Orthogeometries and AW*-algebras

TL;DR

This paper establishes a connection between AW*-algebras and a geometric structure called orthogeometries, revealing categorical relationships and properties of morphisms, especially excluding certain type factors.

Contribution

It introduces a functor linking AW*-algebras to orthogeometries, showing it is injective, full, and faithful under specific conditions, expanding the understanding of their categorical relationship.

Findings

The functor from AW*-algebras to orthogeometries is injective on non-trivial objects.

The functor is full and faithful for morphisms excluding type I_2 factors.

Orthogeometries are constructed from commutative AW*-subalgebras with limited Boolean algebra of projections.

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.