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27/05/2014, 13:30 — 14:30 — Room P3.10, Mathematics Building

Andreas Döring, *Friedrich-Alexander-Universität Erlangen-Nürnberg*

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The Spectral Presheaf as the Spectrum of a Noncommutative Operator
Algebra

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
the category of unital C*-algebras to a category of presheaves that
contains the 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 can lead to a full
reconstruction of nonabelian C*- or von Neumann algebra from its
spectral presheaf.