For each $x$ in a Hilbert $A$-module $X$, there exists a unique $y\in X$ such that $x=y\langle y,y \rangle $.

Question

How do I prove this lemma?

Lemma: Suppose that $X$ is a Hilbert $A$-module. For each $x\in X$ there exists a unique $y\in X$ such that $x=y \langle y,y \rangle $.

I don't know if this is needed but we already know Cohen's factorization theorem that states If $A$ is a Banach algebra with a bounded left or right approximate identity, then for all $a\in A$ there exist $b,c \in A$ such that $a=bc$.

Answer 1

If you have access to the book Morita equivalence and continuous trace C*-algebras by I. Raeburn and D. Williams, you will find this proved there as Proposition 2.31 on page 21 (the proof is on the next page). In this 6 MB pdf file of the article "Déformations de C*-algèbres de Hopf" by E. Blanchard, the result is on the sixth page (labeled as page 145 in the publication).