  We set up a general structure for the analysis of ``frustration-free ground
states'', or ``zero-energy states'', i.e., states minimizing each term in a
lattice interaction individually. The nesting of the finite volume ground state
spaces is described by a generalized inductive limit of observable algebras.
The limit space of this inductive system has a state space which is canonically
isomorphic (as a compact convex set) to the set of zero-energy states. We show
that for Heisenberg ferromagnets, and for generalized valence bond solid
states, the limit space is an abelian C*-algebra, and all zero-energy states
are translationally invariant or periodic. For the $q$-deformed spin-$1/2$
Heisenberg ferromagnet in one dimension (i.e., the XXZ-chain with
S$_q$U(2)-invariant boundary conditions) the limit space is an extension of the
non-commutative algebra of compact operators by two points, corresponding to
the ``all spins up'' and the ``all spins down'' states, respectively. These are
the only translationally invariant zero-energy states. The remaining ones are
parametrized by the density matrices on a Hilbert space, and converge weakly to
the ``all up'' (resp.\ ``all down'') state for shifts to $-\infty$ (resp.\
$+\infty$).
