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We study pointwise-generalized-inverses of linear maps betweenC$^*$-algebras. Let $\Phi$ and $\Psi$ be linear maps between complex Banachalgebras $A$ 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 apointwise-generalized-inverse of $\Psi$ and the latter is apointwise-generalized-inverse of $\Phi$. We study the basic properties of thismaps in connection with Jordan homomorphism, triple homomorphisms and stronglypreservers. We also determine conditions to guarantee the automatic continuityof the pointwise-generalized-inverse of continuous operator betweenC$^*$-algebras. An appropriate generalization is introduced in the setting ofJB$^*$-triples.

Pointwise-generalized-inverses of linear maps between C$^*$-algebras and JB$^*$-triples

Pointwise-generalized-inverses of linear maps between C$^$-algebras and JB$^$-triples