Pointwise-generalized-inverses of linear maps between C${}^{\ast }$-algebras and JB${}^{\ast }$-triples

We study pointwise-generalized-inverses of linear maps between C${}^{\ast }$-algebras. Let $\mathrm{\Phi }$ and $\mathrm{\Psi }$ be linear maps between complex Banach algebras $A$ and $B$. We say that $\mathrm{\Psi }$ is a pointwise-generalized-inverse of $\mathrm{\Phi }$ if $\mathrm{\Phi }(aba)=\mathrm{\Phi }(a)\mathrm{\Psi }(b)\mathrm{\Phi }(a),$ for every $a,b\in A$. The pair $(\mathrm{\Phi },\mathrm{\Psi })$ is Jordan-triple multiplicative if $\mathrm{\Phi }$ is a pointwise-generalized-inverse of $\mathrm{\Psi }$ and the latter is a pointwise-generalized-inverse of $\mathrm{\Phi }$. 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${}^{\ast }$-algebras. An appropriate generalization is introduced in the setting of JB${}^{\ast }$-triples.