nLab
poset of commutative subalgebras

Context

$(0, 1)$ -Category theory

Algebra

Operator algebra

Idea
Properties
References
- General
- Relation to Jordan algebras

Idea

General

For $A$ a an associative algebra, not necessarily commutative, its collection $ComSub (A)$ of commutative subalgebras $B ↪ A$ 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 $Σ (A)$ of $A$ may be understood as a commutative geometry internal to a sheaf topos $𝒯_{A}$ over $ComSub (A)$ or its opposite $ComSub (A)^{op}$ . An advantage of the latter is that $Σ$ becomes a compact regular locale.

As a site for noncommutative phase spaces

Specifically, consider the case that the algebra $A = B (ℋ)$ is that of bounded operators on a Hilbert space.

Applying Bohrification to this situation (see there for more discussion), one finds that the locale $Σ (A)$ internal to $𝒯_{A}$ 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 $𝒯_{A}$ .

Notably the Kochen-Specker theorem says that $Σ (B (ℋ))$ , 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 $𝒯_{A}$ the quantum mechanical kinematics encoded by $B (ℋ)$ looks like classical mechanics kinematics internal to $𝒯_{A}$ (HeunenLandsmannSpitters, following DöringIsham):

the open subsets of $Σ (A)$ are identified with the quantum states on $A$ . Their collection forms the Heyting algebra of quantum logic.
observables are morphisms of internal locales $Σ (A) \to IR$ , where $IR$ is the interval domain?.

The assignment to a noncommutative algebra $A$ of a locale ${\underset{̲}{Σ}}_{A}$ internal to $𝒯_{A}$ 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

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 $A_{J} ≃ (A^{op})_{J}$ .

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 (HardingDöring).

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

Remark

Compare this 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

𝒯_{A} : = [ComSub (A), Set]

for the presheaf topos on $ComSub (A)^{op}$ .

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 DoeringIsham and co-workers, may be seen as the co-Kripke model. This model give the so-called coarse-graining semantics of physics.

Observation

The topos $𝒯_{A}$ is a localic topos.

Because $ComSub (A)$ is a posite.

The locale $Σ (A)$

Proposition

The presheaf

(𝔸 : B \mapsto U (B)) \in 𝒯_{A},

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

This is (HeunenLandsmanSpitters, theorem 5).

Corollary

By the constructive Gelfand duality theorem there is uniquely a locale $Σ (A)$ internal to $𝒯_{A}$ such that $𝔸$ is the internal commutative $C^{*}$ -algebra of functions on $Σ (A)$ .

This observation is amplified in (HeunenLandsmanSpitters).

Proposition

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

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

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)

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 $C^{*}$ -subalgebras of $C^{*}$ -algebras, J. Math. Anal. Appl. 383 (2011) 391–399 (journal)

Revised on April 28, 2013 00:49:11 by Ingo Blechschmidt (46.244.223.210)

nLab poset of commutative subalgebras

Context

(0,1)-Category theory

Theorems

Algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Operator algebra

Definitions

Theorems

States and observables

Operator algebra

Local QFT

Euclidean QFT

Contents

Idea

General

As a site for noncommutative geometry

As a site for noncommutative phase spaces

Properties

General

Observation

Relation to Jordan algebras

Observation

Proof

Proposition

Remark

The presheaf topos over ComSub(A) op

Definition

Remark

Observation

The locale Σ(A)

Proposition

Corollary

Proposition

References

General

Relation to Jordan algebras

nLab
poset of commutative subalgebras

$(0, 1)$ -Category theory

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

The locale $Σ (A)$