Abelian Subalgebras and the Jordan Structure of a Von Neumann Algebra
نویسندگان
چکیده
For von Neumann algebras M,N not isomorphic to C C and without type I2 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 C and without type I2 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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