Wednesday, December 12, 2012

1212.2613 (Andreas Doering)

Generalised Gelfand Spectra of Nonabelian Unital $C^*$-Algebras I:
Categorical Aspects, Automorphisms and Jordan Structure

Andreas Doering
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 present some categorical aspects and 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. In a companion paper, we consider time evolution of quantum systems in the Heisenberg picture and the Schr\"odinger picture in a formulation based on flows on the spectral presheaf.
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