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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 anddomain 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 contractivelinear mapping between C$^*$-algebras. We prove that if $T$ preserves domainabsolutely compatible elements (i.e., $a\triangle_d b\RightarrowT(a)\triangle_d T(b)$) then $T$ is a triple homomorphism. A similar statementis proved when $T$ preserves range absolutely compatible elements. It isfinally shown that $T$ is a triple homomorphism if, and only if, $T$ preservesabsolutely compatible elements.

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