The Lattice of Weak*-Closed Inner Ideals¶in a W*-Algebra

Abstract:

The family of centrally equivalent pairs of projections in a W^*<;-algebra A forms a complete ^*<;-lattice that is ^*<;-order isomorphic to the complete ^*<;-lattice of weak^*<;-closed inner ideals in A and to the complete ^*<;-lattice of structural projections on A. Furthermore, the set of symmetric elements of is order isomorphic to the complete orthomodular lattice of projections in A. 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 (e ₁,f ₁) and (e ₂,f ₂) 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 may be considered to represent the set consisting of a particular kind of sub-system, and the central orthogonality and rigid collinearity of pairs of elements of may be regarded as representing two different types of disjointness of the corresponding 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 A has no weak^*<;-closed ideal of Type I ₂, such measures are precisely those that are the restrictions of bounded centrally symmetric sesquilinear functionals φ _m on A×A. Furthermore, m is an hermitian measure on the complete ^*<;-lattice if and only if the sesquilinear functional φ _m is hermitian and m is a normal measure if and only if φ _m is separately weak*-continuous. These results can be regarded as Gleason-type theorems.