Spectral order isomorphisms and AW*-factors

TL;DR

This paper characterizes spectral order isomorphisms in AW*-factors, showing they are induced by continuous functions and lattice isomorphisms, and solves an open problem for automorphisms on infinite-dimensional Hilbert spaces.

Contribution

It establishes a canonical form for spectral order isomorphisms in AW*-factors of Type I, addressing an open question in the field.

Findings

Spectral order isomorphisms are induced by continuous functions and projection lattice isomorphisms.

Solved the open problem of spectral order automorphisms on infinite-dimensional Hilbert spaces.

Characterized orthogonality-preserving spectral order isomorphisms.

Abstract

The paper deals with spectral order isomorphisms in the framework of AW*-algebras. We establish that every spectral order isomorphism between sets of all self-adjoint operators (or between sets of all effects, or between sets of all positive operators) in 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.