Roughly speaking, linear preserver problems consist on the characterization of linear maps between operator algebras that leave invariant certain quantities, properties or subsets.

In the last years, many mathematicians have studied linear preserver problems not only on operator algebras but on more general Banach algebras. In particular, a substantial attention has been paid to Kaplansky's problem concerned with the characterization of linear maps preserving invertibility, and also the related problem of spectrum preserving linear maps.

Based on some several partial positive results and some counterexamples, the Kaplansky's problem nowadays asks when a surjective unital invertibility preserving linear map between unital semisimple Banach algebras is a Jordan isomorphism. The problem is still open even for C*-algebras. Partial solutions are known for real rank zero C*-algebras, and semisimple Banach algebras with essential socle.

New important contributions to the study of linear preserver problems in the algebra L(H) of all bounded linear maps on an infinite dimensional complex Hilbert space, have been recently made by Mbekhta, Mbekhta, Rodman and Semrl, and Mebkhta and Semrl. They characterize unital surjective linear maps on L(H), preserving the set of Fredholm and semi-Fredholm elements in both directions.

In this talk we present some background on the Kaplansky's problem, and linear preserver problems, making emphasis in the two more favorable settings, that is, C*-algebras of real rank zero, and semisimple Banach algebras with essential socle.

We explain the theory on Fredholm and Atkinson elements in Banach algebras needed to present our main contribution on this topic. We generalize the results of Mbekhta et al. to the more general context of unital surjective linear maps from a C*-algebra of real rank zero to a semisimple complex Banach
algebra.

(This is part of a joint work with M. Bendoud, A. Bourhim and M. Sarih.)