The Spectral Presheaf of an Orthomodular Lattice Some steps towards generalized Stone duality

In the topos approach to quantum physics, a functor known as the spectral presheaf of a von Neumann algebra plays the role of a generalized state space. Mathematically, the spectral presheaf also provides an interesting generalization of the Gelfand spectrum, which is only defined for abelian von Neumann algebras, to the nonabelian case. A partial duality result, analogous to Gelfand duality, exists for this spectral presehaf. This dissertation will begin to explore generalizations of the notion of a spectral presheaf and work towards a duality theory for certain nondistributive lattices. Specifically, we will define the spectral presheaf of an orthomodular lattice, which is a generalization of the Stone space of a Boolean algebra, and prove that it is a complete invariant: two orthomodular lattices are isomorphic if and only if their spectral presheaves are. The analogous result also holds for complete orthomodular lattices. We will map the elements of a complete orthomodular lattice L to the algebra of clopen subobjects of the spectral presheaf of L; using the right adjoint of this map, we show that these clopen subobjects, modulo an equivalence relation, form a complete lattice isomorphic to L. This can be seen as a generalization of Stone’s representation theorem for Boolean algebras. We conclude by discussing some other possible generalizations of the spectral presheaf, including Lie groups.