symmetric monoidal (∞,1)-category of spectra
Various authors have proposed (Butterfield-Hamilton-Isham, Döring-Isham, Heunen-Landsmann-Spitters) that for the case that is a C-star algebra the noncommutative geometry of the formal dual space of may be understood as a commutative geometry internal to a sheaf topos over or its opposite . An advantage of the latter is that becomes a compact regular locale.
Specifically, consider the case that the algebra is that of bounded operators on a Hilbert space. This is interpreted as an algebra of quantum observables and the commutative subalgebras are then “classical contexts”.
Applying Bohrification to this situation (see there for more discussion), one finds that the locale internal to behaves like the noncommutative phase space of a system of quantum mechanics, which however internally looks like an ordinary commutative geometry. Various statements about operator algebra then have geometric analogs in .
Notably the Kochen-Specker theorem says that , while nontrivial, has no points/no global elements. (This topos-theoretic geometric reformulation of the Kochen-Specker theorem had been the original motivation for considering in the first place in ButterfieldIsham).
The assignment to a noncommutative algebra of a locale internal to has been called Bohrification, in honor of Nils Bohr whose heuristic writings about the nature of quantum mechanics as being probed by classical (= commutative) context one may argue is being formalized by this construction.
The poset of commutative subalgebras is always an (unbounded) meet-semilattice. If itself is commutative then it is a bounded meet semilattice, with itself being the top element.
There exist von Neumann algebras , such that there exists a Jordan algebra isomorphism but not an algebra isomorphism .
there is a von Neumann algebra factor with no algebra isomorphism to its opposite algebra . But clearly .
This is the theorem in (Harding-Döring).
For more on this see at Harding-Döring-Hamhalter theorem.
This opposite order on commutative subalgebras may be seen as the information order from Kripke semantics: a larger subalgebra contains more information. In this light the presheaf topos on , as used by (Döring-Isham 07) and co-workers, may be seen as the co-Kripke model. This model is also referred to as the coarse-graining semantics of quantum mechanics. See also at spectral presheaf.
The topos is a localic topos.
Because is a posite.
This is (HeunenLandsmanSpitters, theorem 5).
This observation is amplified in (HeunenLandsmanSpitters).
The proposal that the the noncommutative geometry of is fruitfully studied via the commutative geometry over goes back to
I. quantum states as generalized valuations International Journal of Theoretical Physics, 37(11):2669–2733, 1998.
II. conceptual aspects and classical analogues International Journal of Theoretical Physics, 38(3):827–859, 1999
III. Von Neumann algebras as the base category International Journal of Theoretical Physics, 39(6):1413–1436, 2000.
The presheaf topos on (Bohr topos) and its internal localic Gelfand dual to is discussed in
See also higher category theory and physics.
The relation to Jordan algebras of is discussed in
See at Harding-Döring-Hamhalter theorem.