The Topos Approach to Quantum Theory And Generalised Gelfand Spectra of Noncommutative Operator Algebras

I will present an outline of the so-called topos approach to quantum theory. This approach provides
a new mathematical formulation of algebraic quantum theory based on structures in suitable presheaf
and sheaf topoi. In particular, to each noncommutative unital C*-algebra or von Neumann algebra,
a spectral presheaf is assigned which generalises the Gelfand spectrum of a commutative algebra.
This assignment is shown to be functorial and contravariant. Physically, the spectral presheaf plays
the role of a state space of the system. The representation of states of a von Neumann algebra by
probability measures on its spectral presheaf is discussed and is related to a new topos-based
form of logic for quantum systems.