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This is the last part of a series of three papers on the subject. In thefirst part we have considered the duality of algebraic quantum groups. In thatpaper, we use the term algebraic quantum group for a regular multiplier Hopfalgebra with integrals. We treat the duality as studied in that theory. In thesecond part we have considered the duality for multiplier Hopf $^*$-algebraswith positive integrals. The main purpose of that paper is to explain how itgives rise to a locally compact quantum group in the sense of Kustermans andVaes. In a preliminary section of the second part we have mentioned theanalytic structure of such a $^*$-algebraic quantum group. We have not goneinto the details, except for that part needed to obtain that the scalingconstant is trivial and for proving that the composition of a positive leftintegral with the antipode gives a right integral that is again positive. Alsothe construction of the square root. of the modular element is discussed andused to obtain the two Haar weights on the associated locally compact quantumgroup. In this paper we treat the analytic structure of the $^*$-algebraicquantum group in detail. The analytic structure is intimately related with themodular structure. We obtain a collection of interesting formulas, relating thetwo structures. Further we compare these results with formulas obtained in theframework of general locally compact quantum groups. As for the first twopapers in this series, also here no new results are obtained. On the other handthe approach is different, more direct and instructive. It might help thereader for understanding the more general theory of locally compact quantumgroups.

Algebraic quantum groups III. The modular and the analytic structure