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

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.

-

1
What precisely do the notations $E\langle E,E\rangle$ and $\langle E,E\rangle$ refer to? In any case, a stronger statement can be found here: math.stackexchange.com/questions/163485/… – Jonas Meyer Feb 4 at 17:14

## 1 Answer

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
For those who don't have that book, could you describe a bit about what that citation says? – robjohn♦ Feb 5 at 7:20
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 17, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9236316680908203, "perplexity_flag": "head"}