Abelian subalgebras and the Jordan structure of a von Neumann algebra

TL;DR

This paper demonstrates that the Jordan structure of certain von Neumann algebras is uniquely determined by the order structure of their abelian subalgebras, with implications for quantum foundations.

Contribution

It establishes a one-to-one correspondence between order-isomorphisms of abelian subalgebras and Jordan *-isomorphisms of the algebras, revealing the algebra's structure from its abelian subalgebras.

Findings

Order-isomorphisms of abelian subalgebras induce Jordan *-isomorphisms

The Jordan structure is determined by the poset of abelian subalgebras

Results have implications for foundational quantum mechanics

Abstract

For von Neumann algebras M, N not isomorphic to C^2 and without type I_2 summands, we show that for an order-isomorphism f:AbSub(M)->AbSub(N) between the posets of abelian von Neumann subalgebras of M and N, there is a unique Jordan *-isomorphism g:M->N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of a von Neumann algebra not isomorphic to C^2 and without type I_2 summands is determined by the poset of its abelian subalgebras, and has implications in recent approaches to foundational issues in quantum mechanics.