On states on orthogonally closed subspaces of an inner product space (Q1868412)

It is known that \(L(H)\), the system of all closed subspaces of a Hilbert space \(H\) with dim\(H \geq 3\), admits no two-valued finitely additive state. In particular, this holds for \(H\) infinite-dimensional. This latter fact is generalized in the paper to infinite-dimensional inner product spaces. More precisely, if \(S\) is a real or complex infinite-dimensional inner product space, and \(s\) is a finitely additive state on the system \(F(S)\) of its orthogonally closed subspaces, then the range of \(s\) coincides with the whole interval \([0,1]\). The main technique used rests upon embedding of \(L(H)\), with \(H\) finite dimensional, into \(F(S)\). The authors notice that this result extends also to quaternionic inner product spaces.