Order isomorphisms in $C^{\ast }$-algebras

We provide a complete description of the order isomorphisms between the self-adjoint parts of $C^{\ast }$-algebras. Furthermore, we characterize such isomorphisms between general operator intervals in $A{W}^{\ast }$-algebras. For the description, we use Jordan ${}^{\ast }$-isomorphisms and closed operators in the regular rings of $A{W}^{\ast }$-algebras. This work generalizes previous results on von Neumann algebras.