nLab
spectral presheaf

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Contents

Idea

Given a C*-algebra AA, not necessarily commutative, write ComSub(A)ComSub(A) for its poset of commutative subalgebras, whose morphisms are the inclusion maps. By Gelfand duality, the presheaf topos PSh(ComSub(A))PSh(ComSub(A)) over this contains a canonical object, namely the presheaf

Σ:CΣ C \Sigma \;\colon\; C \mapsto \Sigma_C

which maps a commutative C*-algebra CAC \hookrightarrow A to (the point set underlying) its Gelfand spectrum Σ C\Sigma_C. This is called the spectral presheaf of AA (Isham-Döring 07)

Properties

The Kochen-Specker theorem of quantum mechanics is equivalent to the statement that for HH a (complex) Hilbert space of dimension greater than 2, then the spectral presheaf of the algebra of bounded operators (H)\mathcal{B}(H) (the quantum observables) has no global element (Butterfield-Hamilton-Isham 98). (This observation motivates the topos-theoretic development in (Isham-Döring 07)).

See at Kochen-Specker theorem and at Bohr topos for more on this.

References

The term “spectral presheaf” was introduced in

Last revised on October 3, 2013 at 10:31:24. See the history of this page for a list of all contributions to it.