nLab Döring-Harding-Hamhalter theorem

Idea

Theorems observed by Harding, Döring and Hamhalter say that under some conditions the Jordan algebra structure of a C*-algebra is effectively captured by its poset of commutative subalgebras.

Via the Alfsen-Shultz theorem, which states that two C*-algebra have the same states precisely if they are isomorphic as Jordan algebras, this is related to Gleason's theorem, which says that states on an algebra of bounded operators are determined by their restriction to commutative subalgebras (if the underlying Hilbert space has dimenion $\gt 2$).

Statements

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 (1975), 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 10).

There is a generalization of this theorem to more general C-star algebras in (Hamhalter 11).

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.

Related theorems

Other theorems about the foundations and interpretation of quantum mechanics include:

• order-theoretic structure in quantum mechanics

  • Kochen-Specker theorem

  • Alfsen-Shultz theorem

• Fell's theorem

• Gleason's theorem

• Wigner theorem

• Bell's theorem

• Bub-Clifton theorem

• Kadison-Singer problem

References

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

• Andreas Döring, John Harding, Abelian subalgebras and the Jordan structure of a von Neumann algebra, Houston Journal of Mathematics 42 2 (2016) [arXiv:1009.4945, issue, pdf]

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 (journal)

• Jan Hamhalter, E. Turilova, Structure of associative subalgebras of Jordan operator algebras (arXiv:1111.7240)