Strongly preserver problems in Banach algebras and C1-algebras

Let A and B be Banach algebras. Assume that A is unital. We prove that an additive map $T:A\to B$ strongly preserves Drazin (or equivalently group) invertibility, if and only if T is a Jordan triple homomorphism. When A and B are C¹-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.