Dye's Theorem and Gleason's Theorem for AW*-algebras

作者：Hamhalter Jan^*

来源：Journal of Mathematical Analysis and Applications, 2015, 422(2): 1103-1115.

DOI：10.1016/j.jmaa.2014.09.040

• Dye's Theorem
• Gleason's Theorem
• AW*-algebras

We prove that any map between projection lattices of AW*-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 *-homomorphism between A and B. This allows us to generalize Dye's Theorem from von Neumann algebras to AW*-algebras. We show that Mackey-Gleason-Bunce-Wright Theorem can be extended to homogeneous AW*-algebras of Type I. The interplay between Dye's Theorem and Gleason's Theorem is shown. As an application we prove that Jordan *-homomorphisms are commutatively determined. Another corollary says that Jordan parts of AW*-algebras can be reconstructed from posets of their abelian subalgebras.

• 出版日期2015-2-15