Isometries of absolute order unit spaces

Abstract

We prove that a unital, bijective 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 a unital, bijective $$*$$ -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) $$\hbox {C}^*$$ -algebras such maps are precisely $$\hbox {C}^*$$ -algebra isomorphisms.