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.