The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra
was introduced as a generalised phase space for a quantum system in the
so-called topos approach to quantum theory. Here, it will be shown that
the spectral presheaf has many features of a spectrum of a
noncommutative operator algebra (and that it can be defined for other
classes of algebras as well). The main idea is that the spectrum of a
nonabelian algebra may not be a set, but a presheaf or sheaf over the
base category of abelian subalgebras. In general, the spectral presheaf
has no points, i.e., no global sections. I will show that there is a
contravariant functor from unital C*-algebras to their spectral
presheaves, and that a C*-algebra is determined up to Jordan
*-isomorphisms by its spectral presheaf in many cases. Moreover, time
evolution of a quantum system can be described in terms of flows on the
spectral presheaf, and commutators show up in a natural way. I will
indicate how combining the Jordan and Lie algebra structures may lead to
a full reconstruction of nonabelian C*- or von Neumann algebra from its
spectral presheaf.