  We investigate the relation between the symmetries of a Schr\"odinger
operator and the related topological quantum numbers. We show that, under
suitable assumptions on the symmetry algebra, a generalization of the
Bloch-Floquet transform induces a direct integral decomposition of the algebra
of observables. More relevantly, we prove that the generalized transform
selects uniquely the set of "continuous sections" in the direct integral
decomposition, thus yielding a Hilbert bundle. The proof is constructive and
provides an explicit description of the fibers. The emerging geometric
structure is a rigorous framework for a subsequent analysis of some topological
invariants of the operator, to be developed elsewhere. Two running examples
provide an Ariadne's thread through the paper. For the sake of completeness, we
begin by reviewing two related classical theorems by von Neumann and Maurin.
