Isomorphisms of ordered structures of abelian C⁎-subalgebras of C⁎-algebras

The aim of this note is to study the interplay between the Jordan structure of C⁎-algebra and the structure of its abelian C⁎-subalgebras. Let Abel(A) be a system of unital C⁎-subalgebras of a unital C⁎-algebra A ordered by set theoretic inclusion. We show that any order isomorphism φ:Abel(A)→Abel(B) can be uniquely written in the form φ(C)=ψ(Csa)+iψ(Csa), where ψ is a partially linear Jordan isomorphism between self-adjoint parts of unital C⁎-algebras A and B. As a corollary we obtain that for certain class of C⁎-algebras (including von Neumann algebras) ordered structure of abelian subalgebras completely determines the Jordan structure. The results extend hitherto known results for abelian C⁎-algebras and may be relevant to foundations of quantum theory.