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.


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 $E\subset E\langle E,E\rangle.$

  • 1
    $\begingroup$ For those who don't have that book, could you describe a bit about what that citation says? $\endgroup$ – robjohn Feb 5 '13 at 7:20

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