cocycle bicrossed product construction, starting from a matched pair of locally compact quantum groups. We define exact sequences and establish a one-to-one correspondence between cocycle bicrossed products and cleft extensions. In this way, we obtain new examples of locally compact quantum groups.

Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood.

The boundary of universal discrete quantum groups, exactness, and factoriality

We study the C -algebras and von Neumann algebras associated with the universal discrete quantum groups. They give rise to full prime factors and simple exact C -algebras. The main tool in our work is the study of an amenable boundary action, yielding the Akemann-Ostrand property. Finally, this boundary can be identified with the Martin or the Poisson boundary of a quantum random walk.

Group measure space decomposition of II1 factors and W*-superrigidity

We prove a “unique crossed product decomposition” result for group measure space II1 factors L ∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family

Non-Semi-Regular Quantum Groups Coming from Number Theory

\mathcal{G}

A simple definition for locally compact quantum groups

, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if Tn denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of

DOUBLE CROSSED PRODUCTS OF LOCALLY COMPACT QUANTUM GROUPS

\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z})

Hopf C ⁄ -algebras

is W∗-superrigid, i.e. any isomorphism between L ∞(X)⋊Γ and an arbitrary group measure space factor L ∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family

Actions of _{∞} whose II₁ factors and orbit equivalence relations have prescribed fundamental group

\mathcal{G}