Spectral Presheaves as Generalised Gelfand Spectra for Nonabelian Unital C*-algebras

To each unital C*-algebra, we associate a generalised Gelfand
spectrum in the form of the spectral presheaf. It is shown that this
assignment is contravariantly functorial, generalising the Gelfand
spectrum functor to nonabelian unital C*-algebras. We show that the
spectral presheaf is 'locally dual' to the Bohrification of a C*-algebra
in the sense of Heunen, Landsman and Spitters. Moreover, it is
demonstrated that the spectral presheaf determines the C*-algebra up to
Jordan isomorphisms in many cases, and that time evolution of a quantum
system can be formulated in terms of flows on the spectral presheaf.