Contractive linear preservers of absolutely compatible pairs between C*-algebras

TL;DR

This paper characterizes contractive linear maps between C*-algebras that preserve certain compatibility relations, proving they are triple homomorphisms, thus linking structure-preserving maps to algebraic properties.

Contribution

It establishes that preserving domain or range absolute compatibility in contractive linear maps implies they are triple homomorphisms, providing a new characterization of such structure-preserving maps.

Findings

Preserving domain absolute compatibility implies the map is a triple homomorphism.

Preserving range absolute compatibility also implies the map is a triple homomorphism.

A map is a triple homomorphism if and only if it preserves absolutely compatible elements.

Abstract

Let $a$ and $b$ be elements in the closed ball of a unital C$^*$-algebra $A$ (if $A$ is not unital we consider its natural unitization). We shall say that $a$ and $b$ are domain (respectively, range) absolutely compatible ($a\triangle_d b$, respectively, $a\triangle_r b$, in short) if $\Big| |a| -|b| \Big| + \Big| 1-|a|-|b| \Big| =1$ (resp., $\Big| |a^*| -|b^*| \Big| + \Big| 1-|a^*|-|b^*| \Big| =1$), where $|a|^2= a^* a$. We shall say that $a$ and $b$ are absolutely compatible ($a\triangle b$ in short) if they are both range and domain absolutely compatible. In general, $a\triangle_d b$ (respectively, $a\triangle_r b$ and $a\triangle b$) is strictly weaker than $ab^*=0 $ (respectively, $a^* b =0$ and $a\perp b$). Let $T: A\to B$ be a contractive linear mapping between C$^*$-algebras. We prove that if $T$ preserves domain absolutely compatible elements (i.e., $a\triangle_d b\Rightarrow T(a)\triangle_d T(b)$) then $T$ is a triple homomorphism. A similar statement is proved when $T$ preserves range absolutely compatible elements. It is finally shown that $T$ is a triple homomorphism if, and only if, $T$ preserves absolutely compatible elements.