Multiplier Hopf Algebras Imbedded in Locally Compact Quantum Groups

Abstract

Let (A,∆) be a locally compact quantum group and (A0,∆0) a regular multiplier Hopf algebra. We show that if (A0,∆0) can be imbedded in (A,∆), then A0 will inherit some of the analytic structure of A. Under certain conditions on the imbedding, we will be able to conclude that (A0,∆0) is actually an algebraic quantum group with a full analytic structure. The techniques used to show this can also be applied to obtain the analytic structure of a ∗-algebraic quantum group in a purely algebraic fashion. Moreover, the reason that this analytic structure exists at all, is that the associated one-parameter groups, such as the modular group and the scaling group, are diagonizable. As an immediate corollary, we will show that the scaling constant μ of a ∗-algebraic quantum group equals 1. This solves an open problem posed in [13].