  We are concerned with the issue of quantization of a scalar field in a
diffeomorphism invariant manner. We apply the method used in Loop Quantum
Gravity. It relies on the specific choice of scalar field variables referred to
as the polymer variables. The quantization, in our formulation, amounts to
introducing the `quantum' polymer *-star algebra and looking for positive
linear functionals, called states. Assumed in our paper homeomorphism
invariance allows to derive the complete class of the states. They are
determined by the homeomorphism invariant states defined on the CW-complex
*-algebra. The corresponding GNS representations of the polymer *-algebra and
their self-adjoint extensions are derived, the equivalence classes are found
and invariant subspaces characterized. In the preceding letter (the part I) we
outlined those results. Here, we present the technical details.
