THE STATE-SPACE OF THE LATTICE OF ORTHOGONALLY CLOSED SUBSPACES

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), the system F(S) 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 S.