Von Neumann Algebras and Hilbert Quantales

When A is a von Neumann algebra, the set of all weakly closed linear subspaces forms a Gelfand quantale, Max_w A. We prove that Max_w A is a von Neumann quantale for all von Neumann algebras A. The natural morphism from Max_w A to the Hilbert quantale on the lattice of weakly closed right ideals of A is, in general, not an isomorphism. However, when A is a von Neumann factor, its restriction to right-sided elements is an isomorphism and this leads to a new characterization of von Neumann factors.