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Harding-Döring-Hamhalter theorem

Contents

Context

Operator algebra and AQFT

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

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 >2\gt 2).

Statements

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).

Observation

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.

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

Proposition

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).

Remark

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:

References

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)

Last revised on December 11, 2017 at 08:48:38. See the history of this page for a list of all contributions to it.