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

Abstract: 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.