nLab poset of commutative subalgebras

Idea

General

For $A$ a an associative algebra, not necessarily commutative, its collection $ComSub(A)$ of commutative subalgebras $B \hookrightarrow A$ 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 $A$ is a C-star algebra the noncommutative geometry of the formal dual space $\Sigma(A)$ of $A$ may be understood as a commutative geometry internal to a sheaf topos $\mathcal{T}_A$ over $ComSub(A)$ or its opposite $ComSub(A)^{op}$. An advantage of the latter is that $\Sigma$ becomes a compact regular locale.

As a site for noncommutative phase spaces

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

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

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

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

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

The assignment to a noncommutative algebra $A$ of a locale $\underline{\Sigma}_A$ internal to $\mathcal{T}_A$ 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.

Properties

General

Observation

The poset of commutative subalgebras $C(A)$ is always an (unbounded) meet-semilattice. If $A$ itself is commutative then it is a bounded meet semilattice, with $A$ itself being the top element.

Relation to Jordan algebras

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

Observation

There exist von Neumann algebras $A$, $B$ such that there exists a Jordan algebra isomorphism $A_J \to B_J$ but not an algebra isomorphism $A \to B$.

Proof

By

• 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 $A$ with no algebra isomorphism to its opposite algebra $A^{op}$. But clearly $A_J \simeq (A^{op})_J$.

Proposition

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

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

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.

Remark

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

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

Definition

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

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

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

Remark

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 $ComSub(A)$, 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.

Observation

The topos $\mathcal{T}_A$ is a localic topos.

Because $ComSub(A)$ is a posite.

The locale $\Sigma(A)$

Proposition

The presheaf

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

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

This is (HeunenLandsmanSpitters, theorem 5).

Corollary

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

This observation is amplified in (HeunenLandsmanSpitters).

Proposition

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

This is (HeunenLandsmanSpitters, theorem 6), following (Butterfield-Hamilton-Isham).

References

General

The proposal that the the noncommutative geometry of $A$ is fruitfully studied via the commutative geometry over $ComSub(A)$ 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

• Andreas Döring, Chris Isham, A topos foundation for theories of physics Journal of Mathematical Physics (2008)

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

• Chris Heunen, Klaas Landsman, Bas Spitters, A topos for algebraic quantum theory (arXiv:0709.4364)

See also higher category theory and physics.

Relation to Jordan algebras

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

• John Harding, Andreas Döring, Abelian subalgebras and the Jordan structure of a von Neumann algebra (arXiv:1009.4945)

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

• Jan Hamhalter, Isomorphisms of ordered structures of abelian $C^\ast$-subalgebras of $C^\ast$-algebras, J. Math. Anal. Appl. 383 (2011) 391–399, doi:10.1016/j.jmaa.2011.05.035

See at Harding-Döring-Hamhalter theorem.