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arXiv · 1009.4945

Abelian subalgebras and the Jordan structure of a von Neumann algebra

Andreas Doering·John Harding
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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.

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Andreas Doering, John Harding. 2010-09-24. Abelian subalgebras and the Jordan structure of a von Neumann algebra. https://arxiv.org/abs/1009.4945

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