The Rocky Mountain Journal of Mathematics vol:40 issue:4 pages:1149-1182
Let (A, Delta) be a locally compact quantum group and (A(0), Delta(0)) a regular multiplier Hopf algebra. We show that if (A(0), Delta(0)) can in some sense be imbedded in (A, Delta), 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), Delta(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 well show that necessarily the scaling constat mu of a *-algebraic quantum group equals 1. This solves an open problem posed in .