Let (A, Δ) be a locally compact quantum group and (A_0, Δ_0 a regular multiplier Hopf algebra. We show that if (A_0, Δ_0 can in some sense be imbedded in (A, Δ), then A_0 will inherit some of the analytic structure of A. Under certain conditions on the imbedding, we will be able to conclude that (A_0, Δ_0) is actually an algebraic quantum group with a full analytic structure. The techniques used to show this can 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 one-parameter groups, such as the modular group and the scaling group, are diagonalizable. In particular, we will show that necessarily the scaling constant μ of a ~*-algebraic quantum group equals 1. This solves an open problem posed in [13].
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