Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups

It is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular latticeX, admits a natural embedding into a group{G(X)}with a lattice ordering so that{G(X)}determinesXup to isomorphism. The embedding{X\hookrightarrow G(X)}appears to be a universal (non-commutative) group-valued measure onX, while states ofXturn into real-valued group homomorphisms on{G(X)}. The existence of completions is characterized by a generalized archimedean property which simultaneously applies toXand{G(X)}. By an extension of Foulis’ coordinatization theorem, the negative cone of{G(X)}is shown to be the initial object among generalized Baer{{}^{\ast}}-semigroups. For finiteX, the correspondence betweenXand{G(X)}provides a new class of Garside groups.

Symmetries in synaptic algebras

A synaptic algebra is a generalization of the Jordan algebra of self-adjoint elements of a von Neumann algebra. We study symmetries in synaptic algebras, i.e., elements whose square is the unit element, and we investigate the equivalence relation on the projection lattice of the algebra induced by finite sequences of symmetries. In case the projection lattice is complete, or even centrally orthocomplete, this equivalence relation is shown to possess many of the properties of a dimension equivalence relation on an orthomodular lattice.

Regular elements in generalized hermitian algebras

A generalized Hermitian (GH) algebra is a special Jordan algebra that is at the same time a spectral order-unit space. In this paper we characterize the von Neumann regular elements in a GH-algebra, relate maximal pairwise commuting subsets of the algebra to blocks in its projection lattice, and prove a Gelfand-Naimark type representation theorem for commutative GH-algebras.

Reflexivity for a class of subspace lattices

The complete lattice generated by a totally atomic CSL ℒ and the projection lattice of a von Neumann algebra ℛ, commuting with ℒ, is reflexive. From this it follows that the strongly closed lattice generated by any CSL ℒ and the projection lattice of a properly infinite von Neumann algebra ℛ, commuting with ℒ, is reflexive.