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

Abstract

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