Multiplier Hopf algebras imbedded in C${}^{\ast }$-algebraic quantum groups

Abstract

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 ${}^{\ast }$-algebraic quantum group {\it in a purely algebraic fashion}. Moreover, the {\it reason} that this analytic structure exists at all, is that the one-parameter groups, such as the modular group and the scaling group, are diagonizable. In particular, we will show that necessarily the scaling constant $\mu$ of a ${}^{\ast }$-algebraic quantum group equals 1. This solves an open problem.

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