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

We prove that any map between projection lattices of AW⁎-algebras A and B, where A has no Type I2 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.