On preservers of strong Birkhoff-James orthogonality between $C^{\ast }$-algebras

It is shown that every linear strong Birkhoff-James isomorphism between unital $C^{\ast }$-algebras is a $\ast$-isomorphism followed by a unitary multiplication. Moreover, as a partial extension of this result to the non-unital case, the form of (possibly nonlinear) strong Birkhoff-James isomorphisms between compact $C^{\ast }$-algebras are determined. A nonlinear characterization of compact $C^{\ast }$-algebras in terms of strong Birkhoff-James orthogonality is also given.