Abstract
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) = (a)Psi(b)Phi(a), for every a,b is an element of 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.