Abelian subalgebras and the Jordan structure of a von Neumann algebra
Abelian subalgebras and the Jordan structure of a von Neumann algebra
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发表时间：
2010-09
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通讯作者：
A. Doering;J. Harding
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作者：
A. Doering;J. Harding
对于不同构于C^2且不含I_2求和的von Neumann代数M，证明了对于M和N的交换von Neumann子代数的偏序集之间的序同构f：AbSub(M)->；AbSub(N)，存在唯一的Jordan*-同构g：M->；N，对M的每个交换von Neumann子代数S，g[S]等于f(S)。这表明von Neumann代数的Jordan结构是由它的阿贝尔子代数的偏序集决定的，它与C^2不同构且不含I_2加法，这在最近的量子力学基本问题的研究中具有重要意义。
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.