Gleason's Theorem for Rectangular JBW-Triples

Abstract

A JBW*-triple B is said to be rectangular if there exists a W*-algebra A and a pair (p,q) of centrally equivalent elements of the complete orthomodular lattice \(\) of projections in A such that B is isomorphic to the JBW*-triple pAq. Any weak*-closed injective operator space provides an example of a rectangular JBW*-triple. The principal order ideal \(\) of the complete *-lattice \(\) of centrally equivalent pairs of projections in a W*-algebra A, generated by (p,q), forms a complete lattice that is order isomorphic to the complete lattice\(\) of weak*-closed inner ideals in B and to the complete lattice \(\) of structural projections on B. Although not itself, in general, orthomodular, \(\) possesses a complementation that allows for definitions of orthogonality, centre, and central orthogonality to be given. A less familiar notion in lattice theory, that is well-known in the theory of Jordan algebras and Jordan triple systems, is that of rigid collinearity of a pair (e2,f2) and (e2,f2) of elements of \(\). This is defined and characterized in terms of properties of \(\). A W*-algebra A is sometimes thought of as providing a model for a statistical physical system. In this case B, or, equivalently, pAq, may be thought of as providing a model for a fixed sub-system of that represented by A. Therefore, \(\) may be considered to represent the set consisting of a particular kind of sub-system of that represented by pAq. Central orthogonality and rigid collinearity of pairs of elements of \(\) may be regarded as representing two different types of disjointness, the former, classical disjointness, and the latter, decoherence, of the two sub-systems. It is therefore natural to consider bounded measures m on \(\) that are additive on centrally orthogonal and rigidly collinear pairs of elements. Using results of J.D.M. Wright, it is shown that, provided that neither of the two hereditary sub-W*-algebras pAp and qAq of A has a weak*-closed ideal of Type I2, such measures are precisely those that are the restrictions of bounded sesquilinear functionals φm on pAp×qAq with the property that the action of the centroid Z(B) of B commutes with the adjoint operation. When B is a complex Hilbert space of dimension greater than two, this result reduces to Gleason's Theorem.