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

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