Characterization of Birkhoff–James orthogonality

The Birkhoff–James orthogonality is a generalization of Hilbert space orthogonality to Banach spaces. We investigate this notion of orthogonality when the Banach space has more structures. We start by doing so for the Banach space of square matrices moving gradually to all bounded operators on any Hilbert space, then to an arbitrary C ∗ -algebra and finally a Hilbert C ∗ -module.