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

Abstract

We prove that any map between projection lattices of $A{W}^{\ast }$-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 $A{W}^{\ast }$-algebras. We show that Mackey-Gleason-Bunce-Wright Theorem can be extended to homogeneous $A{W}^{\ast }$-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 $A{W}^{\ast }$-algebras can be reconstructed from posets of their abelian subalgebras.

17 pages

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