Isomorphisms of ordered structures of abelian -subalgebras of -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 ( C s a ) + i ψ ( C s a ) , 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.