Generalised Gelfand Spectra of Nonabelian Unital C*-Algebras

To each unital C*-algebra A we associate a presheaf \Sigma^A, called the spectral presheaf of A, which can be regarded as a generalised Gelfand spectrum. We develop a categorical notion of local duality and show that there is a contravariant functor from the category of unital C*-algebras to a suitable category of presheaves containing the spectral presheaves. We clarify how much algebraic information about a C*-algebra is contained in its spectral presheaf. A nonabelian unital C*-algebra A that is neither isomorphic to C^2 nor to B(C^2) is determined by its spectral presheaf up to quasi-Jordan isomorphisms. For a particular class of unital C*-algebras, including all von Neumann algebras with no type I_2 summand, the spectral presheaf determines the Jordan structure up to isomorphisms.