The Haar measure on some locally compact quantum groups

A locally compact quantum group is a pair (A,Φ) of a C-algebra A and a -homomorphism Φ from A to the multiplier algebra M(A ⊗ A) of the minimal C-tensor product A ⊗ A satisfying certain assumptions (see [K-V1] and [K-V2]). One of the assumptions is the existence of the Haar weights. These are densely defined, lower semi-continuous faithful KMS-weights satisfying the correct invariance properties. Many examples of C-algebras with a comultiplication arise from quantizations of classical locally compact groups. These are first obtained on the Hopf -algebra level and then lifted to the C-algebra context. This step is usually rather complicated (but interesting analysis is involved). It is necessary if one wants to have the Haar weights. It is Woronowicz who has done remarkable work in this direction. However, his technique to pass from the Hopf -algebra level to the C-level does not give the Haar weights. In this paper, we will study a recent example of Woronowicz, the quantum az+ b-group, and obtain the Haar weights. We use a technique that is useful in other cases as well (as we in fact show at the end of our paper). An important feature of the example we study here is that the Haar weights are not invariant, but only relatively invariant with respect to the scaling group (coming from the polar decomposition of the antipode). This phenomenon was expected from the theory, but up to now, no such example existed. It is another indication that the notion of a locally compact quantum group, as introduced and studied by Kustermans and Vaes in [K-V1] and [K-V2], is the correct one. March 2001 (Preliminary version) Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, B-3001 Heverlee (Belgium). E-mail address : [email protected]