Strongly preserver problems in Banach algebras and C∗-algebras

Let A and B be Banach algebras. Assume that A is unital. We prove that an additive map T:A→B strongly preserves Drazin (or equivalently group) invertibility, if and only if T is a Jordan triple homomorphism. When A and B are C∗-algebras, we characterize the linear maps strongly preserving generalized invertibility (in the Jordan systems’ sense), and as consequence we determine the structure of selfadjoint linear maps strongly preserving Moore–Penrose invertibility.