nLab Döring-Harding-Hamhalter theorem



Operator algebra and AQFT

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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 >2\gt 2).


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 (1975), 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 10).

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


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.

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


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)

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

Last revised on July 15, 2023 at 20:22:22. See the history of this page for a list of all contributions to it.