  We study pointwise-generalized-inverses of linear maps between
C$^*$-algebras. Let $\Phi$ and $\Psi$ be linear maps between complex Banach
algebras $A$ and $B$. We say that $\Psi$ is a pointwise-generalized-inverse of
$\Phi$ if $\Phi(aba)=\Phi(a)\Psi(b)\Phi(a),$ for every $a,b\in A$. The pair
$(\Phi,\Psi)$ is Jordan-triple multiplicative if $\Phi$ is a
pointwise-generalized-inverse of $\Psi$ and the latter is a
pointwise-generalized-inverse of $\Phi$. We study the basic properties of this
maps in connection with Jordan homomorphism, triple homomorphisms and strongly
preservers. We also determine conditions to guarantee the automatic continuity
of the pointwise-generalized-inverse of continuous operator between
C$^*$-algebras. An appropriate generalization is introduced in the setting of
JB$^*$-triples.
