Quantum sets and Gelfand spectra (Ortho-sets and Gelfand spectra)

Motivated by quantum states with zero transition probability, we introduce the notion of ortho-set which is a set equipped with a relation $\neq_\mathrm{q}$ satisfying: $x\neq_\mathrm{q} y$ implies both $x\neq y$ and $y \neq_\mathrm{q} x$. For an ortho-set, a canonical complete ortholattice is constructed. Conversely, every complete ortholattice comes from an ortho-set in this way. Hence, the theory of ortho-sets captures almost everything about quantum logics. For a quantum system modeled by the self-adjoint part $B_\mathrm{sa}$ of a $C^*$-algebra $B$, we also introduce a "semi-classical object" called the Gelfand spectrum. It is the ortho-set, $P(B)$, of pure states of $B$ equipped with an "ortho-topology", which is a collection of subsets of $P(B)$, defined via a hull-kernel construction with respects to closed left ideals of $B$. We establish a generalization of the Gelfand theorem by showing that a bijection between the Gelfand spectra of two quantum systems that preserves the respective ortho-topologies is induced by a Jordan isomorphism between the self-adjoint parts of the underlying $C^*$-algebras (i.e. an isomorphism of the quantum systems), when the underlying $C^*$-algebras satisfy a mild condition.