From norm derivatives to orthogonalities in Hilbert C*-modules

Let (X,<.,.>) be a Hilbert C*-module over a C*-algebra A and let S(A) be the set of states on A. In this paper, we first compute the norm derivative for nonzero elements x and y of X as follows: lim(t -> 0+) parallel to x + ty parallel to - parallel to x parallel to/t = 1/parallel to x parallel to max {Re phi(< x,y >) : phi is an element of S(A), phi(< x, x >) = parallel to x parallel to(2)}. We then apply it to characterize different concepts of orthogonality in X. In particular, we present a simpler proof of the classical characterization of Birkhoff-James orthogonality in Hilbert C*-modules. Moreover, some generalized Daugavet equation in the C*-algebra B(H) of all bounded linear operators acting on a Hilbert space H is solved.