Strong Birkhoff–James orthogonality in Hilbert $C^*$-modules

We say that two elements of a Hilbert $C^*$-module are orthogonal if their $C^*$-valued inner product is $0$. In a Hilbert $C^*$-module, besides this type of orthogonality, we can study all other orthogonalities defined in a general normed space. One which is most frequently used is Birkhoff–James orthogonality - if $x, y$ are elements of a normed linear space $X,$ then $x$ is orthogonal to $y$ in the BJ sense if $\|x+\lambda y\|\ge \|x\|$ for all scalars $\lambda$. As we usually do in Hilbert $C^*$-modules, we study analogous relations obtained by replacing scalars with elements of the underlying $C^*$-algebra, or the norm with the $C^*$-valued "norm". It often happens that these relations are very strong and coincide with (the first mentioned) orthogonality in a Hilbert $C^*$-module, but not always. This leads to the notion of the strong (also called modular) BJ orthogonality which is the main topic of this talk. This is a joint work with R. Rajić.