On the C*-valued triangle equality and inequality in Hilbert C*-modules

We prove that for two elements x, y in a Hilbert C*-module V over a C*-algebra \( \mathcal{A} \) the C*-valued triangle equality |x + y| = |x| + |y| holds if and only if 〈x, y〉 = |x| |y|. In addition, if \( \mathcal{A} \) has a unit e, then for every x, y ∊ V and every ɛ > 0 there are contractions u, υ ∊ \( \mathcal{A} \) such that |x + y| ≦ u|x|u* + υ|y|υ* + ɛe.