  Van Daele and Wang developed a purely algebraic notion of weak multiplier
Hopf algebras, which extends the notions of Hopf algebras, multiplier Hopf
algebras, and weak Hopf algebras. With an additional requirement of an
existence of left or right integrals, this framework provides a self-dual class
of algebraic quantum groupoids. The aim of this paper is to show that from this
purely algebraic data, with only some relatively minimal additional
requirements (``quasi-invariance''), one can construct a $C^*$-algebraic
quantum groupoid of separable type, recently defined by the author with Van
Daele. The $C^*$-algebraic quantum groupoid is represented as an operator
algebra on the Hilbert space constructed from the left integral, and the
comultiplication is determined by means of a certain multiplicative partial
isometry $W$, which is no longer unitary. In the last section (Appendix), we
obtain some results in the purely algebraic setting, which have not appeared
elsewhere.
