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

2014/08/20 by Hamhalter, Jan
#46L30 #46L51 #81P10 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1408.4597

Abstract

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 ∗-homomorphims are commutatively determined. Another corollary says that Jordan parts of AW^∗-algebras can be reconstructed from posets of their abelian subalgebras.

Related

• Linear maps preserve the covariance sets
• Extension of projection mappings
• Abelian subalgebras and the Jordan structure of a von Neumann algebra
• Dye's theorem for tripotents in von Neumann algebras and JBW* triples
• On Defining AW*-algebras and Rickart C*-algebras