Abelian subalgebras and the Jordan structure of a von Neumann algebra

Details Abelian subalgebras and the Jordan structure of a von Neumann algebra Abelian subalgebras and the Jordan structure of a von Neumann algebra

2010-09-24

arxiv.org/abs/1009.4945v1

Physics

Mathematical Physics

8 pages, no figures

Scientific paper

For von Neumann algebras $\mathcal{M},\mathcal{N}$ without type $I_2$ summands, we show that for an order-isomorphism $f:AbSub \mathcal{M}\to AbSub \mathcal{N}$ between the posets of abelian von Neumann subalgebras of $\mathcal{M}$ and $\mathcal{N}$, there is a unique Jordan *-isomorphism $g:\mathcal{M}\to \mathcal{N}$ with the image $g[\mathcal{S}]$ equal to $f(\mathcal{S})$ for each abelian von Neumann-subalgebra $\mathcal{S}$ of $\mathcal{M}$. This shows the Jordan structure of a von Neumann algebra without type $I_2$ summand 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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