Quasi-Discrete Locally Compact Quantum Groups

Let A be a C *-algebra. Let A ⊗ A be the minimal C *-tensor product of A with itself and let M (A ⊗ A) be the multiplier algebra of A ⊗ A. A comultiplication on A is a non-degenerate *-homomorphism ∆ : A → M (A ⊗ A) satisfying the coassociativity law (∆ ⊗ ι)∆ = (ι ⊗ ∆)∆ where ι is the identity map and where ∆ ⊗ ι and ι ⊗ ∆ are the unique extensions to M (A ⊗ A) of the obvious maps on A ⊗ A. We think of a pair (A, ∆) as a 'locally compact quantum semi-group'. When these notes where written, in 1993, it was not at all clear what the extra conditions on ∆ should be for (A, ∆) to be a 'locally compact quantum group'. This only became clear in 1999 thanks to the work of Kustermans and Vaes. In the compact case however, that is when A has an identity, rather natural conditions can be formulated and so there was a good notion of a 'compact quantum group' already at the time these notes have been written. These compact quantum groups have been studied by Woronowicz. In these notes, we consider another class of locally compact quantum groups. We assume the existence of a non-zero element h in A such that ∆(a)(1 ⊗ h) = a ⊗ h for all a ∈ A. With some extra, but also natural conditions, the element h is unique. We speak of a quasi-discrete locally compact quantum group. We also discuss the discrete case and we show that, in that case, there exists such an element h. So, the quasi-discrete case is, at least in principle, more general than the discrete case. Later however, it has been shown by Kustermans that a quasi-discrete locally compact quantum group has to be a discrete quantum group. We prove the existence of the Haar measure, the regular representation, the fundamental unitary that satifies the Pentagon equation and we obtain the reduced dual. These notes have not been published. Nevertheless, some of the results and techniques seem to be useful and in recent work, we came across similar settings. Therefore, we have decided to publish these notes in the archive. We have added some comments at the end of the introduction, and also updated the reference list. But apart from these minor changes, the notes …