AW *-代数的Dye定理和Gleason定理
Dye's Theorem and Gleason's Theorem for AW*-algebras
摘要：
证明了在$AW^\ast$ -代数$A$和$B$的投影格之间的任何保持任意元素的正交补和上确界的映射是$A$和$B$之间的正规Jordan $\ast$-同态的限制，其中$A$没有$I_2$型直和项.这使得我们可以将Dye定理从von Neumann代数推广到$AW^\ast$ -代数。我们证明了Mackey-Gleason-Bunce-Wright定理可以推广到I型齐性$AW^\ast$ -代数。Dye定理和Gleason定理之间的相互作用。作为应用，我们证明了Jordan $\ast$ -同态是可交换的.另一个推论是$AW^\ast$ -代数的Jordan部分可以由交换子代数的偏序集重构。
英文摘要：
We prove that any map between projection lattices of $AW^\ast$-algebras $A$ and $B$, where $A$ has no Type $I_2$ direct summand, that preserves orthocomplementation and suprema of arbitrary elements, is a restriction of a normal Jordan $\ast$-homomorphism between $A$ and $B$. This allows us to generalize Dye's Theorem from von Neumann algebras to $AW^\ast$-algebras. We show that Mackey-Gleason-Bunce-Wright Theorem can be extended to homogeneous $AW^\ast$-algebras of Type I. The interplay between Dye's Theorem and Gleason's Theorem is shown. As an application we prove that Jordan $\ast$-homomorphims are commutatively determined. Another corollary says that Jordan parts of $AW^\ast$-algebras can be reconstructed from posets of their abelian subalgebras.
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