  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.
