Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values; dually, states can be modelled as functions from the algebra of observables to outcome values. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be jointly measurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state. Here, we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical (deterministic) state spaces to the quantum setting and argue that these generalised state spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton–Isham–Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry à la Connes et al. To each unital C*- algebra
A we associate a geometric object—a diagram of topological spaces collating quotient spaces of the noncommutative space underlying
A —that performs the role of a generalised Gel'fand spectrum. We show how 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
F~ acting on all unital C*-algebras. This procedure is used to give a novel formulation of the operator K0-functor via a finitary variant
K~f of the extension
K~ of the topological K-functor. We then delineate a C*-algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital C*-algebra the Zariski topological lattice of its primitive ideal spectrum, i.e. its lattice of closed two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture.
Another axiom has to be introduced to make the formalism given in Part I physically equivalent to the conventional Hilbert-space formalism. Then it is shown that, given a certain fundamental set of observables, a B*-algebra A can be built into which the set 𝒪b of all bounded observables can be mapped injectively. The closure of the algebra generated by the image of 𝒪b into A is A itself. A Hilbert space ℋ exists into which the set 𝒪0 of all pure physical states can be mapped injectively. The closure of the subset of ℋ which is the image of this mapping is ℋ itself. The algebra A can be mapped in such a fashion into the C*-algebra B(ℋ) of all bounded observables that these mappings provide, essentially, just a translation of the original formalism in a Hilbert-space formalism.
Group representations on the Liouville representation spaces are considered. It is shown that the state space ℐ1(H) of trace class operators on Hilbert space H and the observable space ℒ(H) of bounded operators are completely reducible under physically induced representations of compact Hausdorff groups when appropriate topologies are used. For state space ℐ1(H) both the norm topology and the weak topology lead to complete reducibility, while for observable space ℒ(H) the weak-* topology—but not the norm topology—suffices. This leads to conservation laws, selection rules, and Wigner–Eckart theorems for the Liouville representation. It is shown that serious difficulties are encountered when a similar theory is attempted on the observable space and state space used in the algebraic approach.