The lattice of weak\(^*\)-closed inner ideals in a \(W^*\)-algebra
The central support of an element \(e\) of a sub-complete orthomodular lattice is the infimum of the set of all central elements of the lattice that dominate \(e\). Two elements \(e\) and \(f\) sharing the same central support are called centrally equivalent, and the collection of all centrally equivalent pairs of elements itself forms a complete \(\ast\)-lattice. If the above construction is applied to the lattice \({\mathcal P}(A)\) of all projections of a \(W^\ast\)-algebra \(A\), then one arrives at the complete \(\ast\)-lattice denoted by \({\mathcal{CP}}(A)\). In general, this lattice is no longer orthomodular, but since it possesses a complementation, such concepts as orthogonality, centre, and central orthogonality remain meaningful nevertheless. The lattice \({\mathcal{CP}}(A)\) viewed in the context of structure theory of general \(W^\ast\)-algebras forms the subject of study of the paper under review. A special attention is paid to revaling links between the structure of lattices of the form \({\mathcal{CP}}(A)\) and theory of Jordan triple algebras, in particular the concept of rigidly collinear pairs is transplanted from the former theory and thoroughly studied for the lattices \({\mathcal{CP}}(A)\). Statistical physics provides motivation for considering bounded measures on \({\mathcal{CP}}(A)\) that are additive on centrally orthogonal and rigidly collinear pairs of elements. Such measures are fully described in the paper under review, whose main result states that they are precisely the restrictions of bounded centrally symmetric sesquilinear functionals on \(A\times A\).
structure theory of general \(W^*\)-algebras
centrally symmetric sesquilinear functionals
scientific article; zbMATH DE number 1294549