# Isometries of absolute order unit spaces

## TL;DR

This paper characterizes isometries in absolute order unit spaces and their matrix analogs, showing they correspond to Jordan and C*-algebra isomorphisms respectively, thus linking geometric and algebraic structures.

## Contribution

It establishes that isometries in absolute order unit spaces are exactly the absolute value preserving maps, extending this characterization to matrix spaces and C*-algebras.

## Key findings

- Isometries are absolute value preserving maps in absolute order unit spaces.
- Such maps are Jordan isomorphisms on JB-algebras.
- In matrix spaces, complete isometries are C*-algebra isomorphisms.

## Abstract

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.