  The lattice model of scalar quantum electrodynamics (Maxwell field coupled to
a complex scalar field) in the Hamiltonian framework is discussed.
  It is shown that the algebra of observables ${\cal O}({\Lambda})$ of this
model is a $C^*$-algebra, generated by a set of gauge-invariant elements
satisfying the Gauss law and some additional relations. Next, the faithful,
irreducible and non-degenerate representations of ${\cal O}({\Lambda})$ are
found. They are labeled by the value of the total electric charge, leading to a
decomposition of the physical Hilbert space into charge superselection sectors.
In the Appendices we give a unified description of spinorial and scalar quantum
electrodynamics and, as a byproduct, we present an interesting example of
weakly commuting operators, which do not commute strongly.
