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 $\skew3\overline S$.(Received September 8 2004)
(Accepted September 29 2004)
Primary 46C05; 46L30; Secondary 03G12.
1 The authors acknowledge the support of the Grant VEGA No 2/3163/23 SAV and of APVT-51-032002, Bratislava, Slovakia.