A REPRESENTATION OF PROJECTION LATTICES AND THEIR STATES IN EUCLIDEAN-SPACE

We propose a representation r:L boolean OR Omega --> R-v, where L is t he collection of closed subspaces of an n-dimensional real, complex, o r quaternionic Hilbert space H or equivalently, the projection lattice of this Hilbert space, where Omega is the set of all states w:L --> [ 0, 1]. The value that w is an element of Omega takes in a is an elemen t of L is given by the scalar product of the representative points (r( a) and r(w)). The representation r(a boolean OR b) of the join of two orthogonal elements a, b is an element of L is equal to r(a) + r(b). T he convex closure of the representation of Sigma, the set of atoms of L, is equal to the representation of Omega. (C) 1998 Academic Press.