Ortho-sets and Gelfand spectra

Journal of Physics A Mathematical and Theoretical, vol. 54, pp. 295301

Abstract

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.

Authors 2

1. Chun Ding corresponding Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands

2. Chi–Keung Ng corresponding

   Nankai University

   Affiliation as printed

   Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China