Inequality in Hilbert modules

Question

I am going through the book 'K-Theory and C^*-algebras' by Wegge-Olsen and at the moment I don't see why would a certain inequality hold. It is below underlined with red.

Answer 1

It is easy, but maybe different order of (in)equalities:
\begin{align*} \Vert xa \Vert^2 &= \Vert a^* (x|x) a \Vert\\ &\leq \Vert a^* \Vert \Vert (x|x) \Vert \Vert a \Vert = \Vert a^* \Vert \Vert x \Vert^2 \Vert a \Vert\\ &= \Vert a \Vert^2 \Vert x \Vert^2 \end{align*}