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

2006/11/28 by De Commer, K., Van Daele, A.
#FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.math/0611872

Abstract

Let (A,Δ) be a locally compact quantum group and (A₀,Δ₀) a regular multiplier Hopf algebra. We show that if (A₀,Δ₀) can in some sense be imbedded in (A,Δ), 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,Δ₀) 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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