  We prove that for a bijective, unital, linear map between absolute order unit
spaces is an isometry if, and only if, it is absolute value preserving. We
deduce that, on (unital) $JB$-algebras, such maps are precisely Jordan
isomorphisms. Next, we introduce the notions of absolutely matrix ordered
spaces and absolute matrix order unit spaces and prove that for a bijective,
unital, linear map between absolute matrix order unit spaces is a complete
isometry if, and only if, it is completely absolute value preserving. We obtain
that on (unital) C$^*$-algebras such maps are precisely C$^*$-algebra
isomorphism.

Functional Analysis