  This is the last part of a series of three papers on the subject. In the
first part we have considered the duality of algebraic quantum groups. In that
paper, we use the term algebraic quantum group for a regular multiplier Hopf
algebra with integrals. We treat the duality as studied in that theory. In the
second part we have considered the duality for multiplier Hopf $^*$-algebras
with positive integrals. The main purpose of that paper is to explain how it
gives rise to a locally compact quantum group in the sense of Kustermans and
Vaes. In a preliminary section of the second part we have mentioned the
analytic structure of such a $^*$-algebraic quantum group. We have not gone
into the details, except for that part needed to obtain that the scaling
constant is trivial and for proving that the composition of a positive left
integral with the antipode gives a right integral that is again positive. Also
the construction of the square root. of the modular element is discussed and
used to obtain the two Haar weights on the associated locally compact quantum
group. In this paper we treat the analytic structure of the $^*$-algebraic
quantum group in detail. The analytic structure is intimately related with the
modular structure. We obtain a collection of interesting formulas, relating the
two structures. Further we compare these results with formulas obtained in the
framework of general locally compact quantum groups. As for the first two
papers in this series, also here no new results are obtained. On the other hand
the approach is different, more direct and instructive. It might help the
reader for understanding the more general theory of locally compact quantum
groups.
