Contextuality and noncommutative geometry in quantum mechanics

It is argued that the geometric dual of a noncommutative operator algebra represents a notion of quantum state space which differs from existing notions by representing observables as maps from states to outcomes rather than from states to distributions on outcomes. A program of solving for an explicitly geometric manifestation of quantum state space by adapting the spectral presheaf, a construction meant to analyze contextuality in quantum mechanics, to derive simple reconstructions of noncommutative topological tools from their topological prototypes is presented. We associate to each unital C-algebraA a geometric object—a diagram of topological spaces representing quotient spaces of the noncommutative space underlying A—meant to serve the role of a generalized Gel’fand spectrum. After showing that any functor F from compact Hausdorff spaces to a suitable target category C can be applied directly to these geometric objects to automatically yield an extension F̃ which acts on all unital C-algebras, we compare a novel formulation of the operator K0 functor to the extension K̃ of the topological K-functor. We then conjecture that the extension of the functor assigning a topological space its topological lattice assigns a unital C-algebra the topological lattice of its primary ideal spectrum and prove the von Neumann algebraic analogue of this conjecture.