Spectral order isomorphisms and AW$AW$*‐factors

Abstract

The paper deals with spectral order isomorphisms in the framework of AW$AW$*‐algebras. We establish that every spectral order isomorphism between sets of all self‐adjoint elements (or between sets of all effects, or between sets of all positive elements) in AW$AW$*‐factors of Type I has a canonical form induced by a continuous function calculus and an isomorphism between projection lattices. In particular, this solves an open question about spectral order automorphisms of the set of all (bounded) self‐adjoint operators on an infinite‐dimensional Hilbert space. We also discuss spectral order isomorphisms preserving, in addition, orthogonality in both directions.