poset of commutative subalgebras


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For AA a an associative algebra, not necessarily commutative, its collection ComSub(A)ComSub(A) of commutative subalgebras BAB \hookrightarrow A is naturally a poset under inclusion of subalgebras.

As a site for noncommutative geometry

Various authors have proposed (Butterfield-Hamilton-Isham, Döring-Isham, Heunen-Landsmann-Spitters) that for the case that AA is a C-star algebra the noncommutative geometry of the formal dual space Σ(A)\Sigma(A) of AA may be understood as a commutative geometry internal to a sheaf topos 𝒯 A\mathcal{T}_A over ComSub(A)ComSub(A) or its opposite ComSub(A) opComSub(A)^{op}. An advantage of the latter is that Σ\Sigma becomes a compact regular locale.

As a site for noncommutative phase spaces

Specifically, consider the case that the algebra A=B()A = B(\mathcal{H}) 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 Σ(A)\Sigma(A) internal to 𝒯 A\mathcal{T}_A 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 𝒯 A\mathcal{T}_A.

Notably the Kochen-Specker theorem says that Σ(B())\Sigma(B(\mathcal{H})), 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 ComSub(A)ComSub(A) in the first place in ButterfieldIsham).

Moreover, inside 𝒯 A\mathcal{T}_A the quantum mechanical kinematics encoded by B()B(\mathcal{H}) looks like classical mechanics kinematics internal to 𝒯 A\mathcal{T}_A (HeunenLandsmannSpitters, following DöringIsham):

  1. the open subsets of Σ(A)\Sigma(A) are identified with the quantum states on AA. Their collection forms the Heyting algebra of quantum logic.

  2. observables are morphisms of internal locales Σ(A)IR\Sigma(A) \to IR, where IRIR is the interval domain?.

The assignment to a noncommutative algebra AA of a locale Σ̲ A\underline{\Sigma}_A internal to 𝒯 A\mathcal{T}_A 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 C(A)C(A) is always an (unbounded) meet-semilattice. If AA itself is commutative then it is a bounded meet semilattice, with AA itself being the top element.

Relation to Jordan algebras

For AA an associative algebra write A JA_J for its corresponding Jordan algebra, where the commutative product :A JA JA J\circ : A_J \otimes A_J \to A_J is the symmetrization of the product in AA: ab=12(ab+ba)a \circ b = \frac{1}{2}(a b + b a).


There exist von Neumann algebras AA, BB such that there exists a Jordan algebra isomorphism A JB JA_J \to B_J but not an algebra isomorphism ABA \to B.



  • Alain Connes, A factor not anti-isomorphic to itself, Annals of Mathematics, 101 (1962), no. 3, 536–554. (JSTOR)

there is a von Neumann algebra factor AA with no algebra isomorphism to its opposite algebra A opA^{op}. But clearly A J(A op) JA_J \simeq (A^{op})_J.


Let A,BA, B be von Neumann algebras without a type I 2I_2-von Neumann algebra factor-summand and let ComSub(A)ComSub(A), ComSub(B)ComSub(B) be their posets of commutative sub-von Neumann algebras.

Then every isomorphism ComSub(A)ComSub(B)ComSub(A) \to ComSub(B) of posets comes from a unique Jordan algebra isomorphism A JB JA_J \to B_J.

This is the theorem in (Harding-Döring).

There is a generalization of this theorem to more general C-star algebras in (Hamhalter).

For more on this see at Harding-Döring-Hamhalter theorem.


This is related to the Alfsen-Shultz theorem, which says that two C *C^*-algebras have the same states precisely if they are Jordan-isomorphic.

The presheaf topos over ComSub(A) opComSub(A)^{op}


For AA a C-star algebra, write ComSub(A)ComSub(A) for its poset of sub-C *C^*-algebras. Write

𝒯 A:=[ComSub(A),Set] \mathcal{T}_A := [ComSub(A),Set]

for the presheaf topos on ComSub(A) opComSub(A)^{op}. This is alse called the Bohr topos.


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 ComSub(A)ComSub(A), 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 𝒯 A\mathcal{T}_A is a localic topos.

Because ComSub(A)ComSub(A) is a posite.

The locale Σ(A)\Sigma(A)


The presheaf

(𝔸:BU(B))𝒯 A, (\mathbb{A} : B \mapsto U(B)) \;\; \in \mathcal{T}_A \,,

where U(B)U(B) is the underlying set of the commutative subalgebra BB, is canonically a commutative C *C^*-algebra internal to 𝒯 A\mathcal{T}_A.

This is (HeunenLandsmanSpitters, theorem 5).


By the constructive Gelfand duality theorem there is uniquely a locale Σ(A)\Sigma(A) internal to 𝒯 A\mathcal{T}_A such that 𝔸\mathbb{A} is the internal commutative C *C^*-algebra of functions on Σ(A)\Sigma(A).

This observation is amplified in (HeunenLandsmanSpitters).


If A=(H)A = \mathcal{B}(H) is the algebra of bounded operators on a Hilbert space HH of dimension >2\gt 2, then then Kochen-Specker theorem implies that Σ(A)\Sigma(A) has no points/no global element.

This is (HeunenLandsmanSpitters, theorem 6), following (ButterfieldIsham).



The proposal that the the noncommutative geometry of AA is fruitfully studied via the commutative geometry over ComSub(A)ComSub(A) goes back to

  • Jeremy Butterfield, John Hamilton, Chris Isham, A topos perspective on the Kochen-Specker theorem

    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 proposal that the non-commutativity of the phase space in quantum mechanics is fruitfully understood in the light of this has been amplified in a series of articles

The presheaf topos on ComSub(A) opComSub(A)^{op} (Bohr topos) and its internal localic Gelfand dual to AA is discussed in

See also higher category theory and physics.

Relation to Jordan algebras

The relation to Jordan algebras of ComSub(A)ComSub(A) is discussed in

for AA a von Neumann algebra and more generally for AA a C*-algebra in

  • Jan Hamhalter, Isomorphisms of ordered structures of abelian C *C^\ast-subalgebras of C *C^\ast-algebras, J. Math. Anal. Appl. 383 (2011) 391–399 (journal)

See at Harding-Döring-Hamhalter theorem.

Revised on October 3, 2013 22:43:00 by Urs Schreiber (