Trying to prove: Let $E$ be a Hilbert $A$-module. Then, $E\langle E,E\rangle $ is norm dense in $E$.

Question

Let $E$ be a Hilbert $A$-module. Then, $E\langle E,E\rangle$ is norm dense in $E$.

I am having trouble proving this. I believe $\langle E,E\rangle $ is a $C^*$-algebra. If I can show this, then the proof is easy since all $C^*$-algebras have an approximate identity.

It seems like it shouldn't be too hard, but I am having trouble showing it.

Thank you.

Answer 1

They are in fact equal See lemma 2.2.3. of book Hilbert C*-Modules by M. Manuilov page20 But I only point out that $\langle E,E\rangle$ is a C*-subalgebra of A and so $E\langle E,E\rangle\subset E$; conversely since any x in E can be written as $x=y\langle y,y\rangle$ we have $E\subset E\langle E,E\rangle.$