Ortho-sets and Gelfand spectra

Abstract

Motivated by the relation on quantum states induced by zero transition probability, we introduce the notion of ortho-set, which is a set equipped with a relation ≠q satisfying: x ≠q y implies both x ≠ y and y ≠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 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.