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Multiplier Hopf algebras imbedded in C$^*$-algebraic quantum groups

K. De Commer·A. Van Daele
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Abstract

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 {\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 $μ$ of a $^*$-algebraic quantum group equals 1. This solves an open problem.

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K. De Commer, A. Van Daele. 2006-11-29. Multiplier Hopf algebras imbedded in C$^*$-algebraic quantum groups. https://arxiv.org/abs/math/0611872

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