Orthogonality preserving linear maps on -algebras with non-zero socles

We study orthogonality preserving surjective linear maps from a unital C ∗ -algebra with non-zero socle A to a C ∗ -algebra. In particular, we prove that, when A has essential socle, every biorthogonality preserving surjective linear map T : A → B is automatically bounded and a Jordan ∗ -isomorphism multiplied by an invertible element.