Orthogeometries and AW*-algebras

Abstract

Based on results of Harding, Heunen, Lindenhovius and Navara, (2019), we give a connection between the category of AW*-algebras and their normal Jordan homomorphisms and a category COG of orthogemetries, which are structures that are somewhat similar to projective geometries, consisting of a set of points and a set of lines, where each line contains exactly 3 points. They are constructed from the commutative AW*-subalgebras of an AW*-algebra that have at most an 8-element Boolean algebra of projections. Morphisms between orthogemetries are partial functions between their sets of points as in projective geometry. The functor from the category of AW*-algebras with normal Jordan homomorphism to COG we create is injective on non-trivial objects, and full and faithful with respect to morphisms that do not involve type $I_2$ factors.

arXiv comment18 pages, 4 figures

References in corpus (6)

• `What is a Thing?': Topos Theory in the Foundations of Physics
• Active lattices determine AW*-algebras
• Abelian subalgebras and the Jordan structure of a von Neumann algebra
• Classifying finite-dimensional C*-algebras by posets of their commutative C*-subalgebras
• Boolean subalgebras of orthoalgebras
• Piecewise Boolean algebras and their domains