Non-Semi-Regular Quantum Groups Coming from Number Theory

Abstract

In this paper, we study C * -algebraic quantum groups obtained through the bicrossed product construction. Examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi-regular: the crossed product of the quantum group acting on itself by translations does not contain any compact operator. We describe all corepresentations of these quantum groups and the associated universal C * -algebras. On the way, we provide several remarks on C * -algebraic properties of quantum groups and their actions.

On low-dimensional locally compact quantum groups

“…As a diffeomorphism, ρ is a Borel isomorphism and so, if k = {0}, θ(W 1 × W 2 ) has measure zero in G. This contradicts the result of [6], saying that θ is automatically a Borel isomorphism. Hence, k = {0}.…”

“…(see [6]) Let G, G 1 and G 2 be (separable) l.c. groups and let a homomorphism i : G 1 → G and an anti-homomorphism j : G 2 → G have closed images and be homeomorphisms onto these images.…”

“…groups, due to Baaj, Skandalis and the first author, is more general than the one studied in [5] and [51]. Indeed, in [6], there is given an example of a matched pair in the sense of the definition above, which does not fit in the definition of [5]. More specifically, consider the map…”

“…In [5] and [51], the map θ is supposed to have a range Ω which is open in G, with complement of measure zero and such that θ is a homeomorphism of G 1 × G 2 onto Ω. In the example of [6], the range of θ has an empty interior. However, the following proposition holds: Proposition 2.2.…”

Strictly Outer Actions of Groups and Quantum Groups

“…We further show that the bicrossed product l.c. quantum groups [38,2] (which need not be Kac algebras) can act strictly outerly on an injective factor if and only if they are co-amenable. Using results of Ueda [36], we show that if the compact quantum group SU q (2) acts strictly outerly on an injective factor, then there exists an irreducible subfactor of the hyperfinite II 1 factor with index (q + q −1 ) 2 .…”