The notion of a strongly dense inner product space is introduced and
it is shown that for such an incomplete space $S$, (in particular,
for each incomplete hyperplane of a Hilbert space) $F(S)$, the
system of all orthogonally closed subspaces of $S$ is not stateless,
and the state-space of $F(S)$ is affinely homeomorphic to the face
consisting of the free states on the projection lattice
corresponding to the completion of $S$. The homeomorphism is
determined by the extension of the states. In particular, when $S$
is complex, the state-space of $F(S)$ is affinely homeomorphic to
the state-space of the Calkin algebra associated with $\overline S$.
