  Let G be a group and let A be the algebra of complex functions on G with
finite support. The product in G gives rise to a coproduct on A making it a
multiplier Hopf algebra. In fact, because there exist integrals, we get an
algebraic quantum group.
  Now let H be a finite subgroup of G and consider the subalgebra of functions
in A that are constant on double cosets of H. The coproduct in general will not
leave this algebra invariant but we can modify it so that it will leave the
subalgebra invariant (in the sense that the image is in the multiplier algebra
of the tensor product of this subalgebra with itself). However, the modified
coproduct on the subalgebra will no longer be an algebra map. So, in general we
do not have an algebraic quantum group but a so-called algebraic quantum
hypergroup.
  Group-like projections in a *-algebraic quantum group A give rise, in a
natural way, to *-algebraic quantum hypergroups, very much like subgroups do as
above for a *-algebraic quantum group associated to a group. In this paper we
push this result further. On the one hand, we no longer assume the *-structure
while on the other hand, we allow the group-like projection to belong to the
multiplier algebra M(A) of A and not only to A itself. Doing so, we not only
get some well-known earlier examples of algebraic quantum hypergroups but also
some interesting new ones.
