The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related modular theory Haar integrals on G and ˆG act level direct integrals. Using these results we derive a web implications between properties such as unimodularity or traciality also in detail two examples: discrete group ˆSUq(2) az+b group.

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability the form, direct integral structure underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including position quantum mechanics invariant under unitary separable locally compact group. ...

We prove that discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed subgroups, and the same holds for free products of amalgamated a common subgroup. Along way we also show [Formula: see text]-algebras again via Fell-topology characterization text]-field continuity, recovering result ...

We introduce and study the concept of a bornological quantum group. This generalizes the theory of algebraic quantum groups in the sense of van Daele from the algebraic setting to the framework of bornological vector spaces. Working with bornological vector spaces, the scope of the latter theory can be extended considerably. In particular, the bornological theory covers smooth convolution algeb...