Gleason's theorem for rectangular \(JBW^*\)-triples
A rectangular \(\text{JB}^*\)-triple \({B}\) is essentially a weak\(^*\)-closed inner ideal of a W\(^*\)-algebra \({A}\), equivalently, there exists a pair \(({p},{q})\) of centrally equivalent projections of the complete orthomodular lattice \(P({A})\) of projections in \({A}\) such that \({B}\) is isomorphic to \({p}{A}{q}\). The principal order ideal \(\text{CP}({A})_{({p},{q})}\) of the complete *-lattice \(\text{CP}({A})\) of centrally equivalent pairs of projections in \({A}\), generated by \(({p},{q})\), forms a complete lattice which, although not itself, in general, orthomodular, possesses a complementation that allows for definition of orthogonality, centre, and central orthogonality to be given. A less familiar notion in lattice theory -- well-known in the theory of Jordan systems -- is that of rigid collinearity of a pair of elements of \(\text{CP}({A})\). Using results of J. D. M. Wright, the authors prove that, provided that neither of the two hereditary sub-W\(^*\)-algebras \({p}{A}{p}\) and \({q}{Q}{q}\) of \({A}\) has a weak\(^*\)-closed ideal of type \(\text{I}_2\), bounded measures on \(\text{CP}({A})_{({p},{q})}\) that are additive on centrally orthogonal and rigidly collinear pairs of elements are precisely the restrictions of bounded sesquilinear functionals on \({p}{A}{p}\times{q}{A}{q}\) with the property that the action on the centroid of \({B}={p}{A}{q}\) commutes with the adjoint operation. When \({B}\) is a complex Hilbert space of dimension greater than two, this result reduces to Gleason's theorem.
weak\(^*\)-closed inner ideal of a W\(^*\)-algebra
scientific article; zbMATH DE number 1334923