The following is a main result in the paper joint with Chi-Keung Ng Ortho-sets and Gelfand spectra .
For a quantum system modeled on the self-adjoint part of a (complex) -algebra , we introduce the semi-classical object of Gelfand spectrum for this system as follows. Let be the set of all pure states on . For any , we denote if and have orthogonal support projections; i.e., and has zero transition probability. For any left closed ideal , we set (see e.g., (Murphy 1990, Theorem 5.3.4)). Then is a q-subset of , and the collection of all such q-subsets form a quantum topology on .
The good point for the Gelfand spectrum of a -algebra is that it captures the self-adjoint part of original algebra up to a Jordan isomorphism (under a mild assumption), which is good enough for the consideration of physical structure modeled on the self-adjoint parts of -algebras. Let us recall that a linear map from the self-adjoint part of a -algebra to that of another -algebra is a Jordan isomorphism if it preserves the Jordan product; i.e. ( ). The following can be seen as a non-commutative generalization of the Gelfand theorem: