The Birkhoff–James orthogonality in Hilbert C*- modules

In this paper we characterize the Birkhoff–James orthogonality for elements of a Hilbert C*-module in terms of states of the underlying C*-algebra. We also show that the Birkhoff–James orthogonality in a Hilbert C*-module over a C*-algebra A and orthogonality with respect to the A-valued inner product coincide if and only if A is isomorphic to C. In addition, some new results concerning the case of equality in the triangle inequality for elements of a Hilbert C*-module are obtained.